PSYC 2200 Elementary Statistics for the Behavioral and Social Sciences. 2021a

BEHAVIORAL 

STATISTICS 

Michelle Oja 

Taft College

Taft College 

PSYC 2200: Elementary Statistics for 

Behavioral and Social Sciences 

Michelle Oja

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This text was compiled on 08/27/2021

TABLE OF CONTENTS 

Welcome to behavioral statistics, a statistics textbook for social science majors! ["Facebook Connections" by mikecogh is licensed with 

CC BY-SA 2.0.] 

ACKNOWLEDGEMENTS 

FOR FACULTY 

UNIT 1: DESCRIPTION 

This first unit will focus on learning the basic vocabulary and how we can use statistics to descriptive groups and answer research 

questions in APA Style. The theme for the chapter is student success and helpful study strategies. 

1: INTRODUCTION TO BEHAVIORAL STATISTICS 

1.1: WHY ARE YOU TAKING THIS COURSE? 

1.2: WHAT IS A STATISTIC? WHAT IS A STATISTICAL ANALYSIS? 

1.3: THE SCIENTIFIC METHOD 

1.4: TYPES OF DATA AND HOW TO MEASURE THEM 

1.4.1: IV AND DV- VARIABLES AS PREDICTORS AND OUTCOMES 

1.4.2: QUALITATIVE VERSUS QUANTITATIVE VARIABLES 

1.4.3: SCALES OF MEASUREMENT 

1.5: POPULATIONS AND SAMPLES 

1.5.1: COLLECTING DATA- MORE PRACTICE WITH POPULATIONS AND SAMPLES 

1.6: "RESEARCH SHOWS THAT..." 

1.7: LEARNING (STATISTICS) 

2: WHAT DO DATA LOOK LIKE? (GRAPHS) 

2.1: INTRODUCTION TO LOOKING AT DATA (THIS IS WHAT TOO MANY NUMBERS LOOKS LIKE) 

2.2: FREQUENCY TABLES 

2.3: APA STYLE TABLES 

2.4: GRAPHING QUALITATIVE VARIABLES- BAR GRAPHS 

2.5: GRAPHING QUALITATIVE VARIABLES- PIE CHARTS 

2.6: GRAPHING QUANTITATIVE VARIABLES 

2.7: SKEW AND KURTOSIS 

2.8: GRAPHING QUANTITATIVE DATE- LINE GRAPHS 

2.9: GRAPHING QUANTITATIVE DATA- HISTOGRAMS 

2.10: GRAPHING QUANTITATIVE DATA- BOXPLOTS 

2.11: GRAPHING QUANTITATIVE DATA- SCATTERPLOTS 

2.12: SUMMARY AND SOME HONESTY 

2.13: APA STYLE CHARTS 

2.14: DESCRIBING DATA USING DISTRIBUTIONS AND GRAPHS EXERCISES 

3: DESCRIPTIVE STATISTICS 

3.1: INTRODUCTION TO DESCRIPTIVE STATISTICS 

3.2: MATH REFRESHER 

3.3: WHAT IS CENTRAL TENDENCY? 

3.3.1: INTRODUCTION TO MEASURES OF CENTRAL TENDENCY 

3.3.2: MEASURES OF CENTRAL TENDENCY- MODE 

3.3.3: MEASURES OF CENTRAL TENDENCY- MEDIAN 

3.3.4: MEASURES OF CENTRAL TENDENCY- MEAN 

3.3.5: SUMMARY OF MEASURES OF CENTRAL TENDENCY 

3.4: INTERPRETING ALL THREE MEASURES OF CENTRAL TENDENCY 

3.5: INTRODUCTION TO MEASURES OF VARIABILITY 

3.6: INTRODUCTION TO STANDARD DEVIATIONS AND CALCULATIONS 

3.7: PRACTICE SD FORMULA AND INTERPRETATION 

3.8: INTERPRETING STANDARD DEVIATIONS 

3.9: PUTTING IT ALL TOGETHER- SD AND 3 M'S 

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3.10: MEASURES OF CENTRAL TENDENCY AND VARIABILITY EXERCISES 

4: DISTRIBUTIONS 

4.1: INTRODUCTION TO DISTRIBUTIONS 

4.2: INTRODUCTION TO PROBABILITY 

4.3: THE BINOMIAL DISTRIBUTION 

4.4: THE LAW OF LARGE NUMBERS 

4.5: NORMAL DISTRIBUTIONS AND PROBABILITY DISTRIBUTIONS 

4.6: SAMPLING DISTRIBUTIONS AND THE CENTRAL LIMIT THEOREM 

4.7: PUTTING IT ALL TOGETHER 

4.8: SUMMARY- THE BIGGER PICTURE 

5: USING Z 

5.1: INTRODUCTION TO Z-SCORES 

5.2: CALCULATING Z-SCORES 

5.2.1: PRACTICE CALCULATING Z-SCORES 

5.3: INTRODUCTION TO THE Z TABLE 

5.3.1: PRACTICE USING THE Z TABLE 

5.3.2: TABLE OF CRITICAL VALUES OF Z 

5.4: PREDICTING AMOUNTS 

5.5: SUMMARY OF Z SCORES 

5.6: THE WRITE-UP 

6: APA STYLE 

6.1: APA AND APA STYLE 

6.2: APA STYLE RESOURCES 

6.3: GENERAL PAPER FORMAT 

6.4: FORMATTING BY SECTION 

6.5: TABLES AND FIGURES 

6.6: SUMMARY OF APA STYLE 

UNIT 2: MEAN DIFFERENCES 

Unit 2 gets into the messy equations that statistically compare means of different groups. But first, let's learn about the hypothesis 

testing process that is most common in statistical analyses and research methods. Theme for this unit is Growth Mindset (described in 

the first section in the first chapter of this unit). 

7: INFERENTIAL STATISTICS AND HYPOTHESIS TESTING 

7.1: GROWTH MINDSET 

7.2: SAMPLES AND POPULATIONS REFRESHER 

7.2.1: CAN SAMPLES PREDICT POPULATIONS? 

7.2.2: DESCRIPTIVE VERSUS INFERENTIAL STATISTICS 

7.3: THE RESEARCH HYPOTHESIS AND THE NULL HYPOTHESIS 

7.4: NULL HYPOTHESIS SIGNIFICANCE TESTING 

7.5: CRITICAL VALUES, P-VALUES, AND SIGNIFICANCE 

7.5.1: CRITICAL VALUES 

7.5.2: SUMMARY OF P-VALUES AND NHST 

7.6: STEPS OF THE HYPOTHESIS TESTING PROCESS 

7.7: THE TWO ERRORS IN NULL HYPOTHESIS SIGNIFICANCE TESTING 

7.7.1: POWER AND SAMPLE SIZE 

7.7.2: THE P-VALUE OF A TEST 

8: ONE SAMPLE T-TEST 

8.1: PREDICTING A POPULATION MEAN 

8.2: INTRODUCTION TO ONE-SAMPLE T-TESTS 

8.3: ONE-SAMPLE T-TEST CALCULATIONS 

8.3.1: TABLE OF CRITICAL T-SCORES 

8.4: REPORTING RESULTS 

8.4.1: DESCRIPTIVE AND INFERENTIAL CALCULATIONS AND CONCLUSION EXAMPLE 

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8.5: CONFIDENCE INTERVALS 

8.5.1: PRACTICE WITH CONFIDENCE INTERVAL CALCULATIONS 

8.6: ONE-SAMPLE T-TESTS AND CIS EXERCISES 

9: INDEPENDENT SAMPLES T-TEST 

9.1: INTRODUCTION TO INDEPENDENT SAMPLES T-TEST 

9.1.1: ANOTHER WAY TO INTRODUCE INDEPENDENT SAMPLE T-TESTS... 

9.2: INDEPENDENT SAMPLES T-TEST EQUATION 

9.3: HYPOTHESES WITH TWO SAMPLES 

9.4: PRACTICE! MOVIES AND MOOD 

9.4.1: MORE PRACTICE! GROWTH MINDSET 

9.5: WHEN TO NOT USE THE INDEPENDENT SAMPLES T-TEST 

9.5.1: NON-PARAMETRIC INDEPENDENT SAMPLE T-TEST 

9.6: TWO INDEPENDENT SAMPLES EXERCISES 

10: DEPENDENT SAMPLES T-TEST 

10.1: INTRODUCTION TO DEPENDENT SAMPLES 

10.2: DEPENDENT SAMPLE T-TEST CALCULATIONS 

10.3: PRACTICE! JOB SATISFACTION 

10.3.1: MORE PRACTICE! CHANGES IN MINDSET 

10.4: NON-PARAMETRIC ANALYSIS OF DEPENDENT SAMPLES 

10.5: CHOOSING WHICH STATISTIC- T-TEST EDITION 

10.6: DEPENDENT T-TEST EXERCISES 

11: BG ANOVA 

11.1: WHY ANOVA? 

11.1.1: OBSERVING AND INTERPRETING VARIABILITY 

11.1.2: RATIO OF VARIABILITY 

11.2: INTRODUCTION TO ANOVA'S SUM OF SQUARES 

11.2.1: SUMMARY OF ANOVA SUMMARY TABLE 

11.3: HYPOTHESES IN ANOVA 

11.4: PRACTICE WITH JOB APPLICANTS 

11.4.1: TABLE OF CRITICAL F-SCORES 

11.5: INTRODUCTION TO PAIRWISE COMPARISONS 

11.5.1: PAIRWISE COMPARISON POST HOC TESTS FOR CRITICAL VALUES OF MEAN DIFFERENCES 

11.6: PRACTICE ON MINDSET DATA 

11.7: ON THE RELATIONSHIP BETWEEN ANOVA AND THE STUDENT T TEST 

11.8: NON-PARAMETRIC ANALYSIS BETWEEN MULTIPLE GROUPS 

11.9: BG ANOVA PRACTICE EXERCISES 

12: RM ANOVA 

12.1: INTRODUCTION TO REPEATED MEASURES ANOVA 

12.1.1: THINGS WORTH KNOWING ABOUT RM ANOVAS 

12.2: ANOVA SUMMARY TABLE 

12.2.1: REPEATED MEASURES ANOVA SUM OF SQUARES FORMULAS 

12.3: PRACTICE WITH RM ANOVA SUMMARY TABLE 

12.3.1: PRACTICE WITH MINDSET 

12.4: NON-PARAMETRIC RM ANOVA 

13: FACTORIAL ANOVA (TWO-WAY) 

13.1: INTRODUCTION TO FACTORIAL DESIGNS 

13.1.1: FACTORIAL NOTATIONS AND SQUARE TABLES 

13.2: INTRODUCTION TO MAIN EFFECTS AND INTERACTIONS 

13.2.1: EXAMPLE WITH MAIN EFFECTS AND INTERACTIONS 

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13.2.2: GRAPHING MAIN EFFECTS AND INTERACTIONS 

13.2.3: INTERPRETING MAIN EFFECTS AND INTERACTIONS IN GRAPHS 

13.2.4: INTERPRETING INTERACTIONS- DO MAIN EFFECTS MATTER? 

13.2.5: INTERPRETING BEYOND 2X2 IN GRAPHS 

13.3: TWO-WAY ANOVA SUMMARY TABLE 

13.3.1: CALCULATING SUM OF SQUARES FOR THE FACTORIAL ANOVA SUMMARY TABLE 

13.4: WHEN SHOULD YOU CONDUCT POST-HOC PAIRWISE COMPARISONS? 

13.5: PRACTICE WITH A 2X2 FACTORIAL DESIGN- ATTENTION 

13.5.1: PRACTICE 2X3 FACTORIAL ANOVA ON MINDSET 

13.6: CHOOSING THE CORRECT ANALYSIS- MEAN COMPARISON EDITION 

UNIT 3: RELATIONSHIPS 

This unit continues Null Hypothesis Significance Testing, but now we are looking to see if two variables have a relationship, rather than 

if the means are similar. 

14: CORRELATIONS 

14.1: REFRESH TO PREPARE 

14.2: WHAT DO TWO QUANTITATIVE VARIABLES LOOK LIKE? 

14.2.1: INTRODUCTION TO PEARSON’S R 

14.3: CORRELATION VERSUS CAUSATION 

14.3.1: CORRELATION VERSUS CAUSATION IN GRAPHS 

14.4: STRENGTH, DIRECTION, AND LINEARITY 

14.5: HYPOTHESES 

14.6: CORRELATION FORMULA- COVARIANCE DIVIDED BY VARIABILITY 

14.7: PRACTICE ON ANXIETY AND DEPRESSION 

14.7.1: TABLE OF CRITICAL VALUES OF R 

14.7.2: PRACTICE ON NUTRITION 

14.8: ALTERNATIVES TO PEARSON'S CORRELATION 

14.9: FINAL CONSIDERATIONS 

14.10: CORRELATION EXERCISES 

15: REGRESSION 

15.1: INTRODUCTION- LINE OF BEST FIT 

15.2: REGRESSION LINE EQUATION 

15.2.1: USING LINEAR EQUATIONS 

15.3: HYPOTHESIS TESTING- SLOPE TO ANOVAS 

15.4: PRACTICE REGRESSION OF HEALTH AND HAPPINESS 

15.4.1: PRACTICE WITH NUTRITION 

15.5: MULTIPLE REGRESSION 

15.6: LINEAR REGRESSION EXERCISES 

16: CHI-SQUARE 

16.1: INTRODUCTION TO CHI-SQUARE 

16.1.1: ASSUMPTIONS OF THE TEST(S) 

16.2: INTRODUCTION TO GOODNESS-OF-FIT CHI-SQUARE 

16.2.1: CRITICAL VALUES OF CHI-SQUARE TABLE 

16.2.2: INTERPRETATION OF THE CHI-SQUARE GOODNESS-OF-FIT TEST 

16.3: GOODNESS OF FIT Χ² FORMULA 

16.4: PRACTICE GOODNESS OF FIT- PINEAPPLE ON PIZZA 

16.5: INTRODUCTION TO TEST OF INDEPENDENCE 

16.6: PRACTICE CHI-SQUARE TEST OF INDEPENDENCE- COLLEGE SPORTS 

16.6.1: PRACTICE- FAST FOOD MEALS 

16.7: RM CHI-SQUARE- THE MCNEMAR TEST 

16.8: CHOOSING THE CORRECT TEST- CHI-SQUARE EDITION 

16.9: CHI-SQUARE EXERCISES 

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UNIT 4: WRAP UP 

You did it! It's time to wrap-up! 

17: WRAP UP 

17.1: INTRODUCTION TO WRAPPING UP 

17.2: CHOOSING THE TEST 

17.3: WHY DID YOU TAKE THIS CLASS? 

BACK MATTER 

COMMON FORMULAS 

COMMON CRITICAL VALUE TABLES 

INDEX 

GLOSSARY 

5 8/27/2021

☰ 

Acknowledgements 

This book was compiled for a sabbatical project during a pandemic, so there is a lot to be thankful for but also many 

challenges. 

I would like to start by thanking Taft College and the college president, Dr. Daniels, for allowing me the time and support to 

learn more about online educational resources (OER) and statistics (and coding of equations!) to support the college and our 

statistics courses (particularly Elementary Statistics for the Behavioral and Social Sciences). From Taft College, and I am also 

indebted to Tabitha Raber and David Wymore for being the first faculty at our college to delve into OER and who gave me 

advice and support. Also, I am thankful to Dr. Eveland for suggesting that we write a behavioral statistics together years ago 

(to which I responding with a resounding "Nope!"). 

I read several openly-licensed statistics textbooks that became the foundation for this compilation. In particular, Foster et al.’s 

(2021) An Introduction to Psychological Statistics was the starting point for many of the chapters. I also loved Navarro’s 

(2020) Learning Statistics with R- A Tutorial for Psychology Students and Other Beginners; she wrote a wonderfully 

enlightening and enjoyable statistics textbook. It would have been perfect if we were looking for a textbook to teach statistical 

software. Finally, Crump's Answering Questions with Data (also found at https://www.crumplab.com/statistics/) provided 

helpful pages, and supported most of the chapter on factorial design ANOVAs. 

Finally, none of this could have happened without my partner supporting me and taking care of the house (and packing and 

moving us in the middle of this sabbatical and pandemic!), and our dog for keeping me grounded and sane while stuck at home 

for months. 

Citations 

Crump, M. J. C. (2021). Answering Questions with 

Data - Introductory Statistics for Psychology Students. 

Retrieved July 3, 2021, from 

https://stats.libretexts.org/@go/page/7883 

Foster, G. C., Lane, D., Scott, D., Hebl, M., Guerra, R., 

Osherson, D., & Zimmer, H. (2021). An Introduction 

to Psychological. Retrieved July 3, 2021, from 


Navarro, D. (2020). Learning Statistics with R - A 

tutorial for Psychology Students and other Beginners. 

Retrieved July 3, 2021, from 


Contributors and Attributions 

Dr. MO (Taft College) 

Figure : My Dog Ate My Students' Homework 

(CC-BY-NC-SA 

Michelle Oja) 

Michelle Oja 8/6/2021 1 CC-BY-SA https://stats.libretexts.org/@go/page/22220

☰ 

For Faculty 

Licensing 

The licensing for this textbook changes based on the webpage (sub-section). As most of the textbook was modified from 

Foster et al. (2021), much of the textbook is CC-BY-NC-SA (a Creative Commons license that allows anyone to use the 

information for non-commercial purposes and the user must license the modified work in the same way). Most of the other 

authors’ work is licensed as CC-BY-SA (a Creative Commons license allowing use that requires that any modifications of the 

current work is licensed in the same way). 

Homework 

Homework related to this textbook were designed as worksheets that can be answered via Canvas Quizzes, and will be housed 

in Canvas Commons. More questions were provided than might be reasonable for a student to complete for one assignment in 

order for faculty to have options to “mix and match” questions to create their own worksheets and Canvas quizzes. Not only 

will that provide a better fit for you and your class, but it will reduce the likelihood of these exact scenarios and results leaking 

into the hands of the internet. If this worksheet "quiz" format or the data sets don't work for you, there are full courses for 

behavioral statistics on Canvas Commons that provide quizzes or labs. You can also look into MyOpenMath, or the Query or 

ADAPT functions in LibreTexts. 

Book Features 

Notation 

Many of the formulas and notation in this textbook are intentionally mathematically incorrect. Although you and I may know 

what they mean, adding a bunch of subscripts and extra symbols can only confuse students. In this introductory textbook, the 

notations and formulas were chosen to be easy to understand for students new to mathematical equations, rather than 

mathematically correct. 

Common Formulas and Critical Value Tables 

In case you don't poke around in the Back Matter, there is a page of Common Formulas which includes all of the formulas 

discussed in this textbook. There's also a Common Critical Value Tables page in the Back Matter that is linked to each page 

with a critical value table in the textbook. 

Articulation 

This book was compiled to fulfill the requirements of the Course Identification descriptor (C-ID) for Math 110 for the 

California Community Colleges and Taft College’s Course Outline of Record for PSYC 2200 (Elementary Statistics for the 

Behavioral and Social Sciences). 

Additional Textbooks 

This textbook is a compilation of Dr. MO’s own work with a lot of support from several CC-BY-SA or CC-BY-NC-SA 

textbooks on LibreTexts website, including: 

Foster, G. C., Lane, D., Scott, D., Hebl, M., Guerra, R., Osherson, D., & Zimmer, H. (2021). An Introduction to 

Psychological. Retrieved July 3, 2021, from https://stats.libretexts.org/@go/page/7077 

Navarro, D. (2020). Learning Statistics with R - A tutorial for Psychology Students and other Beginners. Retrieved July 3, 

2021, from https://stats.libretexts.org/@go/page/3936 

Crump, M. J. C. (2021). Answering Questions with Data - Introductory Statistics for Psychology Students. Retrieved July 

3, 2021, from https://stats.libretexts.org/@go/page/7883 

Additional textbooks that might be useful are: 

Poritz, J. A (2021). Lies, Damned Lies, or Statistics - How to Tell the Truth with Statistics (Poritz). Retrieved July 3, 2021, 

from https://stats.libretexts.org/@go/page/7784 

Illowsky, B., & Dean, S. (2019). Book: Introductory Statistics (OpenStax) With Multimedia and Interactivity. Retrieved 

July 3, 2021, from https://stats.libretexts.org/@go/page/6885 

Michelle Oja 8/6/2021 1 CC-BY https://stats.libretexts.org/@go/page/22221

Finally, this support course to help statistics students with basic math is also listed in Learning (Statistic) section of this 

textbook: ☰ Green, L. (2021). Support Course for Elementary Statistics. Retrieved July 3, 2021, from 


Enjoy! 



Michelle Oja 8/6/2021 2 CC-BY https://stats.libretexts.org/@go/page/22221

SECTION OVERVIEW 

UNIT 1: DESCRIPTION 

This first unit will focus on learning the basic vocabulary and how we can use statistics to 

descriptive groups and answer research questions in APA Style. The theme for the chapter is student 

success and helpful study strategies. 


You might be tempted to skip this first chapter, but it includes lots of new vocabulary (or vocabulary 

that you know, but now they have technical definitions), so don't fall for that tempation! 













Humans aren't good at looking at a bunch of numbers and figuring out what they mean. Instead, we should first look at the numbers as 

pictures. This chapter will discuss graphing. 

2.1: INTRODUCTION TO LOOKING AT DATA (THIS IS WHAT TOO MANY NUMBERS LOOKS LIKE) 















This chapter will show you how to describe a distribution of data with just a few numbers. 











1 8/27/2021







There are kinds of distributions of data that are so common that they have a name. Some distributions have special characteristics that 

are the underpinning of all of statistics. 









5: USING Z 

Our first statistical analysis to answer a research question! 










6: APA STYLE 

You might know how to format a paper in MLA, but you're in the social sciences now! 

6.1: APA AND APA STYLE 

6.2: APA STYLE RESOURCES 

6.3: GENERAL PAPER FORMAT 

6.4: FORMATTING BY SECTION 

6.5: TABLES AND FIGURES 

6.6: SUMMARY OF APA STYLE 

2 8/27/2021

CHAPTER OVERVIEW 


You might be tempted to skip this first chapter, but it includes lots of new vocabulary (or vocabulary 

that you know, but now they have technical definitions), so don't fall for that tempation! 


Read this section if you are wondering why a statistics class is required for your major. 


This page has a basic introduction to the field of statistics. 


This pages discusses the steps of the scientific method, which will lead into the next sections of 

this chapter. 


Here's a quick and dirty explanation of a bunch of new vocabulary about variables. The next sub-sections go into more detail about 

most of these, too. 


Learn about independent variables and dependent variables in research designs. 


Scales of Measurement collapsed into two types of variables? Yes, please! 


How do researchers and statistician categorize different kinds of variables? Find out here! 


But who do the data describe? 



Why should I care about research or statistics? What do I do when I hear about research or statistics that don't support my ideas? 

Who can I trust? 


What's the best way to learn the material and pass the class? 

1 8/27/2021

1.1: Why are you taking this course? 

Why must you take a stats class when your major is in the social sciences (probably), and you just want to help people? There 

are several answers to that. The one that is probably most compelling to you right now is: 

Understanding statistics will help make you better in your future career. 

Whether you become a therapist, a social worker, a police officer, a nurse, or run a business, you should read research from 

your field, and understand the statistical results. You are being trained to think like a manager or supervisor. To make 

decisions in your career, you’ll need to understand and apply the results of statistical analyses of others in your field. 

Understanding statistical analyses will help you make decisions based on evidence. What you will learn in this class can help 

you show that your program (or your decision) is statistically significantly better than the alternatives. 

What you might not know right now, but will know when you start in a career field in the social sciences, is that much of what 

you will be doing is documenting what you did, and how it worked. In other words, you will be describing data that you 

collect on your clients based on their different groups. That leads us to a second major reason that you have to take this 

course: 

You will need to report evidence to show that what you are doing helps your 

clients/patients/customers. 

This relates to the first reason because know what works and what doesn’t will help you figure out how to be more successful 

in the future. It also turns out that the organizations that fund workers in many helping fields only want to continue funding 

programs and services that are effectives. If will be much easier to get funding if you understand the best way to organize and 

present your results. This evidence comes from collecting, analyzing, and reporting data from your program. 

For more on why your major may require a statistics class, watch this short TED Talk by Arthur Benjamin: 

Arthur Benjamin: Teach statistics… 

Figure : Arthur Benjamin's TED Talk on statistics before calculus. (CC-BY-NC-ND TED Talks via YouTube or TED 

Talks directly) 

Finally, some of you will fall in love with statistics, and become a researcher for a living! This isn’t a reason why you have to 

take the course, but it is a happy accident for those of you who want to help people, but tend to be more logical rather than 

emotional (like me!). According to this American Psychological Association (APA) article, there are jobs out there for folks 

who like statistics, including analyzing data for big companies like Netflix and Hulu (as shown in this optional TED Talk by 

Sebastian Wernicke about using data to make a hit TV show). 


This page was extensively adapted by Michelle Oja (Taft College) from work by Danielle Navarro (University of New 

South Wales). 

Michelle Oja 8/20/2021 1.1.1 CC-BY-SA https://stats.libretexts.org/@go/page/21995

Arthur Benjamin's TED Talk: YouTube 


1.2: What is a statistic? What is a statistical analysis? 

The Crash Course video series are great, and there’s one for statistics! Check out this Crach Course Statistics Preview video: 

Crash Course Statistics Preview 

Figure : Crash Course Statistics Preview. (Crash Course: Statistics via YouTube ) 

You should follow the whole Crash Course Statistics playlist as you go through this textbook's chapters; the chapters won't 

completely align with the videos, but they will be similar. The Khan Academy also has a great series on statistics (website 

addres: https://www.khanacademy.org/math/statistics-probability). 

Statistics are the results of statistical analyses, such as the percentage of psychology majors at your school; the analysis was 

calculating the percentage, while the result is a statistic that you could share. 

Definition: Statistics 

The results of statistical analyses 

Definition: Statistical Analyses 

Procedures to organize and interpret numerical information 

To learn more about what statistics are, you can watch Alan Smith's TED Talk about how so many people are bad at estimating 

statistics. 

Why you should love statistics | A… 

Figure 

Talks) 

: Alan Smith's TED Talk on loving statistics. (CC-BY-NC-ND TED Talks via YouTube or directly from TED 


I have my students watch this video because it shows them that statistics aren't something to be afraid of, and that statistics are 

really about each of us. 

Learning statistical analysis is actually not that similar to learning mathematical analyses. It’s more like learning to read and 

interpret another language. Social scientists know that a language is one of the components of a culture; the culture of 

statistics is the scientific method and probability. We’ll talk about the scientific method next, and will touch on probably when 

talking about distributions. 



Crash Course- Statistics Preview: https://youtu.be/zouPoc49xbk 

Alan Smith's TED Talk: https://www.ted.com/talks/alan_smith...ove_statistics 


1.3: The Scientific Method 

I can hear you now, “What? You just convinced me that I will need to know something about statistics to be good in my future 

career, but, seriously, the scientific method?” 

Yup. It turns out that the social sciences are a science. And each step of the scientific method is related to what you’ll learn in 

this course. 

Note 

Steps of the Scientific Method: 

1. Generate research question. 

2. Formulate a Research Hypothesis. 

3. Collect data to test the Research Hypothesis. 

4. Analyze data. 

5. Publicize the results. 

The following example of using the scientific method (Table ) is based off of Part 2 (What Students Should Know 

about 

How People Learn) and Part 3 (Cognitive Principles for Optimizing Learning) of Dr. Chew's five-part video series on the best 

way to learn material and pass your classes. Dr. Chew is a cognitive psychologist, so his career is to learn how people think, 

process material, and learn. 

Steps of the Scientific Method 

Table 

: Scientific Method 

Description of the Step of the Scientific 

Method 

 

Example of the Step of the Scientific Method 

… 

1. Generate research question. This might be pretty broad "What should I do to pass this class?" 

… 

2. Formulate a Research Hypothesis 

This is a sentence that states a predicted 

relationship between two or more groups. This 

should not be a question, and it should not 

include the word “because” or else you 

actually have two hypotheses (the predicted 

relationship, and the reason why the 

relationship is that way). 

"Students who re-organize their notes will earn 

more points on their first paper than students 

who re-read their notes." 

… 

3. Collect data to test your Research 

Hypothesis. 

We’ll learn more about this step later, but the 

important part now is to count or measure the 

variables in your Research Hypothesis. 

"At the end of each class session, half of the 

class will spend 20 minutes rewriting and 

organizing their notes, while the other half of 

the class will spend 20 minutes reviewing (rereading) 

their notes. We will then compare 

their points earned on the first paper between 

these two groups." 

… 

4. Analyze data. 

Remember the second reason that you have to 

take this class? That’s this part of the 

Scientific Method. You would describe what 

you found, and state whether the Research 

Hypothesis was supported or not supported. 

(We don’t say that a Research Hypothesis is 

“proven;” we’ll discuss why not when we talk 

about Null Hypothesis Significant Testing). 

That's what the rest of this book will be about! 

… 

5. Publicize the results. Let people know what you found! 

Typically, researchers present at a conference 

or publish a research article. In this example, 

we could also share our findings with the 

student newspaper or the tutors. 

Danielle Navarro 7/30/2021 1.3.1 CC-BY-SA https://stats.libretexts.org/@go/page/21997

To the surprise of many students, research and statistics is a fairly significant part of the social sciences. To the surprise of no 

one, statistics is very rarely the favorite part of one’s education. After all, if you really loved the idea of doing statistics, you’d 

probably be enrolled in a statistics class right now, not a behavioral statistics class. So, not surprisingly, there’s a pretty large 

proportion of the student base that isn’t happy about the fact that their major has so much research and statistics in it. By the 

end of this chapter, if not this textbook, hopefully you'll understand why research and statistics are so important! 


This page was extensively adapted by Michelle Oja (Taft College) from work by Danielle Navarro (University of New South 

Wales). 


Example 

Example 

Exercise 

1.4: Types of Data and How to Measure Them 

In order to use statistics, we need data to analyze. Data come in an amazingly diverse range of formats, and each type gives us 

a unique type of information. In virtually any form, data represent the measured value of variables. A variable is simply a 

characteristic or feature of the thing we are interested in understanding. In psychology, we are interested in people, so we 

might get a group of people together and measure their levels of stress (one variable), anxiety (a second variable), and their 

physical health (a third variable). Once we have data on these three variables, we can use statistics to understand if and how 

they are related. Before we do so, we need to understand the nature of our data: what they represent and where they came 

from. 

Types of Variables 

When conducting research, experimenters often manipulate variables. For example, an experimenter might compare the 

effectiveness of four types of antidepressants. In this case, the variable is “type of antidepressant.” When a variable is 

manipulated by an experimenter, it is called an independent variable. The experiment seeks to determine the effect of the 

independent variable on relief from depression. In this example, relief from depression is called a dependent variable. In 

general, the independent variable is manipulated by the experimenter and its effects on the dependent variable are measured. 

Can blueberries slow down aging? A study indicates that antioxidants found in blueberries may slow down the process of 

aging. In this study, -month-old rats (equivalent to -year-old humans) were fed either their standard diet or 

a 

diet 

supplemented by either blueberry, strawberry, or spinach powder. After eight weeks, the rats were given memory and 

motor skills tests. Although all supplemented rats showed improvement, those supplemented with blueberry powder 

showed the most notable improvement. 

a. What is the independent variable? 

b. What are the dependent variables? 

Solution 

a. IV: dietary supplement: none, blueberry, strawberry, and spinach 

b. DVs: memory test and motor skills test 

One more together, and then one on your own: 

Does beta-carotene protect against cancer? Beta-carotene supplements have been thought to protect against cancer. 

However, a study published in the Journal of the National Cancer Institute suggests this is false. The study was conducted 

with 39,000 women aged 45 and up. These women were randomly assigned to receive a beta-carotene supplement or a 

placebo, and their health was studied over their lifetime. Cancer rates for women taking the betacarotene supplement did 

not differ systematically from the cancer rates of those women taking the placebo. 

a. What is the independent variable? 

b. What is the dependent variable? 

Solution 

a. IV: Supplements: beta-carotene or placebo 

b. DV: Cancer occurrence 

Your turn! 

Foster et al. 8/20/2021 1.4.1 CC-BY-NC-SA https://stats.libretexts.org/@go/page/21998

How bright is right? An automobile manufacturer wants to know how bright brake lights should be in order to minimize 

the time required for the driver of a following car to realize that the car in front is stopping and to hit the brakes. 

1. What is the independent variable? 

2. What is the dependent variable? 

Answer 

1. IV: Brightness of brake lights 

2. DV: Time to hit brakes 

Levels of an Independent Variable 

If an experiment compares an experimental treatment with a control treatment, then the independent variable (type of 

treatment) has two levels: experimental and control. If an experiment were comparing five types of diets, then the independent 

variable (type of diet) would have levels. In general, the number of levels of an independent variable is the number 

of 

experimental conditions. 

Qualitative and Quantitative Variables 

An important distinction between variables is between qualitative variables and quantitative variables. Qualitative variables 

are those that express a qualitative attribute such as hair color, eye color, religion, favorite movie, gender, and so on. The 

values of a qualitative variable do not imply a numerical ordering. Values of the variable “religion” differ qualitatively; no 

ordering of religions is implied. Qualitative variables are sometimes referred to as categorical variables. Quantitative variables 

are those variables that are measured in terms of numbers. Some examples of quantitative variables are height, weight, and 

shoe size. 

In the study on the effect of diet discussed previously, the independent variable was type of supplement: none, strawberry, 

blueberry, and spinach. The variable “type of supplement” is a qualitative variable; there is nothing quantitative about it. In 

contrast, the dependent variable “memory test” is a quantitative variable since memory performance was measured on a 

quantitative scale (number correct). 

Discrete and Continuous Variables 

Variables such as number of children in a household are called discrete variables since the possible scores are discrete points 

on the scale. For example, a household could have three children or six children, but not children. Other 

variables such as 

“time to respond to a question” are continuous variables since the scale is continuous and not made up of discrete steps. The 

response time could be seconds, or it could be seconds. Of course, the practicalities 

of 

measurement 

preclude most measured variables from being truly continuous. 

Levels of Measurement 

Before we can conduct a statistical analysis, we need to measure our dependent variable. Exactly how the measurement is 

carried out depends on the type of variable involved in the analysis. Different types are measured differently. To measure the 

time taken to respond to a stimulus, you might use a stop watch. Stop watches are of no use, of course, when it comes to 

measuring someone's attitude towards a political candidate. A rating scale is more appropriate in this case (with labels like 

“very favorable,” “somewhat favorable,” etc.). For a dependent variable such as “favorite color,” you can simply note the 

color-word (like “red”) that the subject offers. 

Although procedures for measurement differ in many ways, they can be classified using a few fundamental categories. In a 

given category, all of the procedures share some properties that are important for you to know about. The categories are called 

“scale types,” or just “scales,” and are described in this section. 

Nominal scales 

When measuring using a nominal scale, one simply names or categorizes responses. Gender, handedness, favorite color, and 

religion are examples of variables measured on a nominal scale. The essential point about nominal scales is that they do not 

imply any ordering among the responses. For example, when classifying people according to their favorite color, there is no 


sense in which green is placed “ahead of” blue. Responses are merely categorized. Nominal scales embody the lowest level of 

measurement. 

Ordinal scales 

A researcher wishing to measure consumers' satisfaction with their microwave ovens might ask them to specify their feelings 

as either “very dissatisfied,” “somewhat dissatisfied,” “somewhat satisfied,” or “very satisfied.” The items in this scale are 

ordered, ranging from least to most satisfied. This is what distinguishes ordinal from nominal scales. Unlike nominal scales, 

ordinal scales allow comparisons of the degree to which two subjects possess the dependent variable. For example, our 

satisfaction ordering makes it meaningful to assert that one person is more satisfied than another with their microwave ovens. 

Such an assertion reflects the first person's use of a verbal label that comes later in the list than the label chosen by the second 

person. 

On the other hand, ordinal scales fail to capture important information that will be present in the other scales we examine. In 

particular, the difference between two levels of an ordinal scale cannot be assumed to be the same as the difference between 

two other levels. In our satisfaction scale, for example, the difference between the responses “very dissatisfied” and 

“somewhat dissatisfied” is probably not equivalent to the difference between “somewhat dissatisfied” and “somewhat 

satisfied.” Nothing in our measurement procedure allows us to determine whether the two differences reflect the same 

difference in psychological satisfaction. Statisticians express this point by saying that the differences between adjacent scale 

values do not necessarily represent equal intervals on the underlying scale giving rise to the measurements. (In our case, the 

underlying scale is the true feeling of satisfaction, which we are trying to measure.) 

What if the researcher had measured satisfaction by asking consumers to indicate their level of satisfaction by choosing a 

number from one to four? Would the difference between the responses of one and two necessarily reflect the same difference 

in satisfaction as the difference between the responses two and three? The answer is No. Changing the response format to 

numbers does not change the meaning of the scale. We still are in no position to assert that the mental step from to (for 

example) is the same as the mental step from to . 

 

 

 

 

 

Interval scales 

Interval scales are numerical scales in which intervals have the same interpretation throughout. As an example, consider the 

Fahrenheit scale of temperature. The difference between degrees and degrees represents the same temperature difference 

as the difference between degrees and degrees. This is because each -degree interval has the same physical meaning 

(in terms of the kinetic energy of molecules). 

 

 

 

 

Interval scales are not perfect, however. In particular, they do not have a true zero point even if one of the scaled values 

happens to carry the name “zero.” The Fahrenheit scale illustrates the issue. Zero degrees Fahrenheit does not represent the 

complete absence of temperature (the absence of any molecular kinetic energy). In reality, the label “zero” is applied to its 

temperature for quite accidental reasons connected to the history of temperature measurement. Since an interval scale has no 

true zero point, it does not make sense to compute ratios of temperatures. For example, there is no sense in which the ratio of 

to degrees Fahrenheit is the same as the ratio of to degrees; no interesting physical property is preserved across 

the two ratios. After all, if the “zero” label were applied at the temperature that Fahrenheit happens to label as degrees, the 

two ratios would instead be to and to , no longer the same! For this reason, it does not make sense to say that 

degrees is “twice as hot” as degrees. Such a claim would depend on an arbitrary decision about where to “start” the 

temperature scale, namely, what temperature to call zero (whereas the claim is intended to make a more fundamental assertion 

about the underlying physical reality). 

Ratio scales 

The ratio scale of measurement is the most informative scale. It is an interval scale with the additional property that its zero 

position indicates the absence of the quantity being measured. You can think of a ratio scale as the three earlier scales rolled up 

in one. Like a nominal scale, it provides a name or category for each object (the numbers serve as labels). Like an ordinal 

scale, the objects are ordered (in terms of the ordering of the numbers). Like an interval scale, the same difference at two 

places on the scale has the same meaning. And in addition, the same ratio at two places on the scale also carries the same 

meaning. 


Exercise 

The Fahrenheit scale for temperature has an arbitrary zero point and is therefore not a ratio scale. However, zero on the Kelvin 

scale is absolute zero. This makes the Kelvin scale a ratio scale. For example, if one temperature is twice as high as another as 

measured on the Kelvin scale, then it has twice the kinetic energy of the other temperature. 

Another example of a ratio scale is the amount of money you have in your pocket right now (25 cents, 55 cents, etc.). Money 

is measured on a ratio scale because, in addition to having the properties of an interval scale, it has a true zero point: if you 

have zero money, this implies the absence of money. Since money has a true zero point, it makes sense to say that someone 

with 50 cents has twice as much money as someone with 25 cents (or that Bill Gates has a million times more money than you 

do). 

Let's practice! 

For each of the following, determine the level of measurement: 

1. T-shirt size 

2. Time taken to run 100 meter race 

3. First, second, and third place in 100 meter race 

4. Birthplace 

5. Temperature in Celsius 

Answer 

1. Ordinal 

2. Ratio 

3. Ordinal 

4. Nominal 

5. Interval 

What level of measurement is used for psychological variables? 

Rating scales are used frequently in psychological research. For example, experimental subjects may be asked to rate their 

level of pain, how much they like a consumer product, their attitudes about capital punishment, their confidence in an answer 

to a test question. Typically these ratings are made on a 5-point or a 7-point scale. These scales are ordinal scales since there is 

no assurance that a given difference represents the same thing across the range of the scale. For example, there is no way to be 

sure that a treatment that reduces pain from a rated pain level of 3 to a rated pain level of 2 represents the same level of relief 

as a treatment that reduces pain from a rated pain level of 7 to a rated pain level of 6. 

In memory experiments, the dependent variable is often the number of items correctly recalled. What scale of measurement is 

this? You could reasonably argue that it is a ratio scale. First, there is a true zero point; some subjects may get no items correct 

at all. Moreover, a difference of one represents a difference of one item recalled across the entire scale. It is certainly valid to 

say that someone who recalled 12 items recalled twice as many items as someone who recalled only 6 items. 

But number-of-items recalled is a more complicated case than it appears at first. Consider the following example in which 

subjects are asked to remember as many items as possible from a list of 10. Assume that there are five easy items and five 

difficult items; Half of the participants are able to recall all the easy items and different numbers of difficult items, while the 

other half of the particiapnts are unable to recall any of the difficult items but they do remember different numbers of easy 

items. Some sample data are shown below. 


Participan 

t 

Easy Item 

1 

Easy Item 

2 

Easy Item 

3 

Easy Item 

4 

Table : Sample Data 

Easy Item 

5 

Difficult 

Item 1 

Difficult 

Item 2 

Difficult 

Item 3 

Difficult 

Item 4 

Difficult 

Item 5 

Total 

Score 

A 0 0 1 1 0 0 0 0 0 0 2 

B 1 0 1 1 0 0 0 0 0 0 3 

C 1 1 1 1 1 1 1 0 0 0 7 

D 1 1 1 1 1 0 1 1 0 1 8 

Let's compare the difference between Participant A's score of 2 and Participant B's score of 3, and the difference between 

Participant C's score of 7 and Participant D's score of 8. The former difference is a difference of one easy item (Participant A 

versus Participant B); the latter difference is a difference of one difficult item (Participant C versus Participant D). Do these 

two differences necessarily signify the same difference in memory? We are inclined to respond “No” to this question since 

only a little more memory may be needed to retain the additional easy item whereas a lot more memory may be needed to 

retain the additional hard item. The general point is that it is often inappropriate to consider psychological measurement scales 

as either interval or ratio. 

Consequences of Scale of Measurement 

Why are we so interested in the type of scale that measures a dependent variable? The crux of the matter is the relationship 

between the variable's level of measurement and the statistics that can be meaningfully computed with that variable. For 

example, consider a hypothetical study in which 5 children are asked to choose their favorite color from blue, red, yellow, 

green, and purple. The researcher codes the results as follows: 

Color 

Table : Favorite Color Data 

Code 

Blue 1 

Red 2 

Yellow 3 

Green 4 

Purple 5 

Code 

This means that if a child said her favorite color was “Red,” then the choice was coded as “2,” if the child said her favorite 

color was “Purple,” then the response was coded as 5, and so forth. Consider the following hypothetical data 

Table : Favorite Color 

Data 

Participant Color Code 

1 Blue 1 

2 Blue 1 

3 Green 4 

4 Green 4 

5 Purple 5 

Each code is a number, so nothing prevents us from computing the average code assigned to the children. The average happens 

to be 3, but you can see that it would be senseless to conclude that the average favorite color is yellow (the color with a code 

of 3). Such nonsense arises because favorite color is a nominal scale, and taking the average of its numerical labels is like 

counting the number of letters in the name of a snake to see how long the beast is. 

Does it make sense to compute the mean of numbers measured on an ordinal scale? This is a difficult question, one that 

statisticians have debated for decades. The prevailing (but by no means unanimous) opinion of statisticians is that for almost 


all practical situations, the mean of an ordinally-measured variable is a meaningful statistic. However, there are extreme 

situations in which computing the mean of an ordinally-measured variable can be very misleading. 


Foster et al. (University of Missouri-St. Louis, Rice University, & University of Houston, 

Downtown Campus) 



Exercise 

1.4.1: IV and DV- Variables as Predictors and Outcomes 

Okay, we’ve got more terminology before moving away from variables. All variables can be defined by their Scale of 

Measurement. Variables in research can also be described by whether the experimenter thinks that they are the cause of a 

behavior (IV), or the effect (DV). The IV is the variable that you use to do the explaining and the DV is the variable being 

explained. It’s important to keep the two roles “thing doing the explaining” and “thing being explained” distinct. So let’s be 

clear about this now. 

Definition: Independent Variable (IV) 

The variable that the researcher thinks is the cause of the effect (the DV). The IV is sometimes also called a "predictor" 

or "predicting variable". 

A true IV is created by the experimenter, but sometimes we measure something that we think is the cause and call it an “IV.” 

If we just measure the groups, then we can’t be sure that the IV is causing the DV, so it’s better to create the groups through 

random assignment. IVs are often qualitative/nominal; for most of this textbook, the IV levels are the groups that we are 

comparing. 

Note 

Look up “independent variable” online. What is that definition? How is it similar or different from the one provided? 

Which makes more sense to you? 

In the Scientific Method example ( (Table 

who only re-read their notes. 

Definition: Dependent Variable (DV) 

 

) ), the IV levels were students who re-organized their notes, and students 

The variable that you think is the effect (the thing that the IV changes). The DV is the outcome variable, the thing that 

you want to improve. 

DVs are always measured. DVs can be qualitative, but for most of this textbook the DV will be quantitative because we will 

be comparing means. 

Note 

Look up “dependent variable” online. What is that definition? How is it similar or different from the one provided? 

Which makes more sense to you? 

The logic behind these names goes like this: if there really is a relationship between the variables that we're looking at, then we 

can say that DV depends on the IV. If we have designed our study well, then the IV isn’t dependent on anything else. 

Practice 

Although the terms IV and DV are misleading, they are still the standard phrasing so that's what we'll work with in the 

following examples. In the Scientific Method example, the DV was the points earned on the first paper. The DV is what we 

want to improve, and the IV is a group that we think will do better on the DV plus at least one comparison group (sometimes 

called a control group). 

1. SAT score & type of high school (public vs. private) 

1. Q: Which variable is the IV? (groups or levels)? 

Michelle Oja 8/20/2021 1.4.1.1 CC-BY-SA https://stats.libretexts.org/@go/page/21999

2. Q: Which variable is the DV? 

2. GPA & type of job obtained 



3. Fail rate & brand of computer 



4. Exam score & method of course delivery (online vs. hybrid vs. face-to-face) 



5. Type of phone and how long the battery lasts 



Answer 

1. SAT score & type of high school (public vs. private) 

1. Q: Which variable is the IV? (groups or levels)? type of high school 

2. Q: Which variable is the DV? SAT score 

2. GPA & type of job obtained 

1. Q: Which variable is the IV? (groups or levels)? type of job obtained 

2. Q: Which variable is the DV? GPA 

3. Fail rate & brand of computer 

1. Q: Which variable is the IV? (groups or levels)? brand of computer 

2. Q: Which variable is the DV? Fail rate 

4. Exam score & method of course delivery (online vs. hybrid vs. face-to-face) 

1. Q: Which variable is the IV? (groups or levels)? course delivery 

2. Q: Which variable is the DV? Exam score 

5. Type of phone and how long the battery lasts 

1. Q: Which variable is the IV? (groups or levels)? type of phone 

2. Q: Which variable is the DV? how long the battery lasts 


This page was extensively adapted by Michelle Oja (Taft College) from work by Matthew J. C. Crump (Brooklyn College of 

CUNY) 


Example 

Exercise 

☰ 

1.4.2: Qualitative versus Quantitative Variables 

Some researchers call the first two scales of measurement (Ratio Scale and Interval Scale) “quantitative” because they 

measure things numerically, and call the last scale of measurement (Nominal Scale) “qualitative” because you count the 

number of things that have that quality. The MooMooMath YouTube series did a short segment on these two types of 

variables. [It turns out that there are a LOT of videos online about statistics! Use them any time you are confused! My only 

caution is that some videos use slightly different formulas than in this textbook, and some use software that will not be 

discussed here, so make sure that the information in the video matches what your professor is showing you.] Ordinal scales 

are sort of in-between these two types, but are more similar in statistical analyses to qualitative variables. 

Quantitative Variables 

Quantitative variables are measured with some sort of scale that uses numbers. For example, height can be measures in the 

number of inches for everyone. Halfway between 1 inch and two inches has a meaning. Anything that you can measure with 

a number and finding a mean makes sense is a quantitative variable. If a decimal makes sense, then the variable is 

quantitative. 

Quantitative variables are usually continuous. 

The data are the weights of backpacks with books in them. You sample the same five students. The weights (in pounds) of 

their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Is the 

weight of the backpacks a quantitative variable? 

Solution 

Yes, the weights are quantitative data because weight is a numerical variable that is measured. 

Qualitative Variables 

Qualitative variables, which are the nominal Scale of Measurement, have different values to represent different categories or 

kinds. Qualitative/nominal variables name or label different categories of objects. Something is either an apple or an orange, 

halfway between an apple and an orange doesn’t mean anything. Qualitative variables are counted, and the counts are used in 

statistical analyses.The name or label of a qualitative variable can be a number, but the number doesn’t mean anything. 

Let’s say I collected data and coded it: 

Women = 1 

Men = 2, 

Non-binary = 3 

Does it make any sense to add these numbers? To find the “mean” of gender? 

Answer 

No. An average gender of 1.75 (or whatever) doesn't tell us much since gender is a qualitative variable (nominal scale 

of measurement), so you can only count it. 

Qualitative variables are discrete. 

Deciding: Quant or Qual? 

Use the following to practice identifying whether variables are quantitative (measured with numbers) or qualitative 

(categories). 

Michelle Oja 8/6/2021 1.4.2.1 CC-BY https://stats.libretexts.org/@go/page/22000

Exercise 

Exercise 

☰ 

1. City 

2. Gender 

3. Weight 

4. Type of degree 

5. College major 

6. Percent correct on Exam 1. 

7. Score on a depression scale (between 0 and 10) 

8. How long it takes you to blink after a puff of air hits your eye. 

9. What is another example of a quantitative variable? 

10. What is another example of a qualitative variable? 

Answer 

1. City: Qualitative (named, not measured) 

2. Gender: Qualitative (named, not measured) 

3. Weight: Quantitative (number measured in ounces, pounds, tons, etc.; decimal points make sense) 

4. Type of degree: Qualitative (named, not measured) 

5. College major: Qualitative (named, not measured) 

6. Percent correct on Exam 1: Quantitative (number measured in percentage points; decimal points make sense) 

7. Score on a depression scale (between 0 and 10): Quantitative (number measured by the scale; decimal points 

make sense) 

8. How long it takes you to blink after a puff of air hits your eye: Quantitative (number measured in milliseconds; 

decimal points make sense) 

9. What is another example of a quantitative variable? (Your answer should be something that was measured, not 

counted, and in which decimal points make sense.) 

10. What is another example of a qualitative variable? (Your answer should be something that is a category or name.) 

Okay, that probably makes it seem like it's easy to know whether your variable is qualitative or quantitative. 

Exercise 

 

shows that variables can be defined in different ways. 

You go to the supermarket and purchase three cans of soup (19 ounces) tomato bisque, 14.1 ounces lentil, and 19 ounces 

Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, 

spinach, and carrots), and two desserts (16 ounces Cherry Garcia ice cream and two pounds (32 ounces chocolate chip 

cookies). 

Name data sets that are quantitative discrete, quantitative continuous, and qualitative. 

Solution 

One Possible Solution: 

The three cans of soup, two packages of nuts, four kinds of vegetables and two desserts are quantitative discrete data 

because you count them. 

The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure 

weights as precisely as possible. 

Types of soups, nuts, vegetables and desserts are qualitative data because they are categorical. 

Try to identify additional data sets in this example. 

As you'll learn in the next chapter, there are types of graphs that are designed for qualitative variables and other graphs that are 

most appropriate for quantitative variables. Before you learn about that, why don't you check out these graphs to see if you 


can figure out whether the variable is qualitative or quantitative. 

☰ 

Exercise 

A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or 

seniors. The data she collects are summarized in the pie chart Figure . What type of data does this graph show? 

Figure 

- Statistics Students (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax) 

Answer 

This pie chart shows the students in each year, which is qualitative data. 

Exercise 

The Registrar keeps records of the number of credit hours students complete each semester. The data she collects are 

summarized in the histogram. 

Figure 

- Credit Hours (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax) 

What type of data does this graph show? 

Answer 

A histogram is used to display quantitative data: the numbers of credit hours completed. 

Final Word on Scales of Measurement 

The type of scale determines what specific statistical analysis you should use. I'm going to share a flow chart now that shows 

how knowing the type and number of variables (IVs and levels, and DVs) and whether they are related (dependent) or not 

related (independent) is how you choose which statistical analysis to choose: Decision Tree PDF I know, that might be a 

little overwhelming right now! Dr. MO isn't sharing this to scare you, but to show how important knowing the type of variable 


will be when analyzing data statistically. Plus, it's easier to learn new material if you can connect it to something that you 

already ☰ know. You might want to print out the Decision Tree, then write notes on it when you learn about each type of 

analysis. That way, you can "hang" your new knowledge on the "tree" that you already have. 

In good news, by the end of this book, you'll be familiar with all of these, and know how to compute most of them! 

In bad news, statistical software will run what you ask, regardless of the measurement scale of the variable. If it’s a number, 

you can analyze it. When this happens with qualitative variables, the results are junk. If you say apple=1 and orange=2, it will 

find the average of an appleorange. Since that's not a thing, you're answer wouldn't mean anything. More reason to 

understand the different kinds of variables! 


This page was extensively adapted by Michelle Oja (Taft College) from work by Barbara Illowsky & Susan Dean (De Anza 

College) from OpenStax. 


☰ 

1.4.3: Scales of Measurement 

Look at the Figure : 

Figure : 3 Beakers. (CC-BY-SA Photo by Jaeger5432. The file was originally uploaded to English Wikipedia. This file 

is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license.) 

Example 

List three ways that the beakers in Figure can be described. 

Solution 

Some examples include: by color, by size (bigger, smaller), and measurement (ounces, milligrams, etc). Your 

descriptions probably map pretty well onto the different Scales of Measurement. 

Scales of Measurement 

A very useful concept for distinguishing between different types of variables is what’s known as scales of measurement. 

Note 

Scales of Measurement 

Ratio: true numerical measurement 

Interval: created numerical scale 

Ordinal: variables with an order 

Nominal: Named variable 

Michelle Oja 8/6/2021 1.4.3.1 CC-BY-SA https://stats.libretexts.org/@go/page/22001

☰ 

Scale of Measurement 

Table - Description & Examples of Each Scale of 

Measurement 

Description of Scale of 

Measurement 

Example from Figure 1.3.1 

Another Example 

… 

Ratio Scale 

A type of variables that is 

measured, and zero means that 

there is none of it. 

Ounces of liquid (and you could 

have no liquid) 

Liters of ice cream (you could 

have zero liters of ice cream) 

… 

Interval Scale 

A type of variable that is 

measured, and the intervals 

between each measurement are 

equal (but zero does not mean the 

absence of the measured item). 

[There are no variables with an 

interval scale in the image.] 

A satisfaction scale in which -5 

means “very unsatisfied”, 0 means 

“neutral,” and +5 means “very 

satisfied.” 

… 

Ordinal Scale 

A type of variables that can be put 

in numerical order, but the 

difference between any pair is not 

the same as the difference between 

any other pair. The variables are in 

ranks (first, second, third, etc.). 

You could order these from biggest 

to smallest; the difference between 

the biggest and the middle might 

be much larger than the difference 

between the middle and the 

smallest. 

The medals earned in the 

Olympics; the difference between 

the person who wins the gold 

medal and the person who wins 

the silver medal might be much 

larger than the difference between 

the person who wins the silver 

medal compared to the person who 

wins the bronze medal. 

… 

Nominal Scale 

Type of variable that has quality or 

name, but not a number that means 

something. “Nom” means 

“name,” so it’s just a name, not 

anything that is measured. This 

variable is counted, rather than 

measured. 

Color of the liquids 

Your major in college. 

Things that are measured use some sort of tool (like a ruler, weight scale, or survey scale), which differs from things that can 

only be counted. 

If you're still a little lost (which is fine, these types of variables are confusing and a little squishy), keep reading. If you're 

good, skip down to the last section in this page, then make sure to read the section on Qualitative and Quantitative variables! 

Ratio Scale 

The fourth and final type of variable to consider is a ratio scale variable, in which zero really means zero, and it’s okay to 

multiply and divide. A good example of a ratio scale variable is response time (RT). In a lot of tasks it’s very common to 

record the amount of time somebody takes to solve a problem or answer a question, because it’s an indicator of how difficult 

the task is. Suppose that Amar takes 2.3 seconds to respond to a question, whereas Becky takes 3.1 seconds. As with an 

interval scale variable, addition and subtraction are both meaningful here. Becky really did take 3.1 - 2.3 = 0.8 seconds longer 

than Amar did. However, notice that multiplication and division also make sense here too: Becky took 3.1 / 2.3 = 1.35 times as 

long as Amar did to answer the question. And the reason why you can do this is that, for a ratio scale variable such as RT, 

“zero seconds” really does mean “no time at all”. 

Interval Scale 

In contrast to nominal and ordinal scale variables, interval scale and ratio scale variables are variables for which the numerical 

value is genuinely meaningful. In the case of interval scale variables, the differences between the numbers are interpretable, 

but the variable doesn’t have a “natural” zero value. A good example of an interval scale variable is measuring temperature in 

o 

o 

degrees celsius. For instance, if it was 15 yesterday and 18∘ today, then the 3 difference between the two is genuinely 

o o o o 

meaningful. Moreover, that 3 difference is exactly the same as the 3 difference between 7 and 10 . In short, addition and 

8 

subtraction are meaningful for interval scale variables. 

o 

However, notice that the 0 does not mean “no temperature at all”: it actually means “the temperature at which fresh water 

freezes”, which is pretty arbitrary. As a consequence, it becomes pointless to try to multiply and divide temperatures. It is 


o o o 

wrong to say that 20 is twice as hot as 10 , just as it is weird and meaningless to try to claim that 20 is negative two times as 

hot ☰ o 

as -10 . 

Again, lets look at a more psychological example. Suppose I’m interested in looking at how the attitudes of first-year 

university students have changed over time. Obviously, I’m going to want to record the year in which each student started. 

This is an interval scale variable. A student who started in 2003 did arrive 5 years before a student who started in 2008. 

However, it would be completely insane for me to divide 2008 by 2003 and say that the second student started “1.0024 times 

later” than the first one. That doesn’t make any sense at all. 

Ordinal Scale 

Ordinal scale variables have a bit more structure than nominal scale variables, but not by a lot. An ordinal scale variable is one 

in which there is a natural, meaningful way to order the different possibilities, but you can’t do anything else. The usual 

example given of an ordinal variable is “finishing position in a race”. You can say that the person who finished first was faster 

than the person who finished second, but you don’t know how much faster. As a consequence we know that 1st > 2nd, and we 

know that 2nd > 3rd, but the difference between 1st and 2nd might be much larger than the difference between 2nd and 3rd. 

I accidently found an ordinal scale when trying to find an example of a nominal scale: highest degree earned. I was looking 

for categories, but there is definitely an order to level of education! 

1. Did not finish high school 

2. High school degree only 

3. Some college 

4. Certificate 

5. Associate's degree 

6. Bachelor's degree 

7. Graduate degree 

Since there is a natural order to education level, it would be very weird to list the options like this… 

1. Did not finish high school 

2. Graduate degree 

3. High school degree only 

4. Some college 

5. Certificate 

6. Bachelor's degree 

7. Associate's degree 

… because it seems against the natural “structure” to the question. Notice that while we can use the natural ordering of these 

items to construct sensible groupings, what we can’t do is average them. The #1 doesn't have any meaningful numerical value, 

and is not really related to the #7 in any measurable way. If you can tell me what the mean of these categories might mean, I’d 

love to know. Because that sounds like gibberish to me! 

Nominal Scale 

A nominal scale variable (also referred to as a categorical variable) is one in which there is no particular relationship between 

the different possibilities: for these kinds of variables it doesn’t make any sense to say that one of them is “bigger’ or”better" 

than any other one, and it absolutely doesn’t make any sense to average them. The classic example for this is “eye color”. Eyes 

can be blue, green and brown, among other possibilities, but none of them is any “better” than any other one. As a result, it 

would feel really weird to talk about an “average eye colour”. In short, nominal scale variables are those for which the only 

thing you can say about the different possibilities is that they are different. That’s it. 

Let’s take a slightly closer look at this. Suppose I was doing research on how students studied. This “study” variable could 

have quite a few possible values,but for now, let’s suppose that these four are the only possibilities, and suppose that when I 

ask 100 students what they did the last time that they studied, and I get this: 

Table - Example of Nominal 

Variable 

Method of Studying 

Number of people 

… 

(1) Read the whole chapter 12 


… 

… 

… 

☰ 



(2) Skimmed the whole chapter, read some sections 30 

(3) Watched YouTube videos on class content 48 

(4) Took notes on the chapter 10 

So, what’s the average way that students studied? The answer here is that there isn’t one. It’s a silly question to ask. You can 

say that watching YouTube is the most popular method, and taking notes is the least popular method, but that’s about all. 

Similarly, notice that the order in which I list the options isn’t very interesting. I could have chosen to display the data like this 


Table - Example of Nominal 

Variable 


(3) Watched YouTube videos on class content 48 

(2) Skimmed the whole chapter, read some sections 30 

(1) Read the whole chapter 12 

(4) Took notes on the chapter 10 

and it doesn't matter. 

Some Complexities 

Okay, I know you’re going to be shocked to hear this, but … the real world is much messier than this little classification 

scheme suggests. Very few variables in real life actually fall into these nice neat categories, so you need to be kind of careful 

not to treat the scales of measurement as if they were hard and fast rules. It doesn’t work like that: they’re guidelines, intended 

to help you think about the situations in which you should treat different variables differently. Nothing more. 

So let’s take a classic example, maybe the classic example, of a psychological measurement tool: the Likert scale. The humble 

Likert scale is the bread and butter tool of all survey design. You yourself have filled out hundreds, maybe thousands of them, 

and odds are you’ve even used one yourself. Suppose we have a survey question that looks like this: 

Which of the following best describes your opinion of the statement that “all pirates are freaking awesome” … 

and then the options presented to the participant are these: 

(1) Strongly disagree 

(2) Disagree 

(3) Neither agree nor disagree 

(4) Agree 

(5) Strongly agree 

This set of items is an example of a 5-point Likert scale: people are asked to choose among one of several (in this case 5) 

clearly ordered possibilities, generally with a verbal descriptor given in each case. However, it’s not necessary that all items be 

explicitly described. This is a perfectly good example of a 5-point Likert scale too: 

(1) Strongly disagree 

(2) 

(3) 

(4) 

(5) Strongly agree 

Likert scales are very handy, if somewhat limited, tools. The question is, what kind of variable are they? They’re obviously 

discrete, since you can’t give a response of 2.5. They’re obviously not nominal scale, since the items are ordered; and they’re 

not ratio scale either, since there’s no natural zero. 

But are they ordinal scale or interval scale? One argument says that we can’t really prove that the difference between “strongly 

agree” and “agree” is of the same size as the difference between “agree” and “neither agree nor disagree”. In fact, in everyday 

life it’s pretty obvious that they’re not the same at all. So this suggests that we ought to treat Likert scales as ordinal variables. 

On the other hand, in practice most participants do seem to take the whole “on a scale from 1 to 5” part fairly seriously, and 


they tend to act as if the differences between the five response options were fairly similar to one another. As a consequence, a 

lot ☰of researchers treat Likert scale data as if it were interval scale. It’s not interval scale, but in practice it’s close enough that 

we usually think of it as being quasi-interval scale. 

These different Scales of Measurement are important because they determine which statistical analysis to use on your data. 

We’ll talk about choosing the correct statistical analysis when we know more specific statistical analyses, but that’s why we 

care about what kind of variables we have so much. 


Danielle Navarro (University of New South Wales) 


Photo by Jaeger5432. The file was originally uploaded to English Wikipedia. This file is licensed under the Creative 

Commons Attribution-Share Alike 2.5 Generic license. 


Exercise 

1.5: Populations and Samples 

Because social scientists want to make people’s lives better, they see what works on a small group of people, and then apply it 

to everyone. The small group of people is the sample, and “everyone” is the population. 

Definition: Sample 

People who participate in a study; the smaller group that the data is gathered from. 

A sample is the small group of people that scientists test stuff on. We want at least 30 people in each group, so a study that has 

two groups will need about 60 people in the sample. 

Definition: Population 

The biggest group that your sample can represent. 

A population is the “everyone” that we want to apply the results to. Sometimes, “everyone” can be a pretty small group; if I 

measured the GPA of one of my Behavioral Statistics classes, then the sample would be the class and the population could be 

students in all Behavioral Statistics classes at the college. (GPA would be the DV.) 

A sample is a concrete thing. You can open up a data file, and there’s the data from your sample. A population, on the other 

hand, is a more abstract idea. It refers to the set of all possible people, or all possible observations, that you want to draw 

conclusions about, and is generally much bigger than the sample. In an ideal world, the researcher would begin the study with 

a clear idea of what the population of interest is, since the process of designing a study and testing hypotheses about the data 

that it produces does depend on the population about which you want to make statements. However, that doesn’t always 

happen in practice: usually the researcher has a fairly vague idea of what the population is and designs the study as best he/she 

can on that basis. 

Examples 

In our Scientific Method example, the sample would be the class from which we got the data from, and the population would 

be the biggest group that they could represent. There’s often more than one possible population, but I might say all college 

students could be a good population for this sample. 

You might have heard the phrase “random sample.” This means that everyone in the population has an equal chance of being 

chosen to be in the sample; this almost never happens. 

Let’s say I want to know if there’s a relationship between intelligence and reading science fiction books. If I survey 100 

of my Introduction to Psychology students on their intelligence and reading of science fiction: 

1. Who is the Sample? 

2. Who could be the population? In other words, what is the biggest group that this sample could represent? 

Answer 

Add texts here. Do not delete this text first. 

1. Who is the Sample? 100 of my Introduction to Psychology students 

2. Who could be the population? In other words, what is the biggest group that this sample could represent? There 

are many possible populations, but all Introduction to Psychology students could work, or all Introduction to 

Psychology students at my college might make sense, too. 

Sometimes it’s easy to state the population of interest. In most situations the situation is much less simple. In a typical a 

psychological experiment, determining the population of interest is a bit more complicated. Suppose Dr. Navarro ran an 

Michelle Oja 7/21/2021 1.5.1 CC-BY-SA https://stats.libretexts.org/@go/page/22002

Exercise 

experiment using 100 undergraduate students as participants. Her goal, as a cognitive scientist, is to try to learn something 

about how the mind works. So, which of the following would count as “the population”: 

All of the undergraduate psychology students her university in Australia? 

Undergraduate psychology students in general, anywhere in the world? 

Australians currently living? 

Australians of similar ages to my sample? 

Anyone currently alive? 

Any human being, past, present or future? 

Any biological organism with a sufficient degree of intelligence operating in a terrestrial environment? 

Any intelligent being? 

Each of these defines a real group of mind-possessing entities, all of which might be of interest to me as a cognitive scientist, 

and it’s not at all clear which one ought to be the true population of interest. Maybe surprisingly for you, there's no "right" 

answer! Although some the suggestions get a little vague, they all could potentially be a population that her sample 

represents. Irrespective of how the population is defined, the critical point is that the sample is a subset of the population. The 

goal of researchers is to use our knowledge of the sample to draw inferences about the properties of the population. More on 

that in later chapters! 

Actual drug use is much higher than drug arrests suggest, so you might want to measure how many people use marijuana. 

If you send out a survey asking about their drug useto everyone with a driver’s license in California, but only 30% fill it 

out: 

1. Who is the Sample? 

2. Who could be the population? In other words, what is the biggest group that this sample could represent? 

Answer 

Add texts here. Do not delete this text first. 

1. Who is the Sample? 30% of Californians with driver's licenses 

2. Who could be the population? In other words, what is the biggest group that this sample could represent? There 

are many possible populations, but Californians who have driver's licenses might make the most sense here. 

This last example shows that sometimes our sample limits who we can generalize our results about, who could be our 

population. 

In almost every situation of interest, what we have available to us as researchers is a sample of data. We might have run 

experiment with some number of participants; a polling company might have phoned some number of people to ask questions 

about voting intentions; etc. Regardless: the data set available to us is finite, and incomplete. We can’t possibly get every 

person in the world to do our experiment; a polling company doesn’t have the time or the money to ring up every voter in the 

country etc. 





Example 

Example 

1.5.1: Collecting Data- More Practice with Populations and Samples 

We are usually interested in understanding a specific group of people. This group is known as the population of interest, or 

simply the population. The population is the collection of all people who have some characteristic in common; it can be as 

broad as “all people” if we have a very general research question about human psychology, or it can be extremely narrow, such 

as “all freshmen psychology majors at Midwestern public universities” if we have a specific group in mind. 

Populations and samples 

In statistics, we often rely on a sample --- that is, a small subset of a larger set of data --- to draw inferences about the larger 

set. The larger set is known as the population from which the sample is drawn. 

You have been hired by the National Election Commission to examine how the American people feel about the fairness of 

the voting procedures in the U.S. Who will you ask? 

Solution 

It is not practical to ask every single American how he or she feels about the fairness of the voting procedures. Instead, 

we query a relatively small number of Americans, and draw inferences about the entire country from their responses. The 

Americans actually queried constitute our sample of the larger population of all Americans. 

A sample is typically a small subset of the population. In the case of voting attitudes, we would sample a few thousand 

Americans drawn from the hundreds of millions that make up the country. In choosing a sample, it is therefore crucial that 

it not over-represent one kind of citizen at the expense of others. For example, something would be wrong with our 

sample if it happened to be made up entirely of Florida residents. If the sample held only Floridians, it could not be used 

to infer the attitudes of other Americans. The same problem would arise if the sample were comprised only of 

Republicans. Inferences from statistics are based on the assumption that sampling is representative of the population. If 

the sample is not representative, then the possibility of sampling bias occurs. Sampling bias means that our conclusions 

apply only to our sample and are not generalizable to the full population. 

We are interested in examining how many math classes have been taken on average by current graduating seniors at 

American colleges and universities during their four years in school. 

Solution 

Whereas our population in the last example included all US citizens, now it involves just the graduating seniors 

throughout the country. This is still a large set since there are thousands of colleges and universities, each enrolling many 

students. (New York University, for example, enrolls 48,000 students.) It would be prohibitively costly to examine the 

transcript of every college senior. We therefore take a sample of college seniors and then make inferences to the entire 

population based on what we find. To make the sample, we might first choose some public and private colleges and 

universities across the United States. Then we might sample 50 students from each of these institutions. Suppose that the 

average number of math classes taken by the people in our sample were 3.2. Then we might speculate that 3.2 

approximates the number we would find if we had the resources to examine every senior in the entire population. But we 

must be careful about the possibility that our sample is non-representative of the population. Perhaps we chose an 

overabundance of math majors, or chose too many technical institutions that have heavy math requirements. Such bad 

sampling makes our sample unrepresentative of the population of all seniors. 

To solidify your understanding of sampling bias, consider the following example. Try to identify the population and the 

sample, and then reflect on whether the sample is likely to yield the information desired. 

Foster et al. 7/21/2021 1.5.1.1 CC-BY-NC-SA https://stats.libretexts.org/@go/page/22003

Example 

Example 

Example 

A substitute teacher wants to know how students in the class did on their last test. The teacher asks the 10 students sitting 

in the front row to state their latest test score. He concludes from their report that the class did extremely well. What is the 

sample? What is the population? Can you identify any problems with choosing the sample in the way that the teacher did? 

Solution 

The population consists of all students in the class. The sample is made up of just the 10 students sitting in the front row. 

The sample is not likely to be representative of the population. Those who sit in the front row tend to be more interested 

in the class and tend to perform higher on tests. Hence, the sample may perform at a higher level than the population. 

A coach is interested in how many cartwheels the average college freshmen at his university can do. Eight volunteers 

from the freshman class step forward. After observing their performance, the coach concludes that college freshmen can 

do an average of 16 cartwheels in a row without stopping. 

Solution 

The population is the class of all freshmen at the coach's university. The sample is composed of the 8 volunteers. The 

sample is poorly chosen because volunteers are more likely to be able to do cartwheels than the average freshman; people 

who can't do cartwheels probably did not volunteer! In the example, we are also not told of the gender of the volunteers. 

Were they all women, for example? That might affect the outcome, contributing to the non-representative nature of the 

sample (if the school is co-ed). 

Simple Random Sampling 

Researchers adopt a variety of sampling strategies. The most straightforward is simple random sampling. Such sampling 

requires every member of the population to have an equal chance of being selected into the sample. In addition, the selection 

of one member must be independent of the selection of every other member. That is, picking one member from the population 

must not increase or decrease the probability of picking any other member (relative to the others). In this sense, we can say 

that simple random sampling chooses a sample by pure chance. To check your understanding of simple random sampling, 

consider the following example. What is the population? What is the sample? Was the sample picked by simple random 

sampling? Is it biased? 

A research scientist is interested in studying the experiences of twins raised together versus those raised apart. She obtains 

a list of twins from the National Twin Registry, and selects two subsets of individuals for her study. First, she chooses all 

those in the registry whose last name begins with Z. Then she turns to all those whose last name begins with B. Because 

there are so many names that start with B, however, our researcher decides to incorporate only every other name into her 

sample. Finally, she mails out a survey and compares characteristics of twins raised apart versus together. 

Solution 

The population consists of all twins recorded in the National Twin Registry. It is important that the researcher only make 

statistical generalizations to the twins on this list, not to all twins in the nation or world. That is, the National Twin 

Registry may not be representative of all twins. Even if inferences are limited to the Registry, a number of problems affect 

the sampling procedure we described. For instance, choosing only twins whose last names begin with Z does not give 

every individual an equal chance of being selected into the sample. Moreover, such a procedure risks over-representing 

ethnic groups with many surnames that begin with Z. There are other reasons why choosing just the Z's may bias the 

sample. Perhaps such people are more patient than average because they often find themselves at the end of the line! The 

same problem occurs with choosing twins whose last name begins with B. An additional problem for the B's is that the 

“every-other-one” procedure disallowed adjacent names on the B part of the list from being both selected. Just this defect 

alone means the sample was not formed through simple random sampling. 


Sample size matters 

Recall that the definition of a random sample is a sample in which every member of the population has an equal chance of 

being selected. This means that the sampling procedure rather than the results of the procedure define what it means for a 

sample to be random. Random samples, especially if the sample size is small, are not necessarily representative of the entire 

population. For example, if a random sample of 20 subjects were taken from a population with an equal number of males and 

females, there would be a nontrivial probability (0.06) that 70% or more of the sample would be female. Such a sample would 

not be representative, although it would be drawn randomly. Only a large sample size makes it likely that our sample is close 

to representative of the population. For this reason, inferential statistics take into account the sample size when generalizing 

results from samples to populations. In later chapters, you'll see what kinds of mathematical techniques ensure this sensitivity 

to sample size. 

More complex sampling 

Sometimes it is not feasible to build a sample using simple random sampling. To see the problem, consider the fact that both 

Dallas and Houston are competing to be hosts of the 2012 Olympics. Imagine that you are hired to assess whether most Texans 

prefer Houston to Dallas as the host, or the reverse. Given the impracticality of obtaining the opinion of every single Texan, 

you must construct a sample of the Texas population. But now notice how difficult it would be to proceed by simple random 

sampling. For example, how will you contact those individuals who don’t vote and don’t have a phone? Even among people 

you find in the telephone book, how can you identify those who have just relocated to California (and had no reason to inform 

you of their move)? What do you do about the fact that since the beginning of the study, an additional 4,212 people took up 

residence in the state of Texas? As you can see, it is sometimes very difficult to develop a truly random procedure. For this 

reason, other kinds of sampling techniques have been devised. We now discuss two of them. 

Stratified Sampling 

Since simple random sampling often does not ensure a representative sample, a sampling method called stratified random 

sampling is sometimes used to make the sample more representative of the population. This method can be used if the 

population has a number of distinct “strata” or groups. In stratified sampling, you first identify members of your sample who 

belong to each group. Then you randomly sample from each of those subgroups in such a way that the sizes of the subgroups 

in the sample are proportional to their sizes in the population. 

Let's take an example: Suppose you were interested in views of capital punishment at an urban university. You have the time 

and resources to interview 200 students. The student body is diverse with respect to age; many older people work during the 

day and enroll in night courses (average age is 39), while younger students generally enroll in day classes (average age of 19). 

It is possible that night students have different views about capital punishment than day students. If 70% of the students were 

day students, it makes sense to ensure that 70% of the sample consisted of day students. Thus, your sample of 200 students 

would consist of 140 day students and 60 night students. The proportion of day students in the sample and in the population 

(the entire university) would be the same. Inferences to the entire population of students at the university would therefore be 

more secure. 

Convenience Sampling 

Not all sampling methods are perfect, and sometimes that’s okay. For example, if we are beginning research into a completely 

unstudied area, we may sometimes take some shortcuts to quickly gather data and get a general idea of how things work before 

fully investing a lot of time and money into well-designed research projects with proper sampling. This is known as 

convenience sampling, named for its ease of use. In limited cases, such as the one just described, convenience sampling is 

okay because we intend to follow up with a representative sample. Unfortunately, sometimes convenience sampling is used 

due only to its convenience without the intent of improving on it in future work. 





1.6: "Research shows that..." 

You may have heard someone say, “Research shows that…” What this means is that an experimenter used the scientific 

method to test a Research Hypothesis, and either supported that Research Hypothesis or did not support that Research 

Hypothesis. However, much of what you read online is not supported by research using the scientific method. Instead, 

people who don’t know what an IV or DV is try to convince you that their ideas are correct. 

As a group, scientists seem to be bizarrely fixated on running statistical tests on everything. In fact, we use statistics so often 

that we sometimes forget to explain to people why we do. It’s a kind of article of faith among scientists – and especially social 

scientists – that your findings can’t be trusted until you’ve done some stats. Undergraduate students might be forgiven for 

thinking that we’re all completely mad, because no one takes the time to answer one very simple question: 

Why do you do statistics? Why don’t scientists just use common sense? 

There’s a lot of good answers to it, 1 but a really simple answer is: we don’t trust ourselves enough. Scientists, especially 

psychologists, worry that we’re human, and susceptible to all of the biases, temptations and frailties that humans suffer from. 

Much of statistics is basically a safeguard. Using “common sense” to evaluate evidence means trusting gut instincts, relying on 

verbal arguments and on using the raw power of human reason to come up with the right answer. Most scientists don’t think 

this approach is likely to work. 

In fact, come to think of it, this sounds a lot like a psychological question. In this behavioral statistics textbook, it seems like a 

good idea to dig a little deeper here. Is it really plausible to think that this “common sense” approach is very trustworthy? 

Verbal arguments have to be constructed in language, and all languages have biases – some things are harder to say than 

others, and not necessarily because they’re false (e.g., quantum electrodynamics is a good theory, but hard to explain in 

words). The instincts of our “gut” aren’t designed to solve scientific problems, they’re designed to handle day to day 

inferences – and given that biological evolution is slower than cultural change, we should say that our gut is designed to solve 

the day to day problems for a different world than the one we live in. Most fundamentally, reasoning sensibly requires people 

to engage in “induction”, making wise guesses and going beyond the immediate evidence of the senses to make 

generalizations about the world. If you think that you can do that without being influenced by various distractors, well, I have 

a bridge in Brooklyn I’d like to sell you. 

The Curse of Belief Bias 

People are mostly pretty smart. Our minds are quite amazing things, and we seem to be capable of the most incredible feats of 

thought and reason. That doesn’t make us perfect though. And among the many things that psychologists have shown over the 

years is that we really do find it hard to be neutral, to evaluate evidence impartially and without being swayed by pre-existing 

biases. 

But first, suppose that people really are perfectly able to set aside their pre-existing beliefs about what is true and what isn’t, 

and evaluate an argument purely on its logical merits. We’d expect 100% of people to say that the valid arguments are valid, 

and 0% of people to say that the invalid arguments are valid. So if you ran an experiment looking at this, you’d expect to see 

data like this: 

Table - Valid Arguments and 

Feelings 

Conclusion feels true 

Conclusion feels false 

Argument is valid 100% say “valid” 100% say “valid” 

Argument is invalid 0% say “valid” 0% say “valid” 

If the data looked like this (or even a good approximation to this), we might feel safe in just trusting our gut instincts. That is, 

it’d be perfectly okay just to let scientists evaluate data based on their common sense, and not bother with all this murky 

statistics stuff. However, you have taken classes, and might know where this is going . . . 

In a classic study, Evans, Barston, and Pollard (1983) ran an experiment looking at exactly this. What they found is show 

in Table ; when pre-existing biases (i.e., beliefs) were in agreement with the structure of the data, everything went 

the 

Danielle Navarro 7/21/2021 1.6.1 CC-BY-SA https://stats.libretexts.org/@go/page/22004

way you’d hope (in bold). Not perfect, but that’s pretty good. But look what happens when our intuitive feelings about the 

truth of the conclusion run against the logical structure of the argument (the not-bold percentages): 

Table - Beliefs Don't Match 

Truth 

Conclusion feels true 

Conclusion feels false 

Argument is valid 92% say “valid” 46% say “valid” 

Argument is invalid 92% say “valid” 8% say “valid” 

Almost all people continue to believe something that isn't true, even with a valid argument against it. Also, almost half of 

people continue to believe that true arguments are false (if they started out not believing it). 

Oh dear, that’s not as good. Apparently, when we are presented with a strong argument that contradicts our pre-existing beliefs, 

we find it pretty hard to even perceive it to be a strong argument (people only did so 46% of the time). Even worse, when 

people are presented with a weak argument that agrees with our pre-existing biases, almost no-one can see that the argument is 

weak (people got that one wrong 92% of the time!). 

Overall, people did do better than a chance at compensating for their prior biases, since about 60% of people’s judgments were 

correct (you’d expect 50% by chance). Even so, if you were a professional “evaluator of evidence”, and someone came along 

and offered you a magic tool that improves your chances of making the right decision from 60% to (say) 95%, you’d probably 

jump at it, right? Of course you would. Thankfully, we actually do have a tool that can do this. But it’s not magic, it’s statistics. 

So that’s reason #1 why scientists love statistics. It’s just too easy for us humans to continue to “believe what we want to 

believe”; so if we want to “believe in the data” instead, we’re going to need a bit of help to keep our personal biases under 

control. That’s what statistics does: it helps keep us honest. 

Before you “like” or share news articles or posts online, try to get to get as close to the original study as you can get, and 

figure out what the IV, DV, sample, and population is so that you can decide if their clam is actually supported by research. 

Was the IV created (not just measured)? Does the way that the DV was measured make sense for that outcome? Does the 

sample really represent the population that they are saying it does? Science is hard to do well, but it’s the best way to learn 

new things about the world. 

1 

Including the suggestion that common sense is in short supply among scientists. 

Reference 

Evans, J. St. B. T., Barston, J. L., & Pollard, P. (1983). On the conflict between logic and belief in syllogistic reasoning. 

Memory & Cognition, 11, 295-306. 




Danielle Navarro 7/21/2021 1.6.2 CC-BY-SA https://stats.libretexts.org/@go/page/22004

1.7: Learning (Statistics) 

As we close this chapter, if you are not at least a little overwhelmed, then you probably didn’t read the chapter closely enough. 

We went over many, many words that you know what they mean, but now they have technical definitions use in statistics and 

research that much more specific than how most people use the words. When learning a new field, you must learn new 

vocabulary. In this case, you are learning statistics for the social sciences. A disadvantage of the social sciences is that so 

much of the vocabulary that we use are technical definitions of words that you are familiar with. 

What can you do to learn? 

Learn How To Learn 

This five-part video series from cognitive psychologist Dr. Chew, first mentioned in 1.3: Scientific Method, provides many 

suggestions for how best to learn in any class. We will be returning to some of these ideas throughout the first unit. You 

might want to watch one video every day for the first week of class to get in the practice of spending time and effort on your 

classes. Plus, the videos have a lot of information in them, so it’s best not to watch them all at once. 

Part 1: Beliefs that Make You Fail 

Part 2: What Students Should Know about How People Learn 

Part 3: Cognitive Principles for Optimizing Learning 

Part 4: Putting the Principles of Optimizing Learning into Practice 

Part 5: I Blew the Exam, Now What? 

Spend Time Before, During, & After Class 

In his videos, Dr. Chew emphasizes how important time and effort is for learning. In addition to some of the practices that he 

suggests for learning, you can also read this article by Zanardelli Sickler (2017) that describes what you can do before your 

class session or lecture video, what you should do during lecture, and what you can do after lecture to make sure that you learn 

and understand the material. 

Another strategy is to decide what activities can be done easily enough on your phone for when you are on the go or don’t 

have access to a bigger screen. For example, if you are reading this textbook (or any online material for your classes, really), 

then your phone might be a great option, especially if you are taking notes in an online document. Your phone might be a 

great way to preview assignments so that you can start thinking about what you'll be submitting. However, maybe taking 

quizzes should be reserved when you have a bigger screen and time to think. Doing practice problems are probably best when 

you have a table or desk to write on. Just planning ahead for what activities can be done when and how will help you organize 

your studying. 

Practice! 

Use the Exercises throughout the chapters as practice. Don't just read them, but try to do them on your own, then check to 

see how close you are to the answers. 

Utilize this support “course” or try to finish the chapters during the first few weeks of the semester: Support Course for 

Elementary Statistics, which is a course to prepare you to take a more math-focused statistics course (rather than social 

science focused like this textbook) 

Your school probably has tutors or a learning lab (math lab?), so set up weekly meetings to go over what you’re learning in 

class. Don't just show up in a panic before an exam or paper is due; learning takes time. 

Work in pairs to check each others’ work. But don’t do exactly what your partner is doing (that looks like cheating). 

What else might work for you? 

Phew! That was a lot! Take a break, and I'll meet you in the next chapter to learn about organizing data! 






Humans aren't good at looking at a bunch of numbers and figuring out what they mean. Instead, we 

should first look at the numbers as pictures. This chapter will discuss graphing. 

2.1: INTRODUCTION TO LOOKING AT DATA (THIS IS WHAT TOO MANY NUMBERS 

LOOKS LIKE) 

Why do we need a whole chapter about charts? 


What's a better way to list numbers? 


How should you format tables? 


What does qualitative data look like in a bar chart? 


What's another common way to look at qualitative data? 


Here's an introduction to graphs used for quantitative data, and how they can be described. 


More new vocabulary to describe the shape of your graph! 


How can you show a distribution with only a line? 


What the heck is a histogram? Hint, it's not a bar chart. 


You might never have heard of box plots, but you might've seen something similar! 


What if you have two quantitative variables? 


Have you heard that quote about lies, damn lies, and statistics? Don't be a liar! 


How are charts or graphs labeled in APA Style? 


Wanna answer some questions about graphs? 

1 8/27/2021

2.1: Introduction to Looking at Data (This is what too many numbers looks like) 

I have scores on a 100-point final exam from one class provided by OpenIntro.org Can you look at them in Table 

quickly tell how the class did? 

Table 

- Final Exam Scores 

79 83 66 81 

 

83 72 89 88 

57 74 78 69 

82 73 81 77 

94 71 78 79 

 

and 

Even if you could get a general idea of how these 20 students did on the final exam, this is not the best way to understand data 

sets. And just imagine if the class had 40 or 100 students in it! 

When you deal with data, you may have so many numbers to you that you will be overwhelmed by them. That is why we need 

ways to describe the data in a more manageable fashion. 

We'll start that in this chapter on charts, then describe whole sets of data in only a few important numbers in the following 

chapter. 



CUNY) 


Example 

☰ 

2.2: Frequency Tables 

A table will have columns and rows. Columns go up and down (think of pillars or Greek columns), and rows go left to right 

on the horizon line. 

A graph (also called a chart) is like a drawing in which the data is plotted on two axes. These will be discussed next. 

Table shows the same data as Table 2.1.1 (scores on a 100-point final exam from one class from an 

unknown college 

provided by OpenIntro.org). The only difference is that they are all in one column in Table , which is typically how 

lists 

of raw data is provided. 

Table 

- Final Exam Scores 

Final Exam Scores 

 

Let's practice with what we learned from the prior chapter on this new set of data. 

79 

83 

57 

82 

94 

83 

72 

74 

73 

71 

66 

89 

78 

81 

78 

81 

88 

69 

77 

79 

1. Who was the sample? 

2. Who could this sample represent (population)? 

3. What was measured? 

4. How many students scored above 90 points on the exam? 

Solution 

1. 20 students from an unknown college 

2. Anything reasonable; maybe students in college? 

3. Scores on a 100-point final exam 

4. One 

Michelle Oja 8/6/2021 2.2.1 CC-BY-NC-SA https://stats.libretexts.org/@go/page/22008

Exercise 

It’s difficult to answer that last question with the raw data shown in Table . It would be easier if the data was 

organized 

from ☰highest score to lowest score, but an even easier way to answer these kinds of “how many” questions is to create a 

frequency table. 

Definition: Frequency Table 

Table showing each score in the “x” column, and how many people earned that score in the “f” column. The “x” stands 

in for whatever the score is, and the “f” stands for frequency. 

Table shows the same data as Table , but in a Frequency Table generated by software (Software 

Package 

for the 

Social Sciences, or SPSS). As you can see in Table , only one person scored each amount listed, except for the scores 

of 

78, 79, 81, and 83 points (in which two students earned those scores). 

x 

Table 

 

- Frequency Table of Final Exam Scores 

57.00 1 

66.00 1 

69.00 1 

71.00 1 

72.00 1 

73.00 1 

74.00 1 

77.00 1 

78.00 2 

79.00 2 

81.00 2 

82.00 1 

83.00 2 

88.00 1 

89.00 1 

94.00 1 

Total 20 

You might notice that the Table does not include final exam scores that no student earned (for example, no scores are 

in 

the “x” column between 57 to 66 points). You can include all of the scores with a frequency (“f” column) of zero, it does 

make the table much complicated to interpret. What can be confusing is frequency tables must include scores of zeros (“x” is 

the score, not the frequency or number of people who earned that score). For example, if a student skipped the final exam, 

then their zero points earned would need to be included in the “x” column, with a frequency (“f” column) of one student. 

The following practice Exercise should be much easier with Table ! 

 

f 

1. How many students scored above 90 points on the exam? 

2. How many students scored below 60 points? 

Answer 


1. One 

☰ 

2. Three 

Did you notice that the x-variable, the final exam scores, was a quantitative variable (ratio scale of measurement)? Frequency 

Tables can also be used with qualitative variables (nominal scale of measurement). Table shows the frequency 

(“f” 

column) of Associate of Arts for Transfer degrees (which are slightly different from Associate of Arts degrees in general) 

earned by California community college students in academic year 2019-2020 (“x” column). This data can be found on the 

California Community Colleges Chancellor’s Office DataMart. 

x 

Table 

Biological Sciences 19 

Education 8 

Family and Consumer Sciences 616 

- AA-T Majors 

Fine and Applied Arts 2,778 

Foreign Language 717 

Health 1,982 

Humanities (Letters) 6,667 

Interdisciplinary Studies 1,034 

Law 36 

Media and Communications 444 

Psychology 10,843 

Public and Protective Services 33 

Social Sciences 13,164 

The majors in Table are listed alphabetically, which makes understanding the numbers a little more confusing. 

Because 

qualitative variables have no natural order, you can list them however you like. To find your own major, an alphabetical order 

works best. But if you'd like emphasize which major has the most graduates, then order from the highest to the lowest number 

of graduates would be best. As we will learn more with charts, the best way to display data depends on what message you are 

trying to tell. 

Most of the graphs shown in the rest of the chapter are derived from frequency tables, so I hope that you find some simple data 

and practice making a few before we starting using them more! 


Foster et al. (University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus) 


 

f 


2.3: APA Style Tables 

You are probably a student in the social sciences. Most fields in the social sciences use the Publication Manual of the 

American Psychological Association (Seventh Edition). This style guide is full of information on formatting and writing in 

APA Style. In this Seventh Edition, you are allowed to put tables in an appendix (which would be the last section of the 

paper), or near where they are being discussed in the appear. 

For our purposes right now, we’ll be looking at how to format the titles for tables (which have rows and columns, and are not 

charts or pictures). In APA Style, tables should be numbered and have a descriptive title. “Table 1” should be in bold, and on 

the line above the title (which should be in italics). They both should be above the table and flush-left. An example is below, 

and you can find more details at the best site to learn about APA Style: Purdue Online Writing Lab. Here is OWL Purdue’s 

page on tables and figures in APA Style. Below is a Note on what the title and number should look like in APA Style. 

Note 

Table 1 


You might notice that this is not at all like how the table titles have been throughout this textbook so far. And there's a good 

reason for this! APA Style formatting is for sending your work to a publisher. Forcing everyone to format their papers the 

same way makes it easier on the reviewers, and makes the playing field more fair for the researchers. With a standard 

formatting style, no one is given preference for fancy graphics or exciting desktop publishing; instead, your work is judged on 

the quality of the science. 

Once a work has been accepted for publishing, the formatting is then modified to fit the formatting of the journal, book, or 

website where it is being published. That's what is happening here! It might be confusing to see the table titles in one format 

in your textbook while your own work should be formatted differently. Sorry! When in doubt with formatting, go to your 

APA Manual or OWL Purdue's APA Style website! 




☰ 

2.4: Graphing Qualitative Variables- Bar Graphs 

Graphing Qualitative Variables 

The first types of charts that will be covered are bar charts and pie charts. Both types of charts should be used on qualitative 

variables only. There's another whole mess of chart types to use on quantitative variables! Review the section on quantitative 

and qualitative variables in the first chapter to refresh yourself on qualitative variables and the nominal scale of measurement. 

describes. 

Bar Charts 

Bar charts are meant to display the frequencies of qualitative variables. Figure 

table in Table 2.2.3, but with bars. 

shows the same data as the frequency 

Figure : AA-T Graduates by Major (Copyright CC0; chart created by Michelle Oja via California Community Colleges 

Chancellor’s Office DataMart) 

Let’s unpack Figure 

a bit. 

First, the x-axis has the same information as the “x” column in Table 2.2.3, the major that the graduating student earned the 

degree in. The x-axis is always the one on the bottom going from left to right. Because we are talking about a variable that 

has categories (nominal scale of measurement; qualitative variable), the information on the x-axis is the category names. They 

can be in any order on the chart because qualitative variables don’t have a natural order. As you can see, they are ordered 

alphabetically. This makes no sense to me, so I usually order from lowest frequency to highest frequency but wanted to let 

you see what is looks like when not ordered numerically. You can choose any order that you’d like since qualitative variables 

have no natural order. 

Michelle Oja 8/6/2021 2.4.1 CC-BY-NC-SA https://stats.libretexts.org/@go/page/22010

Exercise 

Second, the y-axis in Figure shows the same information as the “f” column in Table 2.2.3, the number of students 

who 

earned ☰ that degree. The y-axis is always going up and down. Stand with your hands touching above your head to show that 

this is the axis that is going up and down, then move your hands out to a “Y” (like in the song YMCA) to remember that the y- 

axis is the one that goes up and down. 

Third, notice that the bars do not touch; this is what makes it a bar chart and not a histogram (which will be covered later in 

this chapter). This chart was created with common commercial spreadsheet software (Excel), but note that this type of chart is 

called a “column bar chart” in the software. There are fancier types of bar charts, but often the fancier a chart looks, the more 

difficult it is to easily and quickly understand what the chart is showing. 

Interpreting Bar Charts 

The following questions will be asked for each type of graph in this chapter. In most cases, the questions will also be 

answered so that you can get an idea of what you should be looking for in graphs. Some of these questions won't make sense 

for specific types of graphs, but I want you to get in the habit of answering them to yourself when reviewing graphs. 

1. What kind of 

graph is Figure ? 

1. This is a bar graph (notice that the bars are not touching) because the variable is a qualitative category (nominal scale of 

measurement). 

2. What 

does the x-axis measure in Figure ? 

1. The x-axis is the one on the bottom, and it was named “Student Major”. This is the major that the student earned an 

Associate of Arts for Transfer degree from a California community college student during the 2019-2020 academic 

year. 

3. What does 

the y-axis measure in Figure ? 

1. The y-axis is the axis that goes up and down. For most charts discussed in this textbook, the y-axis will be frequencies. 

In this figure, that means that the y-axis shows how many students earned the Associate of Arts for Transfer degree 

from each major. 

4. What do you notice from Figure ? What pops out to you? 

1. What I noticed is that most students are earning the general “Social Science” degree, rather than something more 

specific like Sociology or Psychology. I also noticed that there were very few students earned an Associate of Arts for 

Transfer degree with an Education major. 

5. What does Figure make you wonder about? 

1. I wonder what kind of degree future teachers are earning, because it does not look like they are earning an Associate of 

Arts for Transfer! Maybe a general Associate of Arts? Or is there a different major that they are in? 

6. What is a catchy headline 

for Figure ? 

1. Be Like the Ten Thousand Psychology Graduates- Go to a California Community College 

7. How could you summarize the info in Figure into one sentence? 

1. The most common Associate of Arts for Transfer degrees were in the Social Sciences, Psychology, Humanities, and 

Fine and Applied Arts. 

8. Who might want to know the information in Figure ? 

1. California community college students? Potential California community college students? Faculty or administrators of 

California community colleges? 

Remember those videos about the best way to learn from Dr. Chew, a cognitive psychologist, first discussed in the first 

chapter? Well, Dr. Chew’s second video has a bar chart (Part 2: What Students Should Know about How People Learn! 

I suggest that you watch the whole video to understand the experiment, but the bar chart shows up around 3:10 minutes. 

Then, answer the same questions as above. 

Answer 


☰ 

1. What kind of graph is in the video? 

1. This is a bar graph (notice that the bars are not touching). 

2. What does the x-axis measure in the video? 

1. The x-axis is the one on the bottom, named “Level of Processing” and it has three levels (shallow, deep, and the 

control group). There are different colored bars, and they represent another variable (memory goal?) that has 

two levels, intentional and incidental. 

3. What does the y-axis measure in the video? 

1. The y-axis is the axis that goes up and down. For most charts discussed in this textbook, the y-axis will be 

frequencies. However, in this video, the y-axis is showing the percentage of words that were successfully 

recalled (remembered). 

4. What do you notice from the graph in the video? What pops out to you? 

1. What I noticed is that shallow processing (looking for the letter “e”) leads to worse recall than not having any 

processing goal (the control group). I also find it interesting that the intent to learn didn’t affect memory really 

at all. But you might have noticed other things! 

5. What does the graph in the video make you wonder about? 

1. I wonder why deep processing (elaborating) isn’t more well known to help students?! But what does the graph 

make you wonder? 

6. What is a catchy headline for the graph in the video? 

1. "Learning is Deep" seems catchy to me, but I bet you came up with something better! 

7. How could you summarize the info in the graph in the video into one sentence? 

1. You could say something like, "Focusing on the meaning of words doesn’t seem to help you remember much 

more than just trying to remember information, but it is way better than shallow processing." 

8. Who might want to know the information from the graph in the video? 

1. I imagine that both students and teachers would find this information very useful. Tutors and other learning 

professionals should also know about it, too. 

Summary 

Bar charts effectively portraying qualitative data. Bar charts are a good option when there are more than just a few categories, 

or for comparing two or more distributions. 


This page was extensively adapted by 


from work by 


. 


2.5: Graphing Qualitative Variables- Pie Charts 

In a pie chart, each category is represented by a slice of the pie. The area of the slice is proportional to the percentage of 

responses in the category. Instead of showing frequencies, a pie chart shows proportions. Figure shows the same 

information as the frequency table in Table 2.2.3 and the bar chart in Figure 2.4.1, but as a pie chart. 

Figure - Pie Chart of Graduate's Majors (Copyright CC0; chart created by Michelle Oja via California Community 

Colleges Chancellor’s Office DataMart) 

Both bar charts and pie charts show information about how many people are in each group, but bar charts can show the exam 

numbers while pie charts show the proportion, or percentage. You can learn different things from each chart, so it’s up to you 

to decide which chart is best for you needs. Remembering that both of these kinds of charts show how many people are in 

each qualitative variable’s group! 

Pie Chart Interpretation 

The following will have similar answers as the bar chart interpretation because it’s the same data. But the answers aren’t 

exactly the same because some things pop out in pie charts but not in bar charts, and vice versa. 

1. What kind of graph is Figure ? 

1. This is a pie chart. Pie charts are used with variables that have qualitative categories (nominal scale of measurement) 

when you are want to compare proportions (percentages). 

2. What does the x-axis measure in Figure ? 

1. Trick questions! There’s no axes in pie charts! What is normally on the x-axis is the category or each score, and that is 

what the colors and labels identify. 


3. What does the y-axis measure in Figure ? 

1. Another trick question! Again, there are no axes in pie charts! What is normally on the y-axis is the frequency; instead 

of having the numbers on an axis, the size of the pie slices indicates the size or portion of each category. In this 

example, I also included the percentage next to the name of the category. 

4. What do you notice fromFigure ? What pops out to you? 

1. The pie chart shows which groups have the most graduates with Associate of Arts for Transfer, and that’s Social 

Sciences, Psychology, Humanities, and Fine and Applied Arts. There are also a bunch of majors that show 0% of the 

pie. They really don’t have 0%, or they wouldn’t be in the chart. Instead, those are the ones with really small 

frequencies, and their percentage is a decimal point (like 0.05% for Biological Sciences). 


1. Because the general “Social Sciences” is such a large proportion (slice of pie), I wonder what kind of careers they go 

into? 



1. Your Rainbow of Choices- Associate of Arts for Transfer from California Community Colleges 


1. I would still say that the most common Associate of Arts for Transfer degrees were in the Social Sciences, Psychology, 

Humanities, and Fine and Applied Arts, but the pie chart format shows how small of a proprotion many of other majors 

earn. 


1. Same as from the bar chart! California community college students? Potential California community college students? 

Faculty or administrators of California community colleges? 

Pie charts can be misleading if they are based on a small number of observations. For example, if just five people had 

graduated with an Associate of Arts for Transfer, and three earned their transfer degree in Psychology, it would be misleading 

to display a pie chart with the Psychology major slice showing 60%. Problems like that can happen with small samples. In this 

case, the slices should be labeled with the actual frequencies observed (e.g., 3) instead of with percentages. 

Summary 

Pie charts and bar charts can both be effective methods of portraying qualitative data. Let's move on to graphing quantitative 

data! 


This page was extensively adapted by Michelle Oja (Taft College) from work by Foster et al. (sourced from University of 

Missouri’s Affordable and Open Access Educational Resources Initiative). 


☰ 

2.6: Graphing Quantitative Variables 

Review the first chapter to refresh yourself on what quantitative variables and the ratio and interval scale of measurement 

describes. 

Briefly, quantitative variables are variables measured on a numeric scale. Height, weight, response time, subjective rating of 

pain, temperature, and score on an exam are all examples of quantitative variables. Quantitative variables are distinguished 

from qualitative (sometimes called nominal or categorical) variables such as favorite color, religion, city of birth, favorite sport 

in which there is no ordering or measuring involved. 

There are many types of graphs that can be used to portray distributions of quantitative variables. The upcoming sections cover 

the following types of graphs: 

1. Line graphs (sometimes called frequency polygons) 

2. Histograms 

3. Box plots 

4. Scatter plots 

Each type of chart highlights different kinds of information. It is up to you to decide which fits your purposes best! 

The Shape of Distribution 

The next section will go more in-depth, and the following chapter will go into even more detail, but here is an introduction to 

describing the shape of graphs of quantitative variables. 

The primary characteristic we are concerned about when assessing the shape of a distribution is whether the distribution is 

symmetrical or skewed. A symmetrical distribution, as the name suggests, can be cut down the center to form two mirror 

images. Although in practice we will never get a perfectly symmetrical distribution, we would like our data to be as close to 

symmetrical as possible. Many types of distributions are symmetrical, but by far the most common and pertinent distribution at 

this point is the normal distribution, shown in Figure . Notice that although the symmetry is not perfect (for instance, the 

bar just to the right of the center is taller than the one just to the left), the two sides are roughly the same shape. The normal 

distribution has a single peak, known as the center, and two tails that extend out equally, forming what is known as a bell shape 

or bell curve. 

Figure 

Statistics) 

: A symmetrical distribution in a histogram (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Symmetrical distributions can also have multiple peaks. Figure shows a bimodal distribution (bimodal means two 

modes), named for the two peaks that lie roughly symmetrically on either side of the center point. This is relatively difficult 

characteristic to detect numerically. Thus, it is important to visualize your data before moving ahead with any statistical 

analyses. 


☰ 

Figure 


: A bimodal distribution in a histogram (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Distributions that are not symmetrical also come in many forms, more than can be described here. The most common 

asymmetry to be encountered is referred to as skew, in which one of the two tails of the distribution is disproportionately 

longer than the other. This property can affect the value of the averages we use in our analyses and make them an inaccurate 

representation of our data, which causes many problems. 

Skew can either be positive or negative (also known as right or left, respectively), based on which tail is longer. It is very easy 

to get the two confused at first; many students want to describe the skew by where the bulk of the data (larger portion of the 

histogram, known as the body) is placed, but the correct determination is based on which tail is longer. You can think of the 

tail as an arrow: whichever direction the arrow is pointing is the direction of the skew. Figures and show positive 

(right) and negative (left) skew, respectively. 

Figure : A positively skewed distribution in a histogram (CC-BY-NC-SA Foster et al. from An Introduction to 

Psychological Statistics) 

Figure : A negatively skewed distribution in a histogram (CC-BY-NC-SA Foster et al. from An Introduction to 


The next section will go into more detail about skew, and introduce you to a new term to describe different shapes of 

symmetrical distributions. 



☰ 




2.7: Skew and Kurtosis 

Statisticians and researchers look at a lot of frequency charts (usually line graphs and histograms), so they know what to look 

for. Khan Academy has a great video previewing what will be discussed in this section. 

Skew 

Since it’s the more interesting of the two, let’s start by talking about the skew. 

The shape of a frequency chart (line graph or histogram) can tell you a lot of about that data set. The simplest things to look 

for are any extreme scores (called outliers) that seem to be much higher or much lower than most of the other scores. These 

outliers may affect the shape of the distribution, making it skewed. A skewed distribution is one in which many scores are 

bunched up to one side, and there are only a few scores on the other side. A distribution can be positively skewed (meaning 

that the scores are bunched to the left, and the thin tail is pointing to the right) or negatively skewed (meaning that the scores 

are bunched to the right, and the thin tail is pointing to the left). Figure shows examples of a positively skewed line 

graph (on the right) and a negatively skewed line graph on the left. 

Figure 

- Diagram of Skew. (CC-BY-SA Rodolfo Hermans (Godot), CC BY-SA 3.0,viaWikimedia Commons) 

Skewness is basically a measure of asymmetry, and the easiest way to explain it is by drawing some pictures. As Figure 

illustrates in histograms, if the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper 

tail) and not so many extremely large values (left panel), then we say that the data are negatively skewed. On the other hand, if 

there are more extremely large values than extremely small ones (right panel) we say that the data are positively skewed. That’s 

the qualitative idea behind skewness. 

Figure - Diagram of Skew and No Skew (CC-BY-SA Danielle Navarro from Learning Statistics with R) 

Kurtosis 

For distributions that are not skewed, you can look at how short and fat the distribution is, or if it is a medium bell-shaped 

curve, or if it is tall and narrow. This way of describing the shape of symmetrical (not-skewed) distributions, how broad the 

distribution is, is called kurtosis. Put simply, kurtosis is a measure of the “tailedness” of the data. As you can see in the line 


graph in Figure , there are three main types: wide, medium, or tall. Wide and flat graphs are called platykurtic. 

Medium, 

bell-shaped graphs are called mesokurtic or a normal distribution. Tall and narrow graphs are called leptokurtic. 

Figure 

 

- Diagram of Three Kurtosis Types in Line Graphs (CC-BY Larry Green) 

Figure shows the three types of kurtosis in histograms. The histogram on the left shows platykurtic data, the 

middle 

histogram shows mesokurtic data, and on the right, we have a leptokurtic data set. By mathematical calculations, the “normal 

curve” (black lines) has zero kurtosis. 

Figure : Diagram of Three Kurtosis Types in Histograms (CC-BY-SA Danielle Navarro from Learning Statistics with R) 

 

The types of kurtosis are summarized in Table . 

 


Technical Name 

Table 

 

- Informal Description of Types of Kurtosis 

Informal Description 

mesokurtic 

leptokurtic 

platykurtic 

just pointy enough 

too pointy 

too flat 

Summary 

In practice, neither skew nor kurtosis is used anywhere near as frequently as the measures of central tendency and variability 

that will be discussed in next chapter. Skew is pretty important, though, so you do see it mentioned a fair bit; but I’ve actually 

never seen kurtosis reported in a scientific article to date 

Contributors 


Peter H. Westfall (Paul Whitfield Horn Professor and James and Marguerite Niver Professor, Texas Tech University) 


Figure Image Attribution: Rodolfo Hermans (Godot), CC BY-SA 3.0, via Wikimedia 

Commons 

Figure Image Attribution: CC-BY Larry Green from Lake Tahoe Community College 

 


2.8: Graphing Quantitative Date- Line Graphs 

Although line graphs are great at showing how data changes through time, we will mostly use line graphs to show frequency 

distributions; line graphs of frequency distributions are sometimes called frequency polygons, but that just reminds me of high 

school geometry. For example, the frequency table in Table 2.2.2 showed the frequency of final exam scores, a quantitative 

variable, while Figure shows the same information but as a graph: 

Figure : Line Graph of Frequency of Final Exam Scores. (Copyright CC-SA, chart by Dr. MO via data provided 

by OpenIntro.org ) 

Let’s unpack Figure , as well. First, the x-axis has the same information as the “x” column in Figure 2.1, each score that 

was earned on the final exam. The x-axis is always the one on the bottom going from left to right. Because we are talking 

about a numerical variable (ratio scale of measurement; quantitative variable), the information on the x-axis is each score. In 

a line graph, you did not create your own categories; for example, you would not create categories based on letter grades. The 

scores should be from lowest to highest. Second, the y-axis shows the same information as the “f” column in Figure 2.2.2, the 

number of students who earned that score. The y-axis is always going up and down. This chart was also created with a 

common commercial spreadsheet software (Excel), and there are again fancier types of line graph. And again, the fancier a 

chart looks, the more difficult it is to easily and quickly understand what the chart is showing. 

Interpreting Line Graphs 

I added a question to your interpretations. 

Note 

Can you see which question was added? Why do you think it was added for line charts? 




1. This is a line graph. Line graphs are used to display quantitative data (or changes through time) 

 


1. The x-axis on line graphs show each score (or each time period). The x-axis on line graphs shows each time period. In 

a line graph showing changes through time, the x-axis will always be some marker of time. 

3. 

What does the y-axis measure in Figure ? 


However, the y-axis in Figure shows the average percentages earned out of 

all points possible in the class each 

semester. 

4. [New question!] Is Figure skewed? If so positively or negatively? If 

not, is the graph tall/narrow, medium/normal, 

or wide/flat? 

1. Because the tail is so long on th left and there seems to be a pile of scores towards the right, I would say that this line 

graph is negatively skewed. This seems backwards to me, so I looked back at Figure 2.xxx (the picture of the skewed 

charts). Because it is skewed, it can’t have kurtosis. 


1. Two things that pop out to me. First, there is a bump of more frequent scores between 78 to 83 points. This suggests 

that most students in this unknown class earned a C+ or a B- on the Final Exam. The second thing I notice is that the 

left tail is pretty long. This means there are also quite a few students who earned low scores on the Final Exam. 


1. This chart makes me wonder if the professor ended up “curving” the exam since the highest score was 94 points, and 

that score was earned by only one student. The next highest score was 89 points, again with only one student earning 

that score on the Final Exam. If I haven’t said this before, THIS IS NOT MY CLASS. 



1. 85% of the Class Passed! [17 out of 20 students earned 70 points or higher] 


1. Almost everyone in the class passed the Final Exam, but only one student earned more than 90 points. 


1. I would guess that students who are going to take this class from this professor might be interested. College 

administrators might also want to know so that they can see that most students are passing their classes. 

Line Graphs Through Time 

But for thoroughness, let’s also look at a line graph that shows data through time. Figure shows the percentage of 

points 

earn out of all points in Dr. MO's behavioral statistics courses in seven consecutive semesters (starting in Fall 2013, and 

ending in Fall 2016 when she started teaching a modified version of the course). 


Figure 

- Line Graph of Percentages Through Time (Copyright CC-BY-SA; Michelle Oja) 

Unpacking Figure , what do you see? First, let’s compare Figure to Figure . In Figure , the x-axis was 

each score, while in Figure the x-axis is the time period. Similarly, in Figure , the y-axis in the frequency, the 

number of students who earned that score. In Figure , the y-axis is the percentage earned. Frequency distributions are 

really important for statistical analyses, but they aren’t the only type of chart that you’ll come across. 

Interpreting Line Graphs 

What kind of graph is Figure ? 

This is a line graph. Line graphs are used to display quantitative data (or changes through time) 

What does the x-axis measure in Figure ? 

The x-axis on line graphs shows each time period. In a line graph showing changes through time, the x-axis will 

always be some marker of time. 


The y-axis is the axis that goes up and down. For most charts discussed in this textbook, the y-axis will be frequencies. 

However, the y-axis in Figure shows the average percentages earned out of all points possible in the class each 

semester. 

Is Figure skewed? If so positively or negatively? If not, is the graph tall/narrow, medium/normal, or wide/flat? 

This question doesn't really make sense when the x-axis isn't scores (quantitative data points). In line graphs through 

time, we are looking for trends not kurtosis or skew. 

What do you notice from Figure ? What pops out to you? 

Spring 2015 was a great year for students! 

What does Figure make you wonder about? 


What happened in Spring 2015?! And why didn't it keep happening? 

What is a catchy headline for Figure ? 

Average is Passing in Dr. MO's Behavioral Statistics Class 

 

How could you summarize the info in Figure 

into one sentence? 

The average percentages were all passing, with Spring 2015 being a particularly high average score. 

 

Who might want to know the information in Figure ? 

I would guess that students who are going to take this class from this professor might be interested. College 

administrators might also want to know so that they can see that most students are passing this professor's class. 

 

Another kind of graph that is used to show quantitative variables is a histogram, discussed next! 



CUNY) 


Exercise 

Exercise 

2.9: Graphing Quantitative Data- Histograms 

Another common type of graph for frequency distributions for quantitative variables (interval or ratio scales of measurement) 

is a histogram. 

What other type of chart shows a frequency distribution for quantitative variables? Hint: We've already talked about it. 

Answer 

Line graph (sometimes called a frequency polygon) 

Histograms look more like bar graphs, but are actually more like line graphs. Why? 

Answer 

Your answer probably talks about how the bars in bar charts don't touch, but the bars (or bins) in histograms do touch. 

Your answer should say something about the type of variable: quantitative variables (ratio or interval scale of 

measurements) should be graphed with histograms, and qualitative variables (nominal scale of measurement) should 

be graphed with bar charts. 

Figure shows a histogram (created by SPSS) of the frequency distribution of the final exam score data from Table 

2.2.2 

(and also shown in the line graph in Figure 2.8.1. 

Figure : Histogram of Frequency of Final Exam Scores. (Copyright CC-SA, chart by Dr. MO via data 

provided 


As with all frequency distributions, the y-axis of a histogram shows the frequency of each score (how many people had each 

score) and the x-axis has each score. Figure shows the number of students who in each score category on the 

y-axis 

(Frequency), and labels the score categories on the x-axis (Final Exam Score, with each 10 points labeled). To make the 

histogram, we just count up the number of data points falling inside each bin, then plot those frequency counts as a function of 

the bins. Voila, a histogram.The 10-point categories are called bins. Bin #1 goes from about 55 to abou 59 points, bin #2 goes 

about 65 to 69 points, and so on until the last bin. The difference in the x-axis between a line graph and a histogram is that the 

scores are combined into ranges, or categories, in a histogram. This makes it easier to quickly understand what a graph is 

showing if there are a lot of different scores (like everyone’s GPA in your class), but it does lose some accuracy. 


Each bar (bin), and can be used to answer questions about the frequency of scores within these bin categories. For example, 

how many people scored between 90 to 100 points on the Final Exam? The seventh bar (or first on the right), the one between 

90 and 100 on the x-axis, tells you how many. Look how tall that bar is. How tall is it? The height is shown on the y-axis, 

which provides the frequency. One person scored between 90 points and 100 points on the Final Exam. 

Just like a line graph, a histogram can show the shape (kurtosis) or skew, and the range of our data set. The shape of the 

histogram refers to how it goes up and down. The shape tells us where the data is. For example, when the bars are low we 

know there isn’t much data there. When the bars are high, we know there is more data there. So, where is most of the data? It 

looks like it’s mostly in the middle two bins, between 75-85 points. The range of the data tells us the minimum (lowest score) 

and the maximum (highest score) of the data. In Figure , most of teh scores are between 65 points and 95 points, but the 

minimum scores is between 55-59 points. 

When you make a histogram you get to choose how wide each bar will be. For example, below are four different histograms of 

the same data (a happiness survey). What changes is the width of the bins. 

Figure : Four histograms of the same data using different bin widths. (CC-BY-SA Matthew J. C. Crump from Answering 

Questions with Data- Introductory Statistics for Psychology Students) 

All of the histograms have roughly the same overall shape: From left to right, the bars start off small, then go up, then get 

small again. In other words, as the numbers get closer to zero, they start to occur more frequently. We see this general trend 

across all the histograms. But, some aspects of the trend fall apart when the bars get really narrow. For example, although the 

bars generally get taller when moving from -1000 to 0, there are some exceptions and the bars seem to fluctuate a little bit. 

When the bars are wider, there are less exceptions to the general trend. How wide or narrow should your histogram be? It’s a 

Goldilocks question. Make it just right for your data. 

Students often get confused between a bar graph and a histogram because they both have bars. There are two differences, 

however. First, in a histogram, the bars must touch; there should not be space between the categories that were created from 


the quantitative data. This leads to the second difference between histograms and bar charts: Histograms show quantitative 

variables, while bar charts show qualitative variables (nominal scale of measurement). 

Histogram Interpretation 

The following are the same questions asked about the Figure 2.8.1, the line graph showing the frequency distribution of this 

same data. You might find it interesting how the answers change a little depending on how the data is presented. 



1. This is a histogram. Histograms are used to display quantitative data, but it combines each score into groups. In Figure 

2.xxx, the groups are 10-point ranges. 


1. The x-axis on histograms show categories based 

on each score. In Figure , the software created 10-point 

categories for each Final Exam Score that a student earned. 

 



In Figure , that means that the y-axis shows how many students earned 

scores on the Final Exam in each 10-point 

range. 

4. Is Figure skewed? If so positively or negatively? If not, 

is the graph tall/narrow, medium/normal, or wide/flat? 

1. Because there is a gap between the lowest score category and the next one, I would say that this histrogram is 

negatively skewed. However, it looks less skewed than the same data in the line graph in Figure 2.8.1. In fact, just by 

looking at Figure and not knowing the exact scores, I might have said that it is 

not skewed and that is has 

approximately normal kurtosis (mesokurtic). 


1. Two things that pop out to me. First, there seemed to be the most scores around the middle of the distribution. The 

second thing I notice is that one column that’s to the left. This shows that one student (frequency = 1) score below 60 

points on the Final Exam. 


1. This histogram still makes me wonder if the professor ended up “curving” the exam so that the highest earning score 

was now like earning 100% on the Final Exam. If I haven’t said this before, THIS IS NOT MY CLASS. 



1. Final Exam Scores are Normal [This is catchy if you know what a Normal Distribution is; more in Ch. 4!] 


1. The class did really well on the Final Exam, although one student scored pretty low. 


1. I still am guessing that students who are going to take this class from this professor might be interested. College 


Summary 

Making a histogram is our first act of officially summarizing the data. We are no longer look at the individual bits of data, 

instead we will see how the numbers group together. Let’s look at a histogram of the happiness data, and then explain it. 


Matthew J. C. Crump (Brooklyn College of CUNY) 



2.10: Graphing Quantitative Data- Boxplots 

You might think that you've never seen a box plot, but you probably have seen something similar. 

An alternative to line graphs and histograms is a boxplot, sometimes called a “box and whiskers” plot. Like line graphs and 

histograms, they’re best suited to quantitative data (interval or ratio scale of measurement). The idea behind a boxplot is to 

provide a simple visual depiction of the score in the exact middle (median), the where about each fourth (quartile) of the scores 

are, and the range of the data. And because boxplots do so in a fairly compact and intuitive way, they have become a very 

popular statistical graphic. When you look at this plot, this is how you should interpret it: the thick line in the middle of the 

box is the median; the box itself spans the range from the 25th percentile to the 75th percentile; and the “whiskers” cover the 

full range from the minimum value to the maximum value. This is summarized in the annotated plot in Figure . 

 

Figure - A basic boxplot (panel a), plus the same plot with annotations added to 

explain 

what aspect of the data set each part of the boxplot corresponds to (panel b). (CC-BY-SA- Danielle 

Navarro from Learning Statistics with R) 

In most applications, the “whiskers” don’t cover the full range from minimum to maximum. Instead, they actually go out to the 

most extreme data point that doesn’t exceed a certain bound. By default, this value is 1.5 times the interquartile range (you 

don't have to know what all of that means). Any observation whose value falls outside this range is plotted as a circle instead 

of being covered by the whiskers, and is commonly referred to as an outlier. Because the boxplot automatically separates out 

those observations that lie outside a certain range, people often use them as an informal method for detecting outliers: 

observations that are “suspiciously” distant from the rest of the data. 

Our Final Exam Score data provides just such an example, as shown in Figure , a boxplot created with SPSS 

software 

(standard spreadsheet software generally does not include boxplot options). If you look closely, there's a little circle under the 

bottom whisker with a "3" next to it. This is showing that three scores were below the range of the whiskers. These three 

scores can be considered outliers. If you go back to Figure 2.2.2, you can easily see what those three scores are (57, 66, 69). 


Figure - Boxplot of Frequency of Final Exam Scores. (Copyright CC-SA, chart by Dr. MO via data 

provided 


Box Plots Interpretation 

Let's answer the same questions that we've been answering, over this same data set, but based on the boxplot graphic in Figure 

. 

1. What kind of graph is Figure Figure ? 

 

1. This box plot. 

 


1. There is no x-axis because there is only one group. If there were more groups, each would have its own box and the 

name of the group would be labeled on the x-axis. 

3. 


1. The y-axis is the axis that goes up and down. In Figure ,the y-axis shows how many students earned different 

scores on the Final Exam by the size and location of the square in the middle. The “whiskers” or bars show information 

about the expected variation. You can also see a circle under the 60.00 line, with the number 3 next to it; this shows 

that 3 students scored that extreme, which is outside of what is expected. 

4. Is Figure Figure ? If so positively or negatively? If not, is 

the graph tall/narrow, medium/normal, or wide/flat? 

1. This is sort of a trick question. Skew is not shown in a box plot in the same way that it is shown in a line graph or 

histogram. However, you can see that the box in the middle is not centered evenly between the whiskers; the top 

whisker is longer than the bottom whisker. This shows that the bulk of scores fall within the box, but that there are 

some that go higher and lower, and the circle shows that there are three scores that are really low. 


1. Those three scores at the bottom jump out at me! Plus, the fact that the top whisker doesn’t go all the way to 100 

suggests that no one scored that high. 


1. This box plot makes me wonder about those three students outside of the whiskers more than the other charts did. 



1. Chances Are You’ll Be One of the 85% [Because 17 out of 20 students earned a passing grade on the Final Exam; 

17/20 = .85 x 100 = 85%] 


1. The class did really well on the Final Exam, although three students did not pass. 


1. I still am guessing that students who are going to take this class from this professor might be interested. College 


That's it on boxplots! Let's move on to the last kind of graph that will be discussed, scatterplots! 






2.11: Graphing Quantitative Data- Scatterplots 

So far, we’ve mostly covered charts that show how many people were in each category or earned each score. Scatterplots are a 

different kind of chart that can show you about how many people scored each variable, but the real purpose is show two a 

person’s score on two quantitative variables (ratio or interval scale of measurement) at once. In scatterplots, the both the x- 

axis and the y-axis are scores on a quantitative variable, and there is no direct measurement of frequency. Figure shows 

an example with GPA and the hours that the student studied (the GPA_Study_Hours file from OpenIntro.org). Each dot 

represents a student, and shows both their GPA and how many hours they studied. Find the dot that’s most right; that student 

who spent 70 hours studying, and did not quite earn a 4.0. In fact, it looks like no one who studied over 35 hours earned a 4.0; 

what do you think that can tell you about time spent studying? On the other hand, only a few people who spent less than 20 

hours studying earned a 4.0; what can that tell you about how the time you spend studying leads to your GPA? 

Figure : Scatterplot of Study Hours and GPA. (Copyright CC-SA, chart by Dr. MO via data provided by OpenIntro.org ) 

Remember IVs and DVs from Ch. 1? In scatterplots, the cause (IV) is usually plotted on the x-axis, and the effect (DV) is 

plotted on the y-axis. What we are looking for is: what happens to the DV when the IV increases? In Figure , as Study 

Hours increase, GPA tends to increase. However, this isn’t a strong relationship because there are a couple students who spend 

a lot of time studying and don’t get the GPA payoff, while there are even more students who spend 5-10 hours studying but 

still earn good grades. In general, however, it looks like spending more time studying is related to higher GPAs. 

Note 

 

Use this Statistics How To to practice making a scatterplot. Once you can make one, with the IV on the bottom and the 

DV on the side, you’ll better understand what they can show. 

Scatterplot Interpretation 

1. What kind of graph is Figure ? 

1. This is a scatterplot. Scatterplots are used to plot two quantitative variables for each person. In Figure , the two 

variables are Study Hours and GPA. 

 


1. The x-axis in Figure shows the hours that each person studied. Lower hours of studying are on the left, 


higher hours of studying are on the right. 

 



However, the y-axis of the scatterplot in Figure shows another quantitative variable, GPA of each person. 

Lower 

GPAs are towards the bottom of the axis, and higher GPAs are towards the top. 

 

4. Is Figure skewed? If so positively or negatively? If not, is the graph tall/narrow, medium/normal, or wide/flat? 

 


Exercise 

1. Trick question! Because scatterplots do not show frequency distributions, they aren’t described as skewed and they 

don’t have kurtosis. Instead, we look to see if there is a general trend; as one variable goes up, does the other go up, go 

down, or are they unrelated? It does look like more studying is related to higher GPAs, but this isn’t quite as strong of a 

relationship as we’d like to see. 


1. What I first noticed is that a lot of students study 20 hours or less, and still have high GPAs. I also noticed that there are 

lots of scores in 5-hour increments (e.g., 5 hours, 10 hours, 15 hours, etc.). This suggests that students were estimated 

the number of hours that they studied, and rounding to the nearest 5-hour mark. 


1. I wonder if the results would change if the hours studied were measured more accurately, rather than having students 

estimate at a later time? I also wonder what was going with those two students who studied more than 60 hours (and 

still didn’t earn a 4.0). What were they doing that they considered “studying”? Or is this a data-entry mistake? Like, 

did someone say that they studied 6 hours but it was entered as 60 hours? 



1. Study Less and Still Get a 4.0! 


1. A lot of students studied about 5 to 20 hours and earned GPAs above 3.0. 


1. I still am guessing that students who might be interested. Maybe tutors and folks working in a Learning Center would 

find this useful? 

Remember those videos about the best way to learn from Dr. Chew, a cognitive psychologist? First discussed in 1.3: The 

Scientific Method and all 5 linked in 1.8 (What can you do to learn material for this class?). Well, Dr. Chew’s first video has a 

scatter plot (Part 1: Beliefs that Make You Fail)! I suggest that you watch the whole video, but the experiment and scatterplot 

show up around 4:30 minutes. 

Watch Dr. Chew’s first video (Part 1: Beliefs that Make You Fail), then answer the same set of questions. I suggest that 

you watch the whole video, but the experiment and scatterplot show up around 4:30 minutes. 

Answer 

1. What kind of graph is in Dr. Chew’s Beliefs that Make you Fail video? 

1. This is a scatterplot. Scatterplots are used to plot two quantitative variables for each person. In the scatterplot 

in the video, the two variables are Estimated Percentage Correct and Actual Percentage Correct on an exam. 

2. What does the x-axis measure in the scatterplot in Dr. Chew’s Beliefs that Make you Fail video? 

1. The x-axis in the scatterplot in Dr. Chew’s Beliefs that Make you Fail video shows the percentage that students 

thought that they got correct on the exam. Lower scores are on the left, and higher percentages are on the right. 

3. What does the y-axis measure iin the scatterplot in Dr. Chew’s Beliefs that Make you Fail video? 

1. The y-axis is the axis that goes up and down. For most charts discussed in this textbook, the y-axis will be 

frequencies. However, the y-axis of the scatterplot in the scatterplot in Dr. Chew’s Beliefs that Make you Fail 

video shows another quantitative variable, the actual percentage correct on the exam. Lower scores are 

towards the bottom of the axis, and higher percentages are towards the top. 

4. Is the scatterplot in Dr. Chew’s Beliefs that Make you Fail video skewed? If so positively or negatively? If not, is 

the graph tall/narrow, medium/normal, or wide/flat? 

1. Trick question! Because scatterplots do not show frequency distributions, they aren’t described as skewed and 

they don’t have kurtosis. Instead, we look to see if there is a general trend; as one variable goes up, does the 

other go up, go down, or are they unrelated? It does look like most students estimated their percentage correct 

pretty close to their actual percentage correct, but were a few students who estimated that they did way better 


than they actually did. Dr. Chew says that these students have low meta-cognition, meaning that they are 

unclear on what material they actually know. 

5. What do you notice from the scatterplot in Dr. Chew’s Beliefs that Make you Fail video? What pops out to you? 

1. What I first noticed was that most students’ estimates were pretty close to their actual percentages earned. Yay, 

most students had good meta-cognition! 

6. What does the scatterplot in Dr. Chew’s Beliefs that Make you Fail video make you wonder about? 

1. I wonder if students get better at estimating what they know throughout the semester? 

7. What is a catchy headline for the scatterplot in Dr. Chew’s Beliefs that Make you Fail video? 

1. Meta-Cognition: Predicting Your Future! 

8. How could you summarize the info in the scatterplot in Dr. Chew’s Beliefs that Make you Fail video into one 

sentence? 

1. Most students’ estimates of their exam grades were relatively close to their actual exam grades, but there were 

a few students who badly overestimated their scores. 

9. Who might want to know the information in the scatterplot in Dr. Chew’s Beliefs that Make you Fail video? 

1. I still am guessing that students might be interested. Maybe tutors and folks working in a Learning Center 

would find this useful? 

Positive or Negative? 

Scatterplots can show when there is a linear relationship., meaning that the two variables vary together and when plotted look 

like a straight line. Scatterplots can show positive linear relationships, negative linear relationships, or show that there is no 

relationship between the two variables. 

Positive linear relationship: When one variable goes up, the other goes up. 

Positive doesn’t mean good!! It means that the two variables change in the same direction. 

Negative linear relationship: 

When one variable goes up, the other variable goes down. 

Negative doesn’t mean bad!!! It means that the two variables change in different directions. 

Figure shows a scatterplot for hypothetical scores on Job Satisfaction (x-axis) and Well-Being of the worker (on the 

y- 

axis). We can see from the axes that each of these variables is measured on a 10-point scale, with 10 being the highest on both 

variables (high satisfaction and good health and well-being) and 1 being the lowest (dissatisfaction and poor health). This 

scale suggests an interval scale of measurement for both variables, which means that there are two quantitative variables. 

When we look at this plot, we can see that the variables do seem to be related. The higher scores on Job Satisfaction tend to 

also be the higher scores on Well-Being, and the same is true of the lower scores. 


Figure - Scatterplot of Job Satifaction & Well-Being (CC-BY-NC-SA Foster et al. from An Introduction to 


Figure demonstrates a positive relation. As Job Satisfacation scores increase, Well-Being scores also tend to increase. 

Although this is not a perfect relation (if it were, the points would form a single straight line), it is nonetheless very clearly 

positive. This is one of the key benefits to scatterplots: they make it very easy to see the direction of the relation. 

As another example, Figure shows a negative relation between Job Satisfacation on the x-axis and Burnout on the y- 

axis. As we can see from Figure , higher scores on Job Satisfacation tend to correspond to lower scores on Burnout, 

which is how stressed, un-energetic, and unhappy someone is at their job. As with Figure , this is not a perfect relation, 

but it is still a clear one. As these figures show, points in a positive relation moves from the bottom left of the plot to the top 

right, and points in a negative relation move from the top left to the bottom right. 

Figure 


- Scatterplot of Job Satisfaction and Burnout (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Or None? 

Scatterplots can also indicate that there is no relation between the two variables. In these scatterplots (an example is shown 

below in Figure plotting Job Satisfaction and Job Performance) there is no interpretable shape or line in the scatterplot. 

The points appear randomly throughout the plot. If we tried to draw a straight line through these points, it would basically be 

flat. The low scores on job satisfaction have roughly the same scores on job performance as do the high scores on job 

satisfaction. Scores in the middle or average range of job satisfaction have some scores on job performance that are about 

equal to the high and low levels and some scores on job performance that are a little higher, but the overall picture is one of 

inconsistency. 


As we can see, scatterplots are very useful for giving us an approximate idea of whether or not there is a relation between the 

two variables and, if there is, if that relation is positive or negative. They are also useful for another reason: they are the only 

way to determine one of the characteristics of correlations that are discussed next: form. 

Figure - Scatterplot of Job Satisfaction and Job Performance (CC-BY-NC-SA Foster et al. from An Introduction to 


Curvilinear Relationships 

Sometimes the fastest way isn't a straight line... 

A linear relation is what we saw in Figure and Figure . If we drew a line through the middle points in the any of 

the scatterplots, we would be best suited with a straight line. The term “linear” comes from the word “line”. The relation 

between two variables can also be curvilinear. As the name suggests, a curvilinear relation is one in which a line through the 

middle of the points will be curved rather than straight. Two examples are presented in Figures and . 

Figure 


: Curvilinear Scatterplot with an Elbow (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 


Figure 

Statistics). 

: Curvilinear Scatterplot with an Inverted-U (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Curvilinear relations can take many shapes, and the two examples above are only a small sample of the possibilities. What 

they have in common is that they both have a very clear pattern but that pattern is not a straight line. 

Sometimes when we look at scatterplots, it is tempting to get biased by a few points that fall far away from the rest of the 

points and seem to imply that there may be some sort of relation. These points are outliers, which we discussed before. 

We’ll return to scatterplots in Chapters 14 and 15 to discuss ways to statistically analyze this relationship to see if what the 

scatterplot looks like it shows is really what is happening. 





2.12: Summary and Some Honesty 

Intentional Lies or Accidental Misleading? 

It’s easy to mislead with graphs. Some of the fancy graphics make it more difficult to see the exact numbers, but there are 

many ways to be misled with graphs. To avoid being fooled by a fancy chart you should also check out the axes. In a 

frequency distribution, the y-axis should generally be one or zero. This article “Lying with Graphs” (website 

address: http://chmullig.com/2011/04/lying-with-graphs/) by Chris Mulligan in 2011 shows an example of both a confusing 

type of chart, and mis-leading axes (plus, he didn’t think that the original article calculated the actual numbers correctly, 

either); always be vigilant! 

Example of Misleading 

We will use data from interviews of computer users to show how graphs in general, but especially bar graphs, can be 

misleading. Here's the research scenario: When Apple Computer introduced the iMac computer in August 1998, the company 

wanted to learn whether the iMac was expanding Apple’s market share. Was the iMac just attracting previous Macintosh 

owners? Was it purchased by newcomers to the computer market> Were previous Windows users switching over? To find out, 

500 iMac customers were interviewed. Each customer was categorized as a previous Macintosh owner, a previous Windows 

owner, or a new computer purchaser. 

Figure shows that three-dimensional bars are usually not as effective as two-dimensional bars because it's unclear 

where the top of the bar is. 

Figure 

: A three-dimensional bar chart (CC-BY-NC-SA Foster et al. from An Introduction to Psychological Statistics) 

Here is another way that fanciness can lead to trouble. Instead of plain bars, it is tempting to substitute meaningful images. For 

example, Figure presents iMac data using pictures of computers. The heights of the pictures accurately represent the 

number of buyers, yet Figure is misleading because the viewer's attention will be captured by the area (width) of the 

computers, not just the height. The areas can exaggerate the size differences between the groups. In terms of percentages, the 

ratio of previous Macintosh owners to previous Windows owners is about 6 to 1. But the ratio of the two areas in Figure 

is about 35 to 1. A biased person wishing to hide the fact that many Windows owners purchased iMacs would be 

tempted to use Figure $\PageIndex{2$! Edward Tufte coined the term “lie factor” to refer to the ratio of the size of the effect 

shown in a graph to the size of the effect shown in the data. He suggests that lie factors greater than 1.05 or less than 0.95 

produce unacceptable distortion. 

Foster et al. 7/30/2021 2.12.1 https://stats.libretexts.org/@go/page/22018

Figure 


: Example with a lie factor greater than 8. (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Another distortion in bar charts results from setting the bottom of the y-axis to a something other than zero. The bottom of the 

Y-axis should represent the least number of cases that could have occurred in a category. Normally, this number should be 

zero. Figure shows the iMac data with a baseline of 50. Once again, the differences in areas suggests a different story 

than the true differences in percentages. The number of Windows-switchers seems minuscule compared to its true value of 

12%. 

Figure 

: Example with a baseline of 50. (CC-BY-NC-SA Foster et al. from An Introduction to Psychological Statistics) 

Note 

Why don't you use this iMac data to play around and create your own bar chart by hand or with computer software, 

without making it misleading? 

Examples of Misleading, or Lies? 

Example 

Go through the examples from this 2014 BuzzFeed report by Katie Notopoulos 

(https://www.buzzfeednews.com/article/katienotopoulos/graphs-that-lied-to-us), and decide one thing that is wrong with 

each chart. 

1. The time an upside down y-axis made "Stand Your Ground" seem much more reasonable. 

2. The time 7 million was 5x more than 6 million. 

3. The Govenor race where one guy's 37% was WAY more than just 37% 


4. This bar graph that shows the devastating drop in this pitcher's speed after one year. 

5. The time when Scotland really gave 110%. 

6. This poll which gives about the same infuriating response as when you ask someone what they want for dinner. 

7. The pollster who really doesn't want people to think A levels are getting harder. 

8. This graph that measure..... units of "innovation"? 

9. The number skipping y-axis on this totally legit graph. 

10. This graph showing the astonishing growth rate to .06 hawks? 

11. Whatever happened in this school paper. 

12. The graph where last year, last week, and today are equally far apart. 

13. This chart showing the giant gulf between 35% and 39.6%. 

Solution 

1. Check out the y-axis. Instead of zero on the bottom, with the higher numbers being higher, the y-axis is “upside 

down” from what is typical. 

2. Check out the y-axis. Not only is it not labeled, but if the y-axis had started with zero, the 1 million difference 

would have looked like much. 

3. The two 37%’s probably look difference chart doesn’t show decimal points, but if the y-axis had started with zero, 

the two 37%’s would be indistinguishable. 

4. It’s the missing y-axis problem again, and not starting the y-axis at zero. 

5. There’s nothing technically wrong with these numbers, but you should always have someone else check your math. 

6. There’s nothing technically wrong with these numbers, but it is a reminder that designing survey questions is harder 

than it seems! 

7. This is an appropriate way to use a pie chart (showing percentages that add up to close to 100%), but the pie slices 

need to represent the percentage, so the blue should only be 49% of the pie. It looks like 60% or about two-thirds… 

8. It’s important to label your y-axis with units that make sense! 

9. The y-axis should have equal intervals, meaning that the difference between the first number and the second 

number (2 points in this example, from 1 to 3) is the same as the difference between the highest number and the second 

highest number (which is 1803-516 = 1287 in this example). 

10. There is nothing technically wrong with this chart, but even going from zero to 1.0 would have seemed more 

accurate when dealing with such small numbers on the y-axis. 

11. The colors in the pie charts should take up the portion of the pie that the numbers show. Also, pie charts show 100% 

of the sample, but 26% + 26% + 26% +26% = 104%, so someone needed a friend to check their math here, as well. 

12. Here is an example of a line graph showing changes through time, except that the x-axis doesn’t have equal 

intervals. It should probably have been a bar chart showing the three time categories, rather than trying to show a change 

through time. 

13. Another example of a missing y-axis that is misleading because the missing y-axis doesn’t start at zero. 

You can also read this great 2014 blog by Ravi Parikh of Heap showing some of the same examples of graphs that seem to be 

trying to intentionally mislead readers, and an explanation about what is inappropriate or missing: 

https://heap.io/blog/data-stories/how-to-lie-with-data-visualization 

Be careful to avoid misleading your audience, or being misled! 

Graphs Summary 


Example 

You've learned about many types of graphs, and why they should be used. You also should have practiced interpreting what 

they can tell you. 

Play around with this interactive website (website address: 

https://www.zoology.ubc.ca/~whitlock/Kingfisher/SamplingNormal.htm), then answer what kind of graphs they are. 

Solution 

Although both have lines to show a Normal Distribution, the actual graphs of the data are histograms because they are 

bars that touch, and the bars show the frequency of different quantitative variables. 

Now that you’ve learned so much about charts and graphs, go through these charts from the New York Times (website 

address: https://www.nytimes.com/column/whats-going-on-in-this-graph) and determine if the x-axis is showing qualitative or 

quantitative variables and if you can describe the shape of any frequency distributions. 





2.13: APA Style Charts 

APA Style, as described in the Publication Manual of the American Psychological Association (Seventh Edition), has specific 

ways to format the title of tables and graphs. We already talked about tables, now it's time to talk about graphs. 

Graphs and pictures are considered figures (not tables). In APA Style, figure labels should be numbered and have a 

descriptive title. “Figure 1” should be in bold, and on the line above the title (which should be in italics). They both should be 

above the figure, flush-left. An example is below, and you can find more details at the best site to learn about APA Style: 

Purdue Online Writing Lab. Here is a link to OWL Purdue’s page on tables and figures in APA Style. 

Example: 

Note 

Figure 1 


Figures should be near where they are being discussed in the appear or in an appendix (which would be the last section of the 

paper). Finally, make sure to always label your axes with a title and with the numbers/categories, not just the figure title. 

That's it for graphs! I'll see you in the next chapter to talk about how to describe these distributions of data with just a few 

numbers! 




Exercise 

Exercise 

2.14: Describing Data using Distributions and Graphs Exercises 

At the end of most of the chapters will be a couple questions that go over some basic information in that chapter. However, 

most chapters will have whole sections to practice on, and those practice scenarios are much more helpful than the few 

problems at the end of the chapter. In this chapter, the real practice is when we interpreted what each graph was telling us. 

The problems at the end of the chapter are more like a review than actually helping you know what you know. 

Either way, enjoy! And good luck! Here are some basic questions about graphs. 

Name some ways to graph quantitative variables and some ways to graph qualitative variables. 

Answer 

Qualitative variables are displayed using pie charts and bar charts. Quantitative variables are displayed as box plots, 

histograms, etc. 

Exercise $\PageIndex{2]$ 

Pretend you are constructing a histogram for describing the distribution of salaries for individuals who are 40 years or 

older, but are not yet retired. 

1. What is on the Y-axis? Explain. 

2. What is on the X-axis? Explain. 

3. What would be the probable shape of the salary distribution? Explain why. 

You do not need to draw the histogram, only describe it below. 

Answer 

1. The Y-axis would have the frequency because this is always the case in histograms. 

2. The X-axis has income, because this is out quantitative variable of interest. 

3. Because most income data are positively skewed, this histogram would likely be skewed positively too. 

Which of the box plots on the graph has a large positive skew? Which has a large negative skew? 


Figure : Three box plots. (CC-BY-NC-SA Foster et al. from An Introduction to 


Answer 

Chart B has the positive skew because the outliers (dots and asterisks) are on the upper (higher) end; chart C has the 

negative skew because the outliers are on the lower end. 

Exercise 

Explain the differences between bar charts and histograms. When would each be used? 

Answer 

In bar charts, the bars do not touch; in histograms, the bars do touch. Bar charts are appropriate for qualitative 

variables, whereas histograms are better for quantitative variables. 








This chapter will show you how to describe a distribution of data with just a few numbers. 


Let's start describing distributions of data, instead of only graphing them! 


Whether you took math last semester or 40 years ago, this refresher will orient you to what you'll 

need to remember to complete the statistical analyses later in your class. 


What is a central tendency? 


What are three numbers that can describe the center of a distribution of data? 


Mode = most (not highest) 


Medians are in the middle of the road. 


A mean is the mathematical average, but what does a mean mean? 


Summary and review of the measures of central tendency and that you just learned. 


If you know all three measures of central tendency, what do you really know? 


Now, measures of how different each score is from the center. 


What's the standard measure of variability? Standard deviations, of course! 


Do you know how to use the standard deviation formula? Let's practice! 


But what can standard deviations show us about a distribution of data? 


What can we learn from looking at all three measures of central tendency AND the standard deviation? How can we use this 

information to predict or understand the shape of our distribution of data? 


A little practice? 

1 8/27/2021

3.1: Introduction to Descriptive Statistics 

The death of one man is a tragedy. The death of millions is a statistic. 

– Josef Stalin, Potsdam 1945 

950,000 – 1,200,000 

– Estimate of Soviet repression deaths, 1937-1938 (Ellman 2002) 

Stalin’s infamous quote about the statistical character death of millions is worth giving some thought. The clear intent of his 

statement is that the death of an individual touches us personally and its force cannot be denied, but that the deaths of a 

multitude are incomprehensible, and as a consequence mere statistics, more easily ignored. I’d argue that Stalin was half right. 

A statistic is an abstraction, a description of events beyond our personal experience, and so hard to visualize. Few if any of us 

can imagine what the deaths of millions is “really” like, but we can imagine one death, and this gives the lone death its feeling 

of immediate tragedy, a feeling that is missing from Ellman’s cold statistical description. 

Yet it is not so simple: without numbers, without counts, without a description of what happened, we have no chance of 

understanding what really happened, no opportunity event to try to summon the missing feeling. And in truth, as I write this, 

sitting in comfort on a Saturday morning, half a world and a whole lifetime away from the Gulags, when I put the Ellman 

estimate next to the Stalin quote a dull dread settles in my stomach and a chill settles over me. The Stalinist repression is 

something truly beyond my experience, but with a combination of statistical data and those recorded personal histories that 

have come down to us, it is not entirely beyond my comprehension. Because what Ellman’s numbers tell us is this: over a two 

year period, Stalinist repression wiped out the equivalent of every man, woman and child currently alive in the city where I 

live. Each one of those deaths had it’s own story, was it’s own tragedy, and only some of those are known to us now. Even so, 

with a few carefully chosen statistics, the scale of the atrocity starts to come into focus. 

Thus it is no small thing to say that the first task of the statistician and the scientist is to summarize the data, to find some 

collection of numbers that can convey to an audience a sense of what has happened. This is the job of descriptive statistics, but 

it’s not a job that can be told solely using the numbers. You are a data analyst, not a statistical software package. Part of your 

job is to take these statistics and turn them into a description. When you analyze data, it is not sufficient to list off a collection 

of numbers. Always remember that what you’re really trying to do is communicate with a human audience. The numbers are 

important, but they need to be put together into a meaningful story that your audience can interpret. That means you need to 

think about framing. You need to think about context. And you need to think about the individual events that your statistics are 

summarizing. 

Reference 

Ellman, M. (2002). Soviet Repression Statistics: Some Comments. Europe-Asia Studies, 54 (7). Taylor & Francis: 1151–72. 





Example 

3.2: Math Refresher 

Statistics is not math. It does, however, use math as a tool. This section will review some of the tools that you'll use in 

statistical equations, with an ending section on calculating percentages. You learned much of this in junior high or early high 

school, but that was a long time ago for most of us! 

Order of Operations 

What does “Please Excuse My Dear Aunt Sally” or “PEMDAS” mean? 

Solution 

It's the order in which you should do each mathematical operation. 

“Please Excuse My Dear Aunt Sally” or PEMDAS tells you what kind of calculations must be completed before other 

calculations. However, this can still be overwhelming. Although you must follow the PEMDAS order, you can start at the top 

left, and go right and down, just like how you read. In fact, PEMDAS is sorta like a sentence structure but with equations. 

Note 

P- You must complete the calculations within parentheses before you can use those numbers in the rest of the 

equation. 

E- You must complete the exponents (which will always just be squaring a number in statistics) before you can use 

that number in the rest of the equation. 

M- You must complete the multiplication before you can add or subtract it with other numbers in the equation. I 

actually think that you're supposed to divide before you multiply? 

D- You must complete the division before you can add or subtract it with other numbers in the equation. I actually 

think that you're supposed to divide before you multiply? 

A- Once all of the other mathematical calculations are completed, you can now add or subtract. I actually think that 

you're supposed to subtract before you add? 

S- Once all of the other mathematical calculations are completed, you can now add or subtract. I actually think that 

you're supposed to subtract before you add? 

What might throw you off in actual equations that we'll use is square rooting. Square rooting is actually the last operation 

because you have to get down to one number. 

Notation & Symbols 

Table will cover some of the major mathematical symbols used in statistical equations. The column on the right is 

where 

you can write explanations or examples in your own words. 


Table 

- Common Mathematical Symbols 

Symbol Read As Example or Description 

> "greater than" 

< "less than" 

* or XY "multiply" or "times" 

2 

 

() "square" or "times itself" 

√ 

|#| 

"square root" or "find the number that, 

multiplied by itself, equals the number in the 

square root sign" 

“absolute value” or "make it a positive 

number" (the lines, not the #-sign) 

Summing (Sigma or ) 

Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. 

This section covers the basics of this summation notation. Let's say we have a variable X that represents the weights (in 

grams) of four grapes show in Table : 

V 

Grape 

 

Table 

 

- Grape Weights in Grams 

A 4.6 

B 5.1 

C 4.9 

D 4.4 

The following formula means to sum up the weights of the four grapes: 

X 

The Greek letter 

indicates summation. The X is a placeholder for all of the scores. Therefore, 

 

† 

Sums of Squares 

Many formulas involve squaring numbers before they are summed. This is indicated as 

Notice that: 

 

because the expression on the left means to sum up all the values of and then square the sum (19² = 361), whereas the 

expression on the right means to square the numbers and then sum the squares (90.54, as shown). 

9 9 9 

Ü 

 

Multiplication (Products) 

Some formulas involve the sum of cross products. Below are the data for variables and . The cross products ( , or 

) are shown in the third column. The sum of the cross products is 3 + 4 + 21 = 28. 

Table - Table of Scores and Products 

Participant 

A 1 3 3 


Example 

Example 

9 : 9: 

Participant 

B 2 2 4 

C 3 7 21 

Percentages 

Providing the percentages is often more useful than merely displaying frequency information, especially when comparing 

distributions that have different sample sizes (different total amount of units/numbers). 

 

1. Having 1 red-head in a group of 10= _____% 

2. Having 1 red-head in a group of 100= _____% 

Solution 

1. Having 1 red-head in a group of 10 

2. Having 1 red-head in a 

group of 100 

In Example , showing the frequency of one doesn’t tell us as much as the percentage does because although 10% isn't 

a 

large proportion, it's way bigger than 1%. The only time when providing frequencies (counts) is better than percentages is 

when you have a very small population. For example, having 1 out of 3 people like your cologne is 33%, but it that doesn’t 

really tell you anything meaningful about how you smell. 

Once you get to a decimal point, you then multiply by 100 to get the percentage. 

 

 

You probably all do this in your head, but try to write it out for now so that you don't skip a step. 

Imagine that there are 35 students in your class and 5 of them had earned full credit on all assignments. What’s the 

percentage of students in your class earning an A+? 

Solution 

 

1. Division: (We would normally round decimal points to 2 

numbers after the decimal, but for 

 

percentages we should include 4 numbers because we'll need to multiply them. 

g 2. 

Multiplication: 

3. Conclusion: The percentage of students in your class earning an A+ was 14.29% 

As we move on to the actual calculations used to describe distributions of data, you can check out this Crash Course video on 

data visualization which has a bit on percentages. 





3.3: What is Central Tendency? 

What is “central tendency,” and why do we want to know the central tendency of a group of scores? Let us first try to answer 

these questions intuitively. Then we will proceed to a more formal discussion. 

Imagine this situation: You are in a class with just four other students, and the five of you took a 5-point pop quiz. Today your 

instructor is walking around the room, handing back the quizzes. She stops at your desk and hands you your paper. Written in 

bold black ink on the front is “3/5.” How do you react? Are you happy with your score of 3 or disappointed? How do you 

decide? You might calculate your percentage correct, realize it is 60%, and be appalled. But it is more likely that when 

deciding how to react to your performance, you will want additional information. What additional information would you like? 

If you are like most students, you will immediately ask your neighbors, “Whad'ja get?” and then ask the instructor, “How did 

the class do?” In other words, the additional information you want is how your quiz score compares to other students' scores. 

You therefore understand the importance of comparing your score to the class distribution of scores. Should your score of 3 

turn out to be among the higher scores, then you'll be pleased after all. On the other hand, if 3 is among the lower scores in the 

class, you won't be quite so happy. 

This idea of comparing individual scores to a distribution of scores is fundamental to statistics. So let's explore it further, using 

the same example (the pop quiz you took with your four classmates). Three possible outcomes are shown in Table . They 

are labeled “Dataset A,” “Dataset B,” and “Dataset C.” Which of the three datasets would make you happiest? In other words, 

in comparing your score with your fellow students' scores, in which dataset would your score of 3 be the most impressive? 

Table 

: Three possible datasets for the 5-point make-up quiz. 

Student Dataset A Dataset B Dataset C 

 

You 3 3 3 

John's 3 4 2 

Maria's 3 4 2 

Shareecia's 3 4 2 

Luther's 3 5 1 

In Dataset A, everyone's score is 3. This puts your score at the exact center of the distribution. You can draw satisfaction from 

the fact that you did as well as everyone else. But of course it cuts both ways: everyone else did just as well as you. 

Now consider the possibility that the scores are described as in Dataset B. This is a depressing outcome even though your 

score is no different than the one in Dataset A. The problem is that the other four students had higher grades, putting yours 

below the center of the distribution. 

Finally, let's look at Dataset C. This is more like it! All of your classmates score lower than you so your score is above the 

center of the distribution. 

Now let's change the example in order to develop more insight into the center of a distribution. Figure shows the results 

of an experiment on memory for chess positions. Subjects were shown a chess position and then asked to reconstruct it on an 

empty chess board. The number of pieces correctly placed was recorded. This was repeated for two more chess positions. The 

scores represent the total number of chess pieces correctly placed for the three chess positions. The maximum possible score 

was 89. 


Figure : Back-to-back stem and leaf display. The left side shows the memory scores of the non-players. The right 

side 

shows the scores of the tournament players. 

Two groups are compared. On the left are people who don't play chess. On the right are people who play a great deal 

(tournament players). It is clear that the location of the center of the distribution for the non-players is much lower than the 

center of the distribution for the tournament players. 

We're sure you get the idea now about the center of a distribution. It is time to move beyond intuition. We need a formal 

definition of the center of a distribution. In fact, we'll offer you three definitions! This is not just generosity on our part. There 

turn out to be (at least) three different ways of thinking about the center of a distribution, all of them useful in various contexts. 

In the remainder of this section we attempt to communicate the idea behind each concept. In the succeeding sections we will 

give statistical measures for these concepts of central tendency. 

Definitions of Center 

Now we explain the three different ways of defining the center of a distribution. All three are called measures of central 

tendency. 

Balance Scale 

One definition of central tendency is the point at which the distribution is in balance. Figure shows the distribution of 


five numbers 2, 3, 4, 9, 16 placed upon a balance scale. If each number weighs one pound, and is placed at its position along 

the number line, then it would be possible to balance them by placing a fulcrum at 6.8. 

Figure 

For another example, consider the distribution shown in Figure 

middle. 

 

: A balance scale. 

 

. It is balanced by placing the fulcrum in the geometric 

Figure 

Figure 

: A distribution balanced on the tip of a triangle. 

illustrates that the same distribution can't be balanced by placing the fulcrum to the left of center. 

 

 


Figure 

: The distribution is not balanced. 

Figure shows an asymmetric distribution. To balance it, we cannot put the fulcrum halfway between the lowest 


highest values (as we did in Figure ). Placing the fulcrum at the “half way” point would cause it to tip towards the left. 

 

 

Figure 

An asymmetric distribution balanced on the tip of a triangle. 

Smallest Absolute Deviation Another way to define the center of a distribution is based on the concept of the sum of the 

absolute deviations (differences). Consider the distribution made up of the five numbers 2, 3, 4, 9, 16. Let's see how far the 

distribution is from 10 (picking a number arbitrarily). Table shows the sum of the absolute deviations of these 

numbers 

from the number 10. 

 

Table 

: An example of the sum of absolute deviations 

Values Absolute Deviations from 10 

 

2 

3 

4 

9 

16 

8 

7 

6 

1 

6 

Sum 28 

The first row of the table shows that the absolute value of the difference between 2 and 10 is 8; the second row shows that the 

absolute difference between 3 and 10 is 7, and similarly for the other rows. When we add up the five absolute deviations, we 

get 28. So, the sum of the absolute deviations from 10 is 28. Likewise, the sum of the absolute deviations from 5 equals 3 + 2 

+ 1 + 4 + 11 = 21. So, the sum of the absolute deviations from 5 is smaller than the sum of the absolute deviations from 10. In 

this sense, 5 is closer, overall, to the other numbers than is 10. 

We are now in a position to define a second measure of central tendency, this time in terms of absolute deviations. Specifically, 

according to our second definition, the center of a distribution is the number for which the sum of the absolute deviations is 

smallest. As we just saw, the sum of the absolute deviations from 10 is 28 and the sum of the absolute deviations from 5 is 21. 

Is there a value for which the sum of the absolute deviations is even smaller than 21? Yes. For these data, there is a value for 

which the sum of absolute deviations is only 20. See if you can find it. 

Smallest Squared Deviation 

We shall discuss one more way to define the center of a distribution. It is based on the concept of the sum of squared 

deviations (differences). Again, consider the distribution of the five numbers 2, 3, 4, 9, 16. Table shows the sum of 


squared deviations of these numbers from the number 10. 

Table : An example of the sum of squared deviations 

 


Values Squared Deviations from 10 

2 

3 

4 

9 

16 

64 

49 

36 

1 

36 

Sum 186 

The first row in the table shows that the squared value of the difference between 2 and 10 is 64; the second row shows that the 

squared difference between 3 and 10 is 49, and so forth. When we add up all these squared deviations, we get 186. Changing 

the target from 10 to 5, we calculate the sum of the squared deviations from 5 as 9 + 4 + 1 + 16 + 121 = 151. So, the sum of 

the squared deviations from 5 is smaller than the sum of the squared deviations from 10. Is there a value for which the sum of 

the squared deviations is even smaller than 151? Yes, it is possible to reach 134.8. Can you find the target number for which 

the sum of squared deviations is 134.8? 

The target that minimizes the sum of squared deviations provides another useful definition of central tendency (the last one to 

be discussed in this section). It can be challenging to find the value that minimizes this sum. 





3.3.1: Introduction to Measures of Central Tendency 

We’ve seen that we can get a sense of data by plotting frequency data on a graph. These frequency charts show us what the 

numbers look like, approximately how big and small they are, and how similar and different they are from another. It is good 

to get a feeling about the numbers in this way. But, these visualization are not very precise. In addition to summarizing 

numbers with graphs, we can summarize numbers using numbers (NO, please not more numbers, we promise numbers can be 

your friend). 

From many numbers to one 

Measures of central have one important summary goal: to reduce a pile of numbers to a single number that we can look at. We 

already know that looking at thousands of numbers is hopeless. Wouldn’t it be nice if we could just look at one number 

instead? We think so. It turns out there are lots of ways to do this. Then, if your friend ever asks the frightening question, “hey, 

what are all these numbers like?”. You can say they are like this one number right here. 

But, just like in Indiana Jones and the Last Crusade (highly recommended movie), you must choose your measure of central 

tendency wisely. 

Each of the following sections will look at the mode, the median, and then the mean. So let's start with the easiest measure of 

central tendency- mode! 




Matthew J. C. Crump 8/20/2021 3.3.1.1 CC-BY-SA https://stats.libretexts.org/@go/page/22025

Example 

3.3.2: Measures of Central Tendency- Mode 

The mode is the most frequently occurring number in your measurement. That is it. How do you find it? You have to count the 

number of times each number appears in your measure, then whichever one occurs the most, is the mode. 

Table 

 

- Mode of One 

Scores 

1 

1 

1 

2 

3 

4 

5 

The mode of the above set is 1, which occurs three times. Every other number only occurs once. 

OK fine. What happens here: 

Table 

Scores 

 

- Bi-Modal 

1 

1 

1 

2 

2 

2 

3 

4 

5 

6 

Hmm, now 1 and 2 both occur three times each. What do we do? 

What do we do when there is more than one mode? 

Solution 

When there are only a couple modes, then we list them. When there are many modes, we don't have to list out each 

mode. 

For Table , the mode would be both scores of 1 and scores of 2. This is called bi-modal. When we have more than 

two 

modes, it is called multi-modal, and we don't have to list out each mode. Only list what is useful to know. 


Why is the mode a measure of central tendency? Well, when we ask, “what are my numbers like”, we can say, “most of the 

numbers are like a 1 (or whatever the mode is)”. 

Is the mode a good measure of central tendency? That depends on your numbers. For example, consider these numbers 

Table 

 

- Mode of One Again 

Scores 

1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Here, the mode is 1 again, because there are two 1s, and all of the other numbers occur once. But, are most of the numbers 

like a 1? No, they are mostly not 1s. 

“Argh, so should I or should I not use the mode? I thought this class was supposed to tell me what to do?”. There is no telling 

you what to do. Every time you use a tool in statistics you have to think about what you are doing and justify why what you 

are doing makes sense. Sorry. (Not sorry). 

Note 

When might the mode be useful? 





3.3.3: Measures of Central Tendency- Median 

The median is the exact middle of the data. After all, we are asking about central tendency, so why not go to the center of the 

data and see where we are. What do you mean middle of the data? Let’s look at these numbers: 

Table - List of Numbers 

 

Scores 

1 

5 

4 

3 

6 

7 

9 

Umm, OK. So, three is in the middle? Isn’t that kind of arbitrary. Yes. Before we can compute the median, we need to order 

the numbers from smallest to largest. 

Table - List of Numbers in Order 

 

Scores 

1 

3 

4 

5 

6 

7 

9 

Now, five is in the middle. And, by middle we mean in the middle. There are three numbers to the higher than five, and three 

numbers lower than five. So, five is definitely in the middle. 

OK fine, but what happens when there aren’t an even number of numbers? Then the middle will be missing right? Let’s see: 

Table - List of Six Scores 

 

Scores 

1 

2 

3 

4 

5 

6 

There is no number between 3 and 4 in the data, the middle is empty. In this case, we compute the median by figuring out the 

number in between 3 and 4. So, the median would be 3.5: 

 

 

 

This calculation is a mean, which we'll talk more about soon, and for the rest of the semseter! 


Is the median a good measure of central tendency? Sure, it is often very useful. One property of the median is that it stays in 

the middle even when some of the other numbers get really weird. For example, consider these numbers: 

Scores 

1 

2 

3 

4 

4 

4 

5 

6 

6 

6 

7 

7 

1000 

Most of these numbers are smallish, but the 1000 is a big old weird number, very different from the rest. The median is still 5, 

because it is in the middle of these ordered numbers. We can also see that five is pretty similar to most of the numbers (except 

for 1000). So, the median does a pretty good job of representing most of the numbers in the set, and it does so even if one or 

two of the numbers are very different from the others. 

Finally, outlier is a term will we use to describe numbers that appear in data that are very different from the rest. 1000 is an 

outlier, because it lies way out there on the number line compared to the other numbers. What to do with outliers is another 

topic we discuss sometimes throughout this course. 





3.3.4: Measures of Central Tendency- Mean 

The mean is also called the average. And, we’re guessing you might already now what the average of a bunch of numbers is? 

It’s the sum of the numbers, divided by the number of number right? How do we express that idea in a formula? Just like this: 

 

c 

In which (read as "x-bar") is the mean of the sample. Sometimes Dr. MO adds parentheses to show that the scores (all c 

of 

the X's) are summed before anything is divided because PEMDAS says that you calculate anything in parentheses first. 

 

 

c 

 

 

 

c 

“That looks like Greek to me”. Yup. The symbol is Greek, called sigma, and it stands for the operation of summing. We 

sum up all of the numbers, then divide the sum by , which is the total number of numbers. Sometimes you will see to 

refer to the mean of all of the numbers. 

In plain English, the formula looks like: 

4VNPGNZOVNCFST 

“Well, why didn’t you just say that?”. We just did! 

Let’s compute the mean for these five numbers: 

Table -Five Scores 

 

Scores 

.FBO 

$PVOUPGNZOVNCFST 

2 

3 

6 

7 

9 

Add 'em up: 

 

Divide 'em by the number 

of scores: 

 

Or, to put the numbers in the formula, it looks like this: 

 

 

c 

 

 

Is the mean a good measure of central tendency? By now, you should know: it depends. 

 

 

What does the mean mean? 

It is not enough to know the formula for the mean, or to be able to use the formula to compute a mean for a set of numbers. We 

believe in your ability to add and divide numbers. What you really need to know is what the mean really “means”. This 

requires that you know what the mean does, and not just how to do it. Puzzled? Let’s explain. 

Can you answer this question: What happens when you divide a sum of numbers by the number of numbers? What are the 

consequences of doing this? What is the formula doing? What kind of properties does the result give us? FYI, the answer is 

not that we compute the mean. 

OK, so what happens when you divide any number by another number? Of course, the key word here is divide. We literally 

carve the number up top in the numerator into pieces. How many times do we split the top number? That depends on the 

bottom number in the denominator. Watch: 


So, we know the answer is 4. But, what is really going on here is that we are slicing and dicing up 12 aren’t we. Yes, and we 

slicing 12 into three parts. It turns out the size of those three parts is 4. So, now we are thinking of 12 as three different pieces 

. I know this will be obvious, but what kind of properties do our pieces have? You mean the fours? Yup. 

Well, 

obviously they are all fours. Yes. The pieces are all the same size. They are all equal. So, division equalizes the numerator by 

the denominator… 

“Umm, I think I learned this in elementary school, what does this have to do with the mean?”. The number on top of the 

formula for the mean is just another numerator being divided by a denominator isn’t it. In this case, the numerator is a sum of 

all the values in your data. What if it was the sum of all of the 500 happiness ratings? The sum of all of them would just be a 

single number adding up all the different ratings. If we split the sum up into equal parts representing one part for each person’s 

happiness what would we get? We would get 500 identical and equal numbers for each person. It would be like taking all of 

the happiness in the world, then dividing it up equally, then to be fair, giving back the same equal amount of happiness to 

everyone in the world. This would make some people more happy than they were before, and some people less happy right. Of 

course, that’s because it would be equalizing the distribution of happiness for everybody. This process of equalization by 

dividing something into equal parts is what the mean does. See, it’s more than just a formula. It’s an idea. This is just the 

beginning of thinking about these kinds of ideas. We will come back to this idea about the mean, and other ideas, in later 

chapters. 

Pro tip: The mean is the one and only number that can take the place of every number 

in the data, such that when you add up all the equal parts, you get back the original 

sum of the data. 





3.3.5: Summary of Measures of Central Tendency 

In the previous section we saw that there are several ways to define central tendency. This section defines the three most 

common measures of central tendency: the mean, the median, and the mode. The relationships among these measures of 

central tendency and the definitions given in the previous section will probably not be obvious to you. 

This section gives only the basic definitions of the mean, median and mode. A further discussion of the relative merits and 

proper applications of these statistics is presented in a later section. 

Arithmetic Mean 

The arithmetic mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the 

number of numbers. The symbol “ ” (pronounced “mew”) is used for the mean of a population. The symbol “ ” (pronounced 

“X-bar”) is used for the mean of a sample. The formula for is shown below: 

where is the sum of all the numbers in the population and is the number of numbers in the population. 

The formula for 

is essentially identical: 

where is the sum of all the numbers in the sample and is the number of numbers in the sample. The only distinction 

between these two equations is whether we are referring to the population (in which case we use the parameter ) or a sample 

of that population (in which case we use the statistic ). 

As an example, the mean of the numbers 1, 2, 3, 6, 8 is 20/5 = 4 regardless of whether the numbers constitute the entire 

population or just a sample from the population. 

Figure shows the number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League 

in the 2000 season. The mean number of touchdown passes thrown is 20.45 as shown below. 

Figure 

: Number of touchdown passes. (CC-BY-NC-SA Foster et al. from An Introduction to Psychological Statistics) 

Although the arithmetic mean is not the only “mean” (there is also a geometric mean, a harmonic mean, and many others that 

are all beyond the scope of this course), it is by far the most commonly used. Therefore, if the term “mean” is used without 

specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic 

mean. 

Median The median is also a frequently used measure of central tendency. The median is the midpoint of a distribution: the 

same number of scores is above the median as below it. For the data in Figure , there are 31 scores. The 16th highest 

score (which equals 20) is the median because there are 15 scores below the 16th score and 15 scores above the 16th score. 

The median can also be thought of as the 50th percentile. 

When there is an odd number of numbers, the median is simply the middle number. For example, the median of 2, 4, and 7 is 

4. When there is an even number of numbers, the median is the mean of the two middle numbers. Thus, the median of the 

numbers 2, 4, 7, 12 is: 

Foster et al. 8/20/2021 3.3.5.1 CC-BY-NC-SA https://stats.libretexts.org/@go/page/22029

When there are numbers with the same values, each appearance of that value gets counted. For example, in the set of numbers 

1, 3, 4, 4, 5, 8, and 9, the median is 4 because there are three numbers (1, 3, and 4) below it and three numbers (5, 8, and 9) 

above it. If we only counted 4 once, the median would incorrectly be calculated at 4.5 (4+5 divided by 2). When in doubt, 

writing out all of the numbers in order and marking them off one at a time from the top and bottom will always lead you to the 

correct answer. 

Mode 

The mode is the most frequently occurring value in the dataset. For the data in Figure , the mode is 18 since 

more teams 

(4) had 18 touchdown passes than any other number of touchdown passes. With continuous data, such as response time 

measured to many decimals, the frequency of each value is one since no two scores will be exactly the same (see discussion of 

continuous variables). Therefore the mode of continuous data is normally computed from a grouped frequency distribution. 

Table shows a grouped frequency distribution for the target response time data. Since the interval with the 

highest 

frequency is 600-700, the mode is the middle of that interval (650). Though the mode is not frequently used for continuous 

data, it is nevertheless an important measure of central tendency as it is the only measure we can use on qualitative or 

categorical data. 

Range 

Table : Grouped Frequency 

Distribution 

Frequency 

500 - 600 3 

600 - 700 6 

700 - 800 5 

800 - 900 5 

900 - 1000 0 

1000 - 1100 1 

More on the Mean and Median 

In the section “What is central tendency,” we saw that the center of a distribution could be defined three ways: 

1. the point on which a distribution would balance 

2. the value whose average absolute deviation from all the other values is minimized 

3. the value whose squared difference from all the other values is minimized. 

The mean is the point on which a distribution would balance, the median is the value that minimizes the sum of absolute 

deviations, and the mean is the value that minimizes the sum of the squared deviations. 

Table shows the absolute and squared deviations of the numbers 2, 3, 4, 9, and 16 from their median of 4 and 

their 

mean of 6.8. You can see that the sum of absolute deviations from the median (20) is smaller than the sum of absolute 

deviations from the mean (22.8). On the other hand, the sum of squared deviations from the median (174) is larger than the 

sum of squared deviations from the mean (134.8). 

Value 

Table : Absolute & squared deviations from the median of 4 and the mean of 

6.8. 

Absolute Deviation from 

Median 

Absolute Deviation from 

Mean 

Squared Deviation from 

Median 

Squared Deviation from 

Mean 

2 2 4.8 4 23.04 

3 1 3.8 1 14.44 

4 0 2.8 0 7.84 

9 5 2.2 25 4.84 

16 12 9.2 144 84.64 

Total 20 22.8 174 134.8 


Figure shows that the distribution balances at the mean of 6.8 and not at the median of 4. The relative advantages and 

disadvantages of the mean and median are discussed in the section “Comparing Measures” later in this chapter. 

Figure : The distribution balances at the mean of 6.8 and not at the median of 4.0. (CC-BY-NC-SA Foster et al. from 

An Introduction to Psychological Statistics) 

When a distribution is symmetric, then the mean and the median are the same. Consider the following distribution: 1, 3, 4, 5, 

6, 7, 9. The mean and median are both 5. The mean, median, and mode are identical in the bell-shaped normal distribution. 

Comparing Measures of Central Tendency 

How do the various measures of central tendency compare with each other? For symmetric distributions, the mean and 

median, as is the mode except in bimodal distributions. Differences among the measures occur with skewed distributions. 

Figure shows the distribution of 642 scores on an introductory psychology test. Notice this distribution has a slight 

positive skew. 

Figure 


: A distribution with a positive skew. (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Measures of central tendency are shown in Table . Notice they do not differ greatly, with the exception that the mode is 

considerably lower than the other measures. When distributions have a positive skew, the mean is typically higher than the 

median, although it may not be in bimodal distributions. For these data, the mean of 91.58 is higher than the median of 90. 

This pattern holds true for any skew: the mode will remain at the highest point in the distribution, the median will be pulled 

slightly out into the skewed tail (the longer end of the distribution), and the mean will be pulled the farthest out. Thus, the 

mean is more sensitive to skew than the median or mode, and in cases of extreme skew, the mean may no longer be 

appropriate to use. 

Table 

Measure 

: Measures of central tendency for the test scores. 

Value 

Mode 84.00 

Median 90.00 

Mean 91.58 

The distribution of baseball salaries (in 1994) shown in Figure 

in Figure . 

has a much more pronounced skew than the distribution 

Figure : A distribution with a very large positive skew. This histogram shows the salaries of major league baseball 

players (in thousands of dollars). (CC-BY-NC-SA Foster et al. from An Introduction to Psychological Statistics) 

Table shows the measures of central tendency for these data. The large skew results in very different values for these 

measures. No single measure of central tendency is sufficient for data such as these. If you were asked the very general 

question: “So, what do baseball players make?” and answered with the mean of $1,183,000, you would not have told the 

whole story since only about one third of baseball players make that much. If you answered with the mode of $250,000 or the 


median of $500,000, you would not be giving any indication that some players make many millions of dollars. Fortunately, 

there is no need to summarize a distribution with a single number. When the various measures differ, our opinion is that you 

should report the mean and median. Sometimes it is worth reporting the mode as well. In the media, the median is usually 

reported to summarize the center of skewed distributions. You will hear about median salaries and median prices of houses 

sold, etc. This is better than reporting only the mean, but it would be informative to hear more statistics. 

Table : Measures of central tendency for baseball salaries (in thousands of 

dollars). 

Measure 

Value 

Mode 250 

Median 500 

Mean 1,183 





Example 

3.4: Interpreting All Three Measures of Central Tendency 

In a symmetrical, unimodal distribution, the mode, median, and mean are all the same number. 

Note 

What does symmetrical mean? 

What does unimodal mean? 

This means that the more symmetrical a distribution is, the closer the median, mode, and mean will be. 

By knowing how similar the median, mode, and mean are, you can guesstimate at the symmetricalness of the distribution. 

What’s skew? Positive skew? Negative skew? 

Solution 

Find these terms in the glossary, or the parts of this chapter when skew was discussed. 

Without having all of the data or seeing the distribution, we can make predictions about what the distribution might look like 

because: 

The mean is pulled in the direction of the extreme scores (or tail of the skew), 

The mode is the highest point in the skew, 

The median is between the mean and mode. 

In particular, we can predict that: 

If the mean is a lot bigger than the median, that the distribution will have a positive skew. 

Mean > Median = positive skew 

If the mean is a lot smaller than the median, that the distribution will have a negative skew. 

Mean < Median = negative skew 

Figure 

: Mean, Median, Mode, and Skew (Copyright CC-BY-SA Diva Jain/Diva Dugar, Source) 

 

Summary 


Although you've learned about the measures of central tendency before, this section shows you how we can use these three 

numbers together to understand what a whole distribution of data might tell us. 

Next, we'll look at measures of variability that, combined with the measures of central tendency, will tell us a LOT about a 

distribution of data. 



Image Attribution: CC-BY-SA; Found on: 

https://commons.wikimedia.org/wiki/File:Relationship_between_mean_and_median_under_different_skewness.png 

Author: Diva Jain/Diva Dugar 

Source: https://codeburst.io/2-important-statistics-terms-you-need-to-know-in-data-science-skewness-and-kurtosis- 

388fef94eeaa 


3.5: Introduction to Measures of Variability 

Variability refers to how “spread out” a group of scores is. To see what we mean by spread out, consider graphs in Figure 

. These graphs represent the GPA and the hours that the student studied (the GPA_Study_Hours file from 

www.OpenIntro.org/data website). You can see that the distributions are quite different. Specifically, the scores on Figure 

are more densely packed and those on Figure are more spread out. The differences among students in the hours 

that they studies were much greater than the differences among the same students on their GPA. These histograms show that 

some scores are closer to the mean than other scores, and that some distributions have more scores that are closer to the mean 

and other distributions have scores that tend to be farther from the mean. Sometimes, knowing one, or all, of the measures of 

central tendency is not enough. 

Figure 

: Histogram of GPA (CC-BY-SA Michelle Oja via data from OpenIntro.org) 

Figure 

: Histogram of Study Hours (CC-BY-SA Michelle Oja via data from OpenIntro.org) 

Figure shows sets of two normal distributions. Notice that some distributions are tall and narrow, and some are wider 

and flatter. The tall and narrow distribution has small variability between each score and the mean, and the wide and flat 

distribution has more variability between each score and the mean. The mean is a better description of the distribution of the 

data when it is tall and narrow since most scores aren't that different from the mean. 


Figure 


 

: Differences between two datasets. (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

The terms variability, spread, and dispersion are synonyms, and refer to how spread out a distribution is. Just as in the section 

on central tendency where we discussed measures of the center of a distribution of scores, in this section we will discuss 

measures of the variability of a distribution. We will focus on two frequently used measures of variability: range and standard 

deviation. We'll start with range now, then move on to standard deviation next. 

Range 

The range is the simplest measure of variability to calculate, and one you have probably encountered many times in your life. 

The range is simply the highest score minus the lowest score. Let’s take a few examples. What is the range of the following 

group of numbers: 10, 2, 5, 6, 7, 3, 4? Well, the highest number is 10, and the lowest number is 2, so 10 - 2 = 8. The range is 8. 

Let’s take another example. Here’s a dataset with 10 numbers: 99, 45, 23, 67, 45, 91, 82, 78, 62, 51. What is the range? The 

highest number is 99 and the lowest number is 23, so 99 - 23 equals 76; the range is 76. Now consider the two quizzes shown 

in Figure and Figure . On Quiz 1, the lowest score is 5 and the highest score is 9. Therefore, the range is 

4. 

The 

range on Quiz 2 was larger: the lowest score was 4 and the highest score was 10. Therefore the range is 6. 

The problem with using range is that it is extremely sensitive to outliers, and one number far away from the rest of the data 

will greatly alter the value of the range. For example, in the set of numbers 1, 3, 4, 4, 5, 8, and 9, the range is 8 (9 – 1). 

However, if we add a single person whose score is nowhere close to the rest of the scores, say, 20, the range more than doubles 

from 8 to 19. 

Contributors 





† 

 

☰ 

3.6: Introduction to Standard Deviations and Calculations 

Sum of Squares 

Variability can also be defined in terms of how close the scores in the distribution are to the middle of the distribution. Using 

the mean as the measure of the middle of the distribution, we can see how far, on average, each data point is from the center. 

The data from a fake Quiz 1 are shown in Table . The mean score is 7.0: 

 

 

 

 

Therefore, the column “ ” contains deviations (how far each Ã score deviates from the mean), here calculated as the score 

minus 7. cc 

The column Ã “ ” has the “Squared Deviations” and is simply the previous column squared. 

 

cc 

There are a few things to note about how Table is formatted, as this is the 

format you will use to calculate standard 

deviation. The raw data scores ( ) are always placed in the left-most column. This column is 9 

then summed at the bottom to 

facilitate calculating the mean (simply divided this number by the number of scores in the table). Once you have the mean, you 

can easily work your way down the middle column calculating the deviation scores. This column is also summed and has a 

very important property: it will always sum to 0 (or close to zero if you have rounding error due to many decimal places). This 

step is used as a check on your math to make sure you haven’t made a mistake. THIS IS VERY IMPORTANT. When you 

mis-calculate an equation, it is often because you did some simple math (adding or subtracting) incorrectly. It is very useful 

when equations have these self-checking points in them, so I encourage you to use them. If this column sums to 0, you can 

move on to filling in the third column of squared deviations. This column is summed as well and has its own name: the Sum 

of Squares (abbreviated as and given the formula ). As we will see, the Sum 

of Squares appears Ã again and 

again cc 

in different formulas – it is a very important value, and this table makes it simple to calculate without error. 

 


Ã cc 

9 

Ã cc 

☰ 

Table : Calculation of Variance for Quiz 1 

scores. 

9 2 4 

9 2 4 

9 2 4 

8 1 1 

8 1 1 

8 1 1 

8 1 1 

7 0 0 

7 0 0 

7 0 0 

7 0 0 

7 0 0 

6 -1 1 

6 -1 1 

6 -1 1 

6 -1 1 

6 -1 1 

6 -1 1 

5 -2 4 

5 -2 4 

† † † 

The calculations in Table can be done by hand, but it is also very easy to set up the data in any spreadsheet program 


learn the simple commands to make the spreadsheet do the simple math. As long as you tell it what to do with the correct 

numbers, then your results will be correct. You can also use the memory function in graphing calculators to save the the data 

set, and run some of the more common mathematical functions. Using spreadsheets and your graphing calculator to do the 

math also saves problems with rounding since the devices keep all of the decimals so you only have to round your final result. 

This statistics textbook will not go into explanations on how to use software (like spreadsheets, calculators, or more 

sophisticated statistical programs), but much that is easily accessible (like spreadsheets on Excel or Google) are relatively easy 

to learn to use 

Variance (of a Sample) 

Now that we have the Sum of Squares calculated, we can use it to compute our formal measure of average distance from the 

mean, the variance. The variance is defined as the average squared difference of the scores from the mean. We square the 

deviation scores because, as we saw in the Sum of Squares table, the sum of raw deviations is always 0, and there’s nothing we 

can do mathematically without changing that. 

The population parameter for variance is (“sigma-squared”) and is calculated as: 

U 

U 

Ã N 

 

 

Notice that the numerator that formula is identical to the formula for Sum of Squares presented above with replaced by . 

Thus, we can use the Sum of Squares table to easily calculate the numerator then simply divide that cc 

N 

value by to get 

variance. If we assume that the values in Table represent the full population, then we can take our value of Sum 

of 

Squares and divide it by to get our population variance: 


☰ 

U 

 

So, on average, scores in this population are 1.5 squared units away from the mean. This measure of spread is much more 

robust (a term used by statisticians to mean resilient or resistant to) outliers than the range, so it is a much more useful value to 

compute. 

But we won't do much with variance of a population. Instead, we'll focus on variance of a sample. The sample statistic used 

to estimate the variance is (“s-squared”): 

/ 

/ Ã cc 

 

Ã 

’ 

’ " 

This formula is very similar to the formula for the population variance with one change: we now divide by instead of . 

The value has a special name: the degrees of freedom (abbreviated as ). You don’t need to understand in depth what 

degrees of freedom are (essentially they account for the fact that we have to use a sample statistic to estimate the mean ( ) 

before we estimate the variance) in order to calculate variance, but knowing that the denominator is cc 

called " provides a nice 

shorthand for the variance formula: ". 

Going back to the values in Table and treating those scores as a sample, 

we can estimate the sample variance as: 

\[s^{2}=\dfrac{30}{20-1}=1.58 \nonumber \] 

Notice that this value is slightly larger than the one we calculated when we assumed these scores were the full population. This 

is because our value in the denominator is slightly smaller, making the final value larger. In general, as your sample size 

gets bigger, the effect of subtracting 1 becomes less and less. Comparing a sample size of 10 to a sample size of 1000; 10 – 1 = 

9, or 90% of the original value, whereas 1000 – 1 = 999, or 99.9% of the original value. Thus, larger sample sizes will bring 

the estimate of the sample variance closer to that of the population variance. This is a key idea and principle in statistics that 

we will see over and over again: larger sample sizes better reflect the population. 

The Big Finish: Standard Deviation 

The standard deviation is simply the square root of the variance. This is a useful and interpretable statistic because taking the 

square root of the variance (recalling that variance is the average squared difference) puts the standard deviation back into the 

original units of the measure we used. Thus, when reporting descriptive statistics in a study, scientists virtually always report 

mean and standard deviation. Standard deviation is therefore the most commonly used measure of spread for our purposes. 

The sample statistic follows the same conventions and is given as /: 

Ã ÃÃ 

Ã ÃÃÃÃÃÃÃÃ Ã 

 

/ 

cc 

Ã cc 

 

" 

The sample standard deviation from Table 

is: 

 

Ã 

 

Ã ÃÃÃÃ Ã 

ÃÃÃ 

 

 

ÃÃ 

 

 

We'll practice calculating standard deviations, then interpreting what the numbers mean. Because in behavioral statistics, it's 

not about the numbers. We never end with a number, we end with a conclusion (which can be as simple as a sentence, or can 

be several paragraphs). Social scientists want to know what the numbers mean because we use statistical analyses to answer 

real questions. 

Ã 

 

Ê 

/ 

Contributors 





3.7: Practice SD Formula and Interpretation 

You may or may not understand the importance of calculating and understanding the variation of your data. In some data sets, 

the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the 

mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that 

measures how far data values are from their mean. 

The Standard Deviation 

provides a numerical measure of the overall amount of variation in a data set, and 

can be used to determine whether a particular data value is close to or far from the mean. 

Answering Questions 

There are a couple common kinds of questions that standard deviations can answer, in addition being foundational for later 

statistical analyses. First, a standard deviation helps understand the shape of a distribution. Second, a standard deviation can 

show if a score is extreme. 

Describing the Shape of a Distribution 

The standard deviation provides a measure of the overall variation in a data set. 

The standard deviation is always positive or zero. The standard deviation is small when the data are all concentrated close to 

the mean, exhibiting little variation or spread. Distributions with small standard deviations have a tall and narrow line graph. 

The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation. 

Distributions with large standard deviations may have a wide and flat lin graph, or they may be skewed (with the outlier(s) 

making the standard deviation bigger). 

Suppose that we are studying the amount of time customers wait in line at the checkout at supermarket A and supermarket B. 

the average wait time at both supermarkets is five minutes. At supermarket A, the standard deviation for the wait time is two 

minutes; at supermarket B the standard deviation for the wait time is four minutes. 

Because supermarket B has a higher standard deviation, we know that there is more variation in the wait times at supermarket 

B. Overall, wait times at supermarket B are more spread out from the average; wait times at supermarket A are more 

concentrated near the average. 

Identifying Extreme Scores 

The standard deviation can be used to determine whether a data value is close to or far from the mean. 

Suppose that Rosa and Binh both shop at supermarket A. Rosa waits at the checkout counter for seven minutes and Binh waits 

for one minute. At supermarket A, the mean waiting time is five minutes and the standard deviation is two minutes. The 

standard deviation can be used to determine whether a data value is close to or far from the mean. 

Rosa waits for seven minutes: 

Seven is two minutes longer than the average of five; two minutes is equal to one standard deviation. 

Rosa's wait time of seven minutes is two minutes longer than the average of five minutes. 

Rosa's wait time of seven minutes is one standard deviation above the average of five minutes. 

Binh waits for one minute. 

One is four minutes less than the average of five; four minutes is equal to two standard deviations. 

Binh's wait time of one minute is four minutes less than the average of five minutes. 

Binh's wait time of one minute is two standard deviations below the average of five minutes. 

A data value that is two standard deviations from the average is just on the borderline for what many statisticians would 

consider to be far from the average. Considering data to be far from the mean if it is more than two standard deviations 

Michelle Oja 8/27/2021 3.7.1 CC-BY https://stats.libretexts.org/@go/page/22033

away is more of an approximate "rule of thumb" than a rigid rule. In general, the shape of the distribution of the data affects 

how much of the data is further away than two standard deviations. (You will learn more about this in later chapters.) 

The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to 

the right of five. We say, then, that seven is one standard deviation to the right of five because . 

If one were also part of the data set, then one is two standard deviations to the left of five because . 

Figure 

- Scale from 0 to 7 (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax) 

In general, a value = mean + (#ofSTDEV)*(standard deviation) 

where #ofSTDEVs = the number of standard deviations 

#ofSTDEV does not need to be an integer 

One is two standard deviations less than the mean of five because: 

touch should be multiplied) 

. (The numbers in parentheses that 

The equation value = mean + (#ofSTDEVs)*(standard deviation) can be expressed for a sample and for a population. 

sample: 

Population: 

The lower case letter s represents the sample standard deviation and the Greek letter (sigma, lower case) represents the 

population standard deviation. 

The symbol is the sample mean and the Greek symbol is the population mean. 

Calculating the Standard Deviation 

If is a number, then the difference " – mean" is called its deviation. In a data set, there are as many deviations as there are 

items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in 

symbols a deviation is . For sample data, in symbols a deviation is . 

The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a 

sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends 

on whether it is calculated from a population or a sample. The lower case letter s represents the sample standard deviation and 

the Greek letter (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics 

as the population, then s should be a good estimate of . 

To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the 

deviations (the values for a sample, or the values for a population). The symbol represents the population 

variance; the population standard deviation is the square root of the population variance. The symbol represents the 

sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard 

deviation as a special average of the deviations. 

If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared 

deviations to find the variance, we divide by , the number of items in the population. If the data are from a sample rather 

than a population, when we calculate the average of the squared deviations, we divide by n – 1, one less than the number of 

items in the sample. 

Formulas for the Sample Standard Deviation 

For the sample standard deviation, the denominator is , that is the sample size MINUS 1. 

Practice! 


Example 

 

 

 

In a fifth grade class at a private school, the teacher was interested in the average age and the sample standard deviation of 

the ages of her students. The following data are the ages for a sample of n = 20 fifth grade students. The ages are rounded 

to the nearest half year in Table , but first let's talk about the 

context. 

1. Who was the sample? Who could this sample represent (population)? 

The sample is the 20 fifth graders from a private school. The population could be all fifth graders from private schools? 


Age, in years, was measured. This is the DV, the outcome variable. 

Table - Ages of a sample of 20 fifth 

graders 

Ages of Sample Fifth Graders 

9 

9.5 

9.5 

10 

10 

10 

10 

10.5 

10.5 

10.5 

10.5 

11 

11 

11 

11 

11 

11 

11.5 

11.5 

11.5 

3. What is the mean? 

 

 

"c 

The average age is 10.53 years, rounded to two places. 

4. What is the standard deviation? 

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the 

variance. We will explain the parts of the table after calculating s. 

Table - Ages of One Fifth Grade 

Class 

Data Deviations Deviations 

2 


Ã 

/ 


x (X – ) (X– 

c c 

 

’ Ã g Ã 

9 

 

’ ’ 

9.5 

’ ’ ’ ’g’ 

 

’ ’ ’ 

9.5 

 

’ ’ ’g’ 

10 

 

’ 

’ 

10 

 

’ ’ 

 

10 

 

’ ’ ’ 

 

10 

 

’ ’ ’ 

 

10.5 

 

’ ’ ’ ’g’ 

10.5 

 

’ ’ ’ 

 

10.5 

’ ’ ’ 

 

10.5 

 

’ ’ 

 

’ 

 

11 

 

’ g 

 

11 

 

’ 

 

’ 11 

 

 

 

’ 11 

 

 

 

’ 11 

 

 

 

11 

’ 

 

g 

11.5 

 

’ 

11.5 

 

’ 

11.5 

’ 

 

0 (basically) 

The first column in Table has the data, the second column has has deviations (each score minus the mean), the 

third 

column has deviations squared. The first row is the row's title, the second row is the symbols for that column, the rest of 

the rows are the scores until the bottom row, which is the sum of each of the rows. 

Take the sum of the last column (9.7375) divided by the total number of data values minus one (20 – 1): 

 

2 

 

The sample standard deviation s is the square root of 

 

: 

" 

Ã ÃÃÃ Ã 

and this is rounded to two decimal places, . The standard deviation of the sample fo 20 fifth graders / is 

0.72 

years. 

Typically, you do the calculation for the standard deviation on your calculator or computer. When calculations are 

completed on devices, the intermediate results are not rounded so the results are more accurate. It's also darned easier. So 

why are spending time learning this outdated formula? So that you can see what's happening. We are finding the 

difference between each score and the mean to see how varied the distribution of data is around the center, dividing it by 

the sample size minus one to make it like an average, then square rooting it to get the final answer back into the units that 

we started with ( age in years). 

Ê 


Exercise 

" "c 

" N U 

4c 

4c 

4c 

For the following problems, recall that value = mean + (#ofSTDEVs)(standard deviation). Verify the mean and 

standard deviation or a calculator or computer. 

For a sample: = + (#ofSTDEVs)(s) 

For a population: = + (#ofSTDEVs) 

For this example, use x = + (#ofSTDEVs)(s) because the data is from a sample 

5. Verify the mean and standard deviation on your own. 

6. Find the value that is one standard deviation above the mean. Find ( + 1s). 

7. Find the value that is two standard deviations below the mean. Find ( – 2s). 

8. Find the values that are 1.5 standard deviations from (below and above) the mean. 

Solution 

a. You should get something close to 0.72 years, but anything from 0.70 to 0.74 shows that you have the general idea. 

b. 4c ( / 

4c Ã/ ’ 

c. 

Ã/ ’ 

4c d. 

4c / 

* * Ã Ã 

* Ã 

Notice that instead of dividing by , the calculation divided by because the data is a sample. For 

the sample, we divide by the sample size minus one ( ). The sample variance is an estimate of the population variance. 

After countless replications, it turns out that when the formula division by only N (the size of the sample) is used on a sample 

to infer the population’s variance, it always under-estimates the variance of the population. 

Which one has the bigger solution, the one with the smaller denominator or the larger denominator? 

 

 

 

 

 

 

Smaller denominators make the resulting product larger. To solve our problem of using the population’s variance formula on 

a sample under-estimating the variance, we make the denominator of our equation smaller when calculating variance for a 

sample. In other words, based on the mathematics that lies behind these calculations, dividing by ( ) gives * a Ã better 

estimate of the population. 

What does it mean? 

The deviations show how spread out the data are about the mean. From Table , The data value 11.5 is 

farther from the 

mean than is the data value 11 which is indicated by the deviations 0.97 and 0.47. A positive deviation occurs when the data 

value (age, in this case) is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean 

(that particular student is younger than the average age of the class) . The deviation is –1.525 for the data value nine. If you add 

the deviations, the sum is always zero, so you cannot simply add the deviations to get the spread of the data. By squaring the 

deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared 

deviation. But the variance is a squared measure and does not have the same units as the data. No one knows what 9.7375 

years squared means. Taking the square root solves the problem! The standard deviation measures the spread in the same units 

as the data. 

The standard deviation, or , is either zero or larger than zero. When the standard deviation is zero, there is no spread; / that U 

is, 

all the data values are equal to each other. The standard deviation is small when the data are all concentrated close to the mean, 

and is larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, 

the data values are very spread out about the mean; outliers can make or very large. 

/ U 

Scenario: Using one baseball professional team as a sample for all professional baseball teams, the ages of each of the 

players are shown in Table . 


cc 

Ã cc 

Table - One Baseball Team's 

Ages 


x (x – "c ) "c (x – 

 

21 

21 

22 

23 

24 

24 

25 

25 

28 

29 

29 

31 

32 

33 

33 

34 

35 

36 

36 

36 

36 

38 

38 

38 

40 

= 767 should be 0 (basically) = ? 

2 

 

If you get stuck after the table, don't forget that: 

cc 

/ Ã c c 

 

Ã 

ÃÃÃÃÃÃÃÃ 

Ã 

Ã 

All of your answers should be complete sentences, not just one word or one number. Behavioral statistics is about 

research, not math. 

1. Who was the sample? Who could this sample represent (population)? 


3. What is the mean? (Get in the practice of including the units of measurement when answering questions; a number is 

usually not a complete answer). 

4. What is the standard deviation? 

Ã 


Ã 

Ã ÃÃ 

 

/ 

Ã 

 

 

" 


4 Ã 4c 

5. Find the value that is two standard deviations above the mean, and determine if there are any players that are more 

than two standard deviations above the mean. 

Answer 

1. The sample is 25 players from a professional baseball team. They were chosen to represent all professional baseball 

players (it says so in the scenario description!). 

2. Age, in years, was measured. 

3. The mean of the sample ( was 30.68 years. 

4. The standard deviation was 6.09 years ( ), although due to rounding differences you c 

could / get 

something 

from about 6.05 to 6.12. Don't worry too much if you don't get exactly 6.09; if you are close, then you did the formula 

correctly! 

5. The age that is two standard deviations above the mean is 42.86 years, and none of the players are older than that. 

[\(\bar{x} + 2s = 30.68 + (2)(6.09) = 42.86 \nonumber \]. 

What standard deviation show us can seem unclear at first. Especially when you are unfamiliar (and maybe nervous) about 

using the formula. By graphing your data, you can get a better "feel" for what a standard deviation can show you. You will find 

that in symmetrical distributions, the standard deviation can be very helpful. Because numbers can be confusing, always graph 

your data. 

Summary 

The standard deviation can help you calculate the spread of data. 

The Standard Deviation allows us to compare individual data or classes to the data set mean numerically. 

 

is the formula for calculating the standard deviation of a sample. 

Ã 

ÃÃÃÃÃÃÃ 

Ã 

* Ã 


Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax 

College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at 

http://cnx.org/contents/30189442-699...b91b9de@18.114. 



3.8: Interpreting Standard Deviations 

Figure shows two symmerical distributions. The frequency distribution on the left (red) has a mean of 40 and a standard 

deviation of 5; the frequency distribution on the right (blue) has a mean of 60 and a standard deviation of 10. For the tall and 

narrow distribution on the left, 68% of the distribution is between 45 and 55; for the shorter distribution on the right, 68% is 

between 50 and 70. Notice that as the standard deviation gets smaller, the distribution becomes much narrower, regardless of 

where the center of the distribution (mean) is. Figure presents several more examples of this effect, discussed below. 

Figure : Symmetrical distributions with standard deviations of 5 and 10. (CC-BY-NC-SA Foster et al. from An 

Introduction to Psychological Statistics) 

Figure 


: Differences between two datasets. (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Figure was Figure 3.8.3 from the Introduction to Measures of Variability. In Figure Panel (a), the mean and 

standard deviations are both the same. You can see where the name “bell curve” comes from: it looks a bit like a bell. In 

Panel (b), the mean of one of the distributions of data (red, on the right) is higher then the mean of the other distribution (blue, 

on the left) but the standard deviations are the same because they are the same shape. these distributions have the same 

“width”. The only difference between them is that they’ve been shifted to the left or to the right. In every other respect they’re 

identical. In contrast, if we increase the standard deviation while keeping the mean constant, the peak of the distribution stays 

in the same place, but the distribution gets taller, as you can see in Panel (c); the means are the same (because the center of 

both distributions are in the same place on the x-axis), but one standard deviation is smaller (red, tall and narrow) then the 

other standard deviation (blue, bell shaped). The smaller standard deviation tells us that the scores, on average, are close to the 


mean. Notice, though, that when a distribution gets taller, the width shrinks (it gets more narrow). Finally, Panel (d) shows 

two distributions of data in which the means and the standard deviations are different. 

Contributors 





Exercise 

3.9: Putting It All Together- SD and 3 M's 

Consider the histogram of a frequency distribution in Figure . 

 

Figure . Example Symmetrical Histogram (Ciara Kidder of Marian University using work by Barbara Illowsky & 

Susan 

Dean (De Anza College) from OpenStax) 

The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some 

point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The 

mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the 

median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a 

symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median. 

The histogram for the data in Figure is not symmetrical. The right-hand side seems "chopped off" compared to the 

left 

side. A distribution of this type is called negatively skewed because the tail is pulled out to the left. 

Figure - Example Negatively Skewed Histogram (Ciara Kidder of Marian University using work by Barbara Illowsky 

& 

Susan Dean (De Anza College) from OpenStax) 

The data used to create Figure 

the mode? 

 

is: 4, 5, 6, 6, 6, 7, 7, 7, 7, and 8. What is the mean? What is the median? What is 

Answer 

The mean is 6.3. The median is 6.5. The mode is 7. 

(We cannnot include units in these because this is example data; we don't know what was measured.) 

Notice that the mean is less than the median, and they are both less than the mode. Distributions of data are negatively skewed 

when the mean is lower than the median. The extreme low scores bring the mean down, but their extremeity doesn't affect the 

median (or mode). 

The histogram in Figure 

is also not symmetrical, but is positively skewed as the tail points to the right. 

 


Exercise 

Example 

Figure - Example Positively Skewed Histogram (Ciara Kidder of Marian University using work by Barbara Illowsky 

& 

Susan Dean (De Anza College) from OpenStax) 

The data used to create Figure 

the mode? 

 

is: 6, 7, 7, 7, 7, 8, 8, 8, 9, and 10. What is the mean? What is the median? What is 

Answer 

The mean is 7.7. The median is 7.5. The mode is 7. 

(We cannnot include units in these because this is example data; we don't know what was measured.) 

In contrast to the negatively skewed distribution in Figure , the positively skewed distribution's mean is larger 

than the 

median. Looking at Figure , you can see how the few high scores pulled the mean up, but extreme scores do not 

affect 

what number is in the middle (the median). 

Generally, if the distribution of data is skewed to the left, the mean is less than the 

median, which is often less than the mode. If the distribution of data is skewed to the 

right, the mode is often less than the median, which is less than the mean. 

Skewness and symmetry become important when we discuss probability distributions in later chapters. 

Note 

Do you think that is bigger in the symmetrical distribution shown in Figure 

? Let's see! 

 

 

or the skewed distribution in Figure 

First, let's work together to practice calculating a standard deviation using the data from Figure . 

 

 

 

The data from Figure is in Table . Use the table to find the Sum of Squares, then finish the standard deviation 

formula for a sample. 


cc 

Ã cc 

cc 

Ã cc 

cc 

Solution 

Table - Finding of the Sums of 

Squares 

Ã cc 

9 

Sums of Squares ( Ã ) 

cc 

cc 

4 

cc 

Ã Ã 

cc 

cc 

 

5 

cc 

Ã Ã 

cc 

cc 

 

Ã 

Ã 

cc 

6 

cc 

Ã Ã 

cc 

cc 

 

Ã 

Ã 

cc 

6 

cc 

Ã Ã 

cc 

cc 

 

Ã 

Ã 

cc 

6 

cc 

Ã Ã 

cc 

cc 

 

Ã 

Ã 

cc 

7 

cc 

Ã 

cc 

cc 

 

Ã 

Ã 

cc 

Ã 

7 

cc 

Ã 

cc 

cc 

 

Ã 

cc 

Ã 

Ã 

7 

cc 

Ã 

cc 

cc 

 

cc 

7 

cc 

Ã 

cc 

cc 

 

Ã 

Ã 

cc 

7 

cc 

Ã 

cc 

cc 

 

Ã 

Ã 

cc 

7 

cc 

Ã 

cc 

cc 

 

Ã 

Ã 

cc 

Ã 

8 

Ã 

cc Ã 

 

cc 

 

cc 

Ã 

 

8 

cc 

Ã 

cc 

cc 

 

cc 

Ã 

 

8 

cc 

Ã 

cc 

cc 

 

cc 

9 

cc 

Ã 

cc 

cc 

 

Ã 

 

cc 

10 

cc 

Ã 

cc 

cc 

 

Ã 

 

cc 

Ã 

 

V † † 

The last column (the one on the right) is the squared deviations. So, when they are summed to 32, you have the Sum of 

Squares. 

Ã 


Ã 

Ã ÃÃ 

 

/ 

" 

Ã 

As this suggests, you then divide the Sum of Squares by N-1 (N is the number of scores), so: 

 

 

Ã Ã 

Then, 

Ã 


Ã 

Ã ÃÃ 

 

 

cc 

cc 

Ã cc 

 

 

 

/ 

And the final step of square rooting to get it back into whatever units we started with: 

" 

Ã 

 

Ã 


Ã 

Ã ÃÃ 

ÃÃ 

 

/ 

The standard deviation of the data from Figure is 1.46! 

" 

Ã 

 

 

 

 

Now, try it on your own! 


Exercise 

 

 

Compute a standard deviation with the data from Figure that is in Table . 

Table - Your Turn to Find the Sums of 

Squares 

Ã cc 

9 

Sums of Squares ( Ã ) 

cc 

cc 

4 

5 

6 

6 

6 

7 

7 

7 

7 

8 

V † † 

Ã 


Ã 

Ã ÃÃ 

cc 

Ã cc 

 

 

/ 

 

" 

 

Ã 

Answer 

The standard deviation for the data from Figure is 1.16 (rounded from 

1.159502). 

If you didn't get something close to this, check that you used the correct N (number of scores) fo 10 in this sample. 

Another thing to check is that your Sum of Squares should be 12.1; often, simple adding or messing up the mean will 

wreck the rest of your calculations! 

So what did we learn from Example and Exercise ? The skewed 

distribution 

had a smaller standard deviation than 

the symmetrical distribution. Why might that be? Maybe looking at the range will help? The range of the symmetrical 

distribution was 6 ( ), while the range of the skewed distribution was 4. Having a smaller Ã range means that 

there 

was no extreme scores, so that could explain the lower standard deviation. 

Quiz Yourself! 

You can use the following to text yourself to see if you understand the relationships between the measures of central tendency 

(mean, median, and mode) and th measures of variability (range and standard deviation) in relation to the shape of 

distributions of data. 

Exercise 

 

When the data are symmetrical, what is the typical relationship between the mean and median? 

Answer 

When the data are symmetrical, the mean and median are close or the same. 

Exercise 

 

Describe the shape of this distribution. 


Figure - Example Histogram (Ciara Kidder of Marian University using work by Barbara Illowsky & Susan 

Dean 

(De Anza College) from OpenStax) 

Answer 

The distribution is skewed right because it looks pulled out to the right. 

Exercise 

Describe the relationship between the mode and the median of this distribution. 

 

Figure - - Example Histogram (Ciara Kidder of Marian University using work by Barbara Illowsky & Susan 

Dean 


Answer 

The mode and the median are the same. In this case, they are both five. 

Exercise 

Describe the shape of this distribution. 

 

Figure - Example Histogram (Ciara Kidder of Marian University using work by Barbara Illowsky & Susan 

Dean 


Answer 

The distribution is skewed left because it looks pulled out to the left. 

Exercise 

Which is the greatest, the mean, the mode, or the median of the data set? 

 

11; 11; 12; 12; 12; 12; 13; 15; 17; 22; 22; 22 


Answer 

The mode is 12, the median is 12.5, and the mean is 15.1. The mean is the largest. 

Putting It All Together: SD & 3 M's 

Let's look at everything that we know about the data for 

Figure . 

First, we know the measures of central tendency. These tells us where the "center" of the distribution might be. For Figure 

, the sample mean was 6.3, the median was 6.5, and the mode was 7. This suggests that the center of the distribution 

of 

data was probably somewhere between 6.3 and 7. Even without seeing the histogram, I might think that the distribution was 

skewed because the mean was smaller than the median. It wasn't that much smaller (0.2), so I wouldn't bet my life on that 

one. 

Second, we know some measures of variability. The range is 4 (the highest score of 8 minus the lowest score of 4), and the 

standard deviation was 1.16. Compared to a mean of 6.3, a standard deviation of 1.16 might be considered medium or slightly 

large. A medium standard deviation would suggest a bell-shaped curve, while a larger standard deviation would suggest a 

wide and flat distribution or a skewed distribution. 

Finally, I can look at the actual Figure 

median, and mean would be on Figure 

below the mean. 

. That shows a negatively skewed distribution. I can find where the mode, 

, and look to see what scores might fall within one standard deviation above and 


Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by 

OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at 

http://cnx.org/contents/30189442-699...b91b9de@18.114. 


 

 


Exercise 

Exercise 

Exercise 

Exercise 

3.10: Measures of Central Tendency and Variability Exercises 

If the mean time to respond to a stimulus is much higher than the median time to respond, what can you say about the 

shape of the distribution of response times? 

Answer 

If the mean is higher, that means it is farther out into the right-hand tail of the distribution. Therefore, we know this 

distribution is positively skewed. 

Your younger sibling comes home one day after taking a science test. He says that someone at school told him that 

“60% of the students in the class scored above the median test grade.” What is wrong with this statement? 

What if they had said “60% of the students scored above the mean?” 

Answer 

The median is defined as the value with 50% of scores above it and 50% of scores below it; therefore, 60% of score 

cannot fall above the median. If 60% of scores fall above the mean, that would indicate that the mean has been pulled 

down below the value of the median, which means that the distribution is negatively skewed. 

Compute the population mean and population standard deviation for the following scores: 5, 7, 8, 3, 4, 4, 2, 7, 1, 6 

Remember,you can use the Sum of Squares table. 

Answer 

U 

N 

 

If two normal distributions have exactly the same mean, but one has a standard deviation of 20 and the other has a 

standard deviation of 10, how would the shapes of the two distributions compare? 

Answer 

If both distributions are normal, then they are both symmetrical, and having the same mean causes them to overlap 

with one another. The distribution with the standard deviation of 10 will be narrower than the other distribution. 


Exercise 

9: 

: 

Use the following dataset for the computations below: 

Table - Sample Data for Two 

Variables 

9 : 

1. 

2. 

3. 

4. 

9 

: 

Answer 

1. 26 

2. 161 

3. 109 

4. 625 

2 8 

3 8 

7 4 

5 1 

9 4 








There are kinds of distributions of data that are so common that they have a name. 

distributions have special characteristics that are the underpinning of all of statistics. 

Some 


Review to set the stage... 


Before you can understand the special types of distributions, you should understand the basic idea 

of probability. 


Since "bi" means "two" and "nom" means "name" (or "category name"), what do you think a binomial distribution is? 


How numbers work... 


Why is being Normal so important? 


Sorta redundant, but sampling distributions are distributions of the means of samples 


Why do we care about all of these distributions? 


Probability distributions are the heart of how inferential statistics work. 

1 8/27/2021

Exercise 

4.1: Introduction to Distributions 

Refresher 

Let’s remind ourselves three important ideas we learned about last chapter: distribution, central tendency, and variance. These 

terms are similar to their everyday meanings (although I suspect most people don’t say central tendency very often). 

Distribution 

When you order something from Amazon, where does it come from, and how does it get to your place? That stuff comes from 

one of Amazon’s distribution centers. They distribute all sorts of things by spreading them around to your doorstep. “To 

Distribute”" is to spread something. Notice, the data in the histogram is distributed, or spread across the bins. We can also talk 

about a distribution as a noun. The histogram is a distribution of the frequency counts across the bins. Distributions are very, 

very, very, very, very important. They can have many different shapes. 

We will start this chapter on distributions talking about frequency distributions, which show the frequency of scores. When 

you hear "distribution," "distribution of data," or "frequency distribution," you should be imagining a histogram or line graph 

(frequency polygon). 

The X-axis has all possible scores. 

The Y-axis has frequencies of occurrences. 

As we learn about other types of distributions, the Y-axis might be measured differently (like in percentages or probability), 

but the idea is the same: How often did that score occur? 

Measures of Central Tendency 

Central Tendency is all about sameness: What is common about some numbers? For example, if we had a distribution in 

which most of scores were near 0, we could say that there is a tendency for most of the numbers to be centered near 0. Notice 

we are being cautious about our generalization about the numbers. We are not saying they are all 0. We are saying there is a 

tendency for many of them to be near zero. There are lots of ways to talk about the central tendency of some numbers. There 

can even be more than one kind of tendency. For example, if lots of the numbers were around -1000, and a similar large 

amount of numbers were grouped around 1000, we could say there was two tendencies. The three common measures of 

central tendency that we discussed last chapter are: 

Mode 

Median 

Mean 

Which measures of central tendency can be used with quantitative data? 

Answer 

Add texts here. Do not delete this text first. All three measures of central tendency can be used with quantitative data. 

The mode is the only measure of central tendency that can be used with qualitative varaibles, though. 

Measures of Variance 

Variance is all about differentness: What is different about some numbers?. For example, is there anything different about all 

of the numbers in the histogram? YES!!! The numbers are not all the same! When the numbers are not all the same, they must 

vary. So, the variance in the numbers refers to how the numbers are different. There are many ways to summarize the amount 

of variance in the numbers, and we discuss these very soon. 

The most common measure of variance that we will use is standard deviation. Some researchers also like to know the range, 

or the maximum and minimum scores (highest score and lowest score). 

Why? 

Matthew J. C. Crump 7/21/2021 4.1.1 CC-BY-SA https://stats.libretexts.org/@go/page/22038

As we begin talking about different kinds of special distributions, you might be asking yourself why these matter. We will go 

through some of these special distributions to show why statistical analyses can be used to predict information about a 

population from information about a sample. This will, ultimately, let us test research hypotheses. I could explain why, but it 

wouldn't make sense because you don't know about the special distributions yet! So, for now, let's learn the basics about some 

of these special distributions so that you can interpret your results later. 

Let's begin! 




Matthew J. C. Crump 7/21/2021 4.1.2 CC-BY-SA https://stats.libretexts.org/@go/page/22038

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4.2: Introduction to Probability 

When we speak of the probability of something happening, we are talking how likely it is that “thing” will happen based on 

the conditions present. For instance, what is the probability that it will rain? That is, how likely do we think it is that it will rain 

today under the circumstances or conditions today? To define or understand the conditions that might affect how likely it is to 

rain, we might look out the window and say, “it’s sunny outside, so it’s not very likely that it will rain today.” Stated using 

probability language: given that it is sunny outside, the probability of rain is low. “Given” is the word we use to state what the 

conditions are. As the conditions change, so does the probability. Thus, if it were cloudy and windy outside, we might say, 

“given the current weather conditions, there is a high probability that it is going to rain.” 

In these examples, we spoke about whether or not it is going to rain. Raining is an example of an event, which is the catch-all 

term we use to talk about any specific thing happening; it is a generic term that we specified to mean “rain” in exactly the 

same way that “conditions” is a generic term that we specified to mean “sunny” or “cloudy and windy.” 

It should also be noted that the terms “low” and “high” are relative and vague, and they will likely be interpreted different by 

different people (in other words: given how vague the terminology was, the probability of different interpretations is high). 

Most of the time we try to use more precise language or, even better, numbers to represent the probability of our event. 

Regardless, the basic structure and logic of our statements are consistent with how we speak about probability using numbers 

and formulas. 

Let’s look at a slightly deeper example. Say we have a regular, six-sided die (note that “die” is singular and “dice” is plural, a 

distinction that Dr. Foster has yet to get correct on his first try) and want to know how likely it is that we will roll a 1. That is, 

what is the probability of rolling a 1, given that the die is not weighted (which would introduce what we call a bias, though that 

is beyond the scope of this chapter). We could roll the die and see if it is a 1 or not, but that won’t tell us about the probability, 

it will only tell us a single result. We could also roll the die hundreds or thousands of times, recording each outcome and 

seeing what the final list looks like, but this is time consuming, and rolling a die that many times may lead down a dark path to 

gambling or, worse, playing Dungeons & Dragons. What we need is a simple equation that represents what we are looking for 

and what is possible. 

To calculate the probability of an event, which here is defined as rolling a 1 on an unbiased die, we need to know two things: 

how many outcomes satisfy the criteria of our event (stated different, how many outcomes would count as what we are looking 

for) and the total number of outcomes possible. In our example, only a single outcome, rolling a 1, will satisfy our criteria, and 

there are a total of six possible outcomes (rolling a 1, rolling a 2, rolling a 3, rolling a 4, rolling a 5, and rolling a 6). Thus, the 

probability of rolling a 1 on an unbiased die is 1 in 6 or 1/6. Put into an equation using generic terms, we get: 

 

We can also using P() as shorthand for probability and A as shorthand for an event: 

 

Using this equation, let’s now calculate the probability of rolling an even number on this die: 

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&WFO/VNCFS 

So we have a 50% chance of rolling an even number of this die. The principles laid out here operate under a certain set of 

conditions and can be elaborated into ideas that are complex yet powerful and elegant. However, such extensions are not 

necessary for a basic understanding of statistics, so we will end our discussion on the math of probability here. Now, let’s turn 

back to more familiar topics. 

PS` 

 

 





4.3: The Binomial Distribution 

The vast majority of the content in this book relies on one of five distributions: the binomial distribution, the normal 

2 

distribution, the t distribution, the F distribution, and the χ (“chi-square”) distribution. Let's start with the binomial 

distribution, since it’s the simplest of the five. The normal distribution will be discussed later this chapter, and the others will 

be mentioned in other chapters. Our discussion of the binomial distribution will start with figuring out some probabilities of 

events happening. 

Let’s imagine a simple “experiment”: in my hot little hand I’m holding 20 identical six-sided dice. On one face of each die 

there’s a picture of a skull; the other five faces are all blank. If I proceed to roll all 20 dice, what’s the probability that I’ll get 

exactly 4 skulls? 

Let's unpack that last sentence. First, in case you missed it, rolling of the 20 dice is your sample. N = 20 (20 scores in this 

distribution). Second, it's asking about a probability, so we'll be looking back at the last section and realizing that we should 

do some division. Finally, this is a question, some might even call it a Research Question. 

Okay, back to the "experiment." Assuming that the dice are fair, we know that the chance of any one die coming up skulls is 1 

in 6; to say this another way, the skull probability for a single die is approximately .1667, or 16.67%. 

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The probability of rolling a skull with on die is 16.67%, but I'm doing this 20 times so that's 

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skulls 

So, if I rolled 20 dice, I could expect about 3 of them to come up skulls. But that's not quite the Research Question is it? Let’s 

have a look at how this is related to a binomial distribution. 

Figure plots the binomial probabilities for all possible values for our dice rolling experiment, from X=0 (no 

skulls) all 

the way up to X=20 (all skulls). On the horizontal axis we have all the possible events (the number of skulls coming up when 

all 20 dice are rolled), and on the vertical axis we can read off the probability of each of those events. If you multiple these by 

100, you'd get the probability in a percentage form. Each bar depicts the probability of one specific outcome (i.e., one possible 

value of X). Because this is a probability distribution, each of the probabilities must be a number between 0 and 1 (or 0% to 

100%), and the heights of the all of the bars together must sum to 1 as well. Looking at Figure , the probability of 

rolling 

4 skulls out of 20 times is about 0.20 (the actual answer is 0.2022036), or 20.22%. In other words, you’d expect to roll exactly 

4 skulls about 20% of the times you repeated this experiment. This means that, if you rolled these 20 dice for 100 different 

repetitions, you'd get exactly 4 skulls in about 20 of your attempts. 

Ç 

 


Figure - Probability of Rolling 4 Skulls out of 20 Dice with One Skull (CC-BY-SA Danielle Navarro from 

Learning 

Statistics with R) 

Sample Size Matters 

We'll be talking a lot about how sample size (N) affects distributions in this chapter, starting now! 

To give you a feel for how the binomial distribution changes when we alter the probability and N, let’s suppose that instead of 

rolling dice, I’m actually flipping coins. This time around, my experiment involves flipping a fair coin repeatedly, and the 

outcome that I’m interested in is the number of heads that I observe. In this scenario, the success probability is now 1/2 (one 

out of two options). Suppose I were to flip the coin N=20 times. In this example, I’ve changed the success probability (1 out of 

2, instead of 1 out of 6 in the dice example), but kept the size of the experiment the same (N=20). What does this do to our 

binomial distribution? Well, as Figure shows, the main effect of this is to shift the whole distribution higher 

(since there's 

more chance of getting a heads (one out of two options) than a 4 (one out of six options). Okay, what if we flipped a coin 

N=100 times? Well, in that case, we get Figure . The distribution stays roughly in the middle, but there’s a bit 

more 

variability in the possible outcomes (meaning that there are more extreme scores); with more tosses, you are more likely to get 

no heads but also more likely to get all heads than if you only flipped the coin 20 times. 


Figure 

with R) 

 

- Probability of Heads from Flipping a Fair Coin 20 Times (CC-BY-SA Danielle Navarro from Learning Statistics 

Figure - Probability of Heads from Flipping a Fair Coin 100 Times (CC-BY-SA Danielle Navarro from 

Learning 


And that's it on binomial distributions! We are building understanding about distributions and sample size, so nothing too 

earth-shattering here. One thing to note is that both the coin flip and the dice toss were measured as discrete variables. Our 

next distributions will cover continuous variables. The difference is that discrete variables can be one or the other (a heads or 

a tails, a four or not-four), while continuous variables can have gradations shown as decimal points. Although the math works 

out to 3.33 skulls when you throw 30 dice, you can't actually get 3.33 skulls, you can only get 3 or 4 skulls. 


We're building our knowledge, so keep going! 





4.4: The Law of Large Numbers 

If we want our sample statistics to be much closer to the population parameters, what can we do about it? 

The answer is to collect more data. Larger samples are a much better approximation to the true population distribution than 

smaller samples. I feel a bit silly saying this, but the thing I want you to take away from this is that large samples generally 

give you better information. It does feel a bit obvious that more data will give you better answers. 

The question is, why is this so? Not surprisingly, this intuition that we all share turns out to be correct, and statisticians refer to 

it as the law of large numbers. The law of large numbers is a mathematical law that applies to many different sample statistics, 

but the simplest way to think about it is as a law about averages. The sample mean is the most obvious example of a statistic 

that relies on averaging (because that’s what the mean is… an average), so let’s look at that. When applied to the sample mean, 

what the law of large numbers states is that as the sample gets larger, the sample mean tends to get closer to the true population 

mean. Or, to say it a little bit more precisely, as the sample size “approaches” infinity (written as N→∞) the sample mean 

approaches the population mean ( →μ). 

I don’t intend to subject you to a proof that the law of large numbers is true, but it’s one of the most important tools c 

for 

statistical theory. The law of large numbers is the thing we can use to justify our belief that collecting more and more data will 

eventually lead us to the truth. For any particular data set, the sample statistics that we calculate from it will be wrong, but the 

law of large numbers tells us that if we keep collecting more data those sample statistics will tend to get closer and closer to 

the true population parameters. 

It's okay if this isn't obvious to you. I'm just mentioning it because it will start to feel obvious, and explains lots about why 

statistical formulas are the way they are. I mean, we've already run across that in how the formula for the standard deviation 

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of a sample ( ) is different from the formula for the standard deviation of the population ( ). 

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☰ 

4.5: Normal Distributions and Probability Distributions 

We will see shortly that the normal distribution is the key to how probability works for our purposes. To understand exactly 

how, let’s first look at a simple, intuitive example using pie charts. 

Probability in Pie 

Recall that a pie chart represents how frequently a category was observed and that all slices of the pie chart add up to 100%, or 

1. This means that if we randomly select an observation from the data used to create the pie chart, the probability of it taking 

on a specific value is exactly equal to the size of that category’s slice in the pie chart. 

Figure 

: Favorite Sports (CC-BY-NC-SA Foster et al. from An Introduction to Psychological Statistics) 

Take, for example, the pie chart in Figure representing the favorite sports of 100 people. If you put this pie chart on a 

dart board and aimed blindly (assuming you are guaranteed to hit the board), the likelihood of hitting the slice for any given 

sport would be equal to the size of that slice. So, the probability of hitting the baseball slice is the highest at 36%. The 

probability is equal to the proportion of the chart taken up by that section. 

We can also add slices together. For instance, maybe we want to know the probability to finding someone whose favorite sport 

is usually played on grass. The outcomes that satisfy this criteria are baseball, football, and soccer. To get the probability, we 

simply add their slices together to see what proportion of the area of the pie chart is in that region: 

. We can also add sections together even if they do not touch. If we want to know the likelihood 

that someone’s favorite sport is not called football somewhere in the world (i.e. baseball and hockey), we can add those slices 

even though they aren’t adjacent or continuous in the chart itself: . We are able to do all of this because 1) 

the size of the slice corresponds to the area of the chart taken up by that slice, 2) the percentage for a specific category can be 

represented as a decimal (this step was skipped for ease of explanation above), and 3) the total area of the chart is equal to 

or 1.0, which makes the size of the slices interpretable. 

Normal Distributions 

The normal distribution is the most important and most widely used distribution in statistics. It is sometimes called the “bell 

curve,” although the tonal qualities of such a bell would be less than pleasing. It is also called the “Gaussian curve” of 

Gaussian distribution after the mathematician Karl Friedrich Gauss. 

Strictly speaking, it is not correct to talk about “the normal distribution” since there are many normal distributions. Normal 

distributions can differ in their means and in their standard deviations. Figure shows three normal distributions. The 

green (left-most) distribution has a mean of -3 and a standard deviation of 0.5, the distribution in red (the middle distribution) 

has a mean of 0 and a standard deviation of 1, and the distribution in black (right-most) has a mean of 2 and a standard 

deviation of 3. These as well as all other normal distributions are symmetric with relatively more values at the center of the 

distribution and relatively few in the tails. What is consistent about all normal distribution is the shape and the proportion of 

scores within a given distance along the x-axis. We will focus on the Standard Normal Distribution (also known as the Unit 

Normal Distribution), which has a mean of 0 and a standard deviation of 1 (i.e. the red distribution in Figure ). 


☰ 

Figure : Symmetrical Distributions with Different Means and Standard Deviations (CC-BY-NC-SA Foster et al. from An 


Standard Normal Distribution 

Important features of normal distributions are listed below. 

1. Normal distributions are symmetric around their mean. 

2. The mean, median, and mode of a normal distribution are equal. In Standard Normal Curves, the mean, median, and mode 

are all 0. 

3. The area under the normal curve is equal to 1.0 (or 100% of all scores will fall somewhere in the distribution). 

4. Normal distributions are denser in the center and less dense in the tails (bell-shaped). 

5. There are known proportions of scores between the mean and each standard deviation. 

1. One standard deviation- 68% of the area of a normal distribution is within one standard deviation of the mean (one 

standard deviation below the mean through one standard deviation above the mean). 

2. Two standard deviations- Approximately 95% of the area of a normal distribution is within two standard deviations of 

the mean (two standard deviations below the mean through two standard deviations above the mean). 

These properties enable us to use the normal distribution to understand how scores relate to one another within and across a 

distribution. 

Probability in Normal Distributions 

Play with this applet to learn about probability distributions (website 

address: http://www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=1) 

Probability distributions are what make statistical analyses work. Each distribution of events from any sample can be 

“translated” into a probability distribution, then used to predict the probability of that event happening! 

Just like a pie chart is broken up into slices by drawing lines through it, we can also draw a line through the normal 

distribution to split it into sections. We know that one standard deviation below the mean to one standard deviation above the 

mean contains 68% of the area under the curve because we know the properties of standard normal curves. 

Figure : There is a 68% chance of selection a Score from the shaded region (CC-BY-NC-SA Foster et al. from An 


This time, let’s find the area corresponding to the extreme tails. For standard normal curves, 95% of scores should be in the 

middle section of Figure , and a total of 5% in the shaded areas. Since there are two shaded areas, that's about 2.5% on 

each side. 


☰ 

Figure 


: Area Beyond 2 Standard Deviations (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

Those important characteristics of a Standard Normal Distribution are deceptively simple, so let's put some of these ideas 

together to see what we get. 

Law of Large Numbers: I know through the Law of Large Numbers that if I have enough data, then my sample will begin to 

become more and more "normal' (have the characteristics of a normally distributed population, including the important 

characteristics above). 

Probability Distributions: I know that I can make predictions from probability distributions. I can use probability 

distributions to understand the likelihood of specific events happening. 

Standard Normal Curve: I know that there should be specific and predictable proportions (percentages) of scores between the 

mean and each standard deviation away from the mean. 

Together, these features will allow you to: 

Predict the probability of events happening. 

Estimate how many people might be affected based on sample size. 

Test Research Hypotheses about different groups. 

This is the magic of the Standard Normal Curve! 

Non-Parametric Distributions 

One last thing... 

There's a lot of heavy lifting for the Standard Normal Distribution. But what if your sample, or even worse, the population, is 

not normally distributed? That's when non-parametric statistics come in. 

A parameter is a statistic that describes the population. Non-parametric statistics don’t require the population data to be 

normally distributed. Most of the analyses that we'll conduct compare means (a measure of central tendency) of different 

groups. But if the data are not normally distributed, then we can’t compare means because there is no center! Non-normal 

distributions may occur when there are: 

Few people (small N) 

Extreme scores (outliers) 

There’s an arbitrary cut-off point on the scale. (Like if a survey asked for ages, but then just said, “17 and below”.) 

The reason that I bring this up is because sometimes you just can't run the statistics that we're going to learn about because the 

population's data is not normally distributed. Since the Standard Normal Curve is the basis for most types of statistical 

inferences, you can't use these (parametric) analyses with data that they don't fit. We'll talk about alternative analyses for some 

of the common (parametric) statistical analyses, but you heard it here first! Non-parametric distributions are when the 

population is not normally distributed. 

Okay, onward to learning more about how the Standard Normal Curve is important... 



☰ 





4.6: Sampling Distributions and the Central Limit Theorem 

The law of large numbers is a very powerful tool, but it’s not going to be good enough to answer all our questions. Among 

other things, all it gives us is a “long run guarantee”. In the long run, if we were somehow able to collect an infinite amount of 

data, then the law of large numbers guarantees that our sample statistics will be correct. But as John Maynard Keynes 

famously argued in economics, a long run guarantee is of little use in real life: 

[The] long run is a misleading guide to current affairs. In the long run we are all 

dead. Economists set themselves too easy, too useless a task, if in tempestuous seasons 

they can only tell us, that when the storm is long past, the ocean is flat again. Keynes 

(1923) 

As in economics, so too in psychology and statistics. It is not enough to know that we will eventually arrive at the right answer 

when calculating the sample mean. Knowing that an infinitely large data set will tell me the exact value of the population 

mean is cold comfort when my actual data set has a sample size of N=100. In real life, then, we must know something about 

the behavior of the sample mean when it is calculated from a more modest data set! 

Sampling Distribution of the Mean 

With this in mind, let’s abandon the idea that our studies will have sample sizes of 10,000, and consider a very modest 

experiment indeed. This time around we’ll sample N=5 people and measure their IQ scores. In a simulated study, the mean IQ 

in this sample turns out to be exactly 95. Not surprisingly, this is much less accurate than the previous experiment with 10,000 

simulated IQ scores. Now imagine that Dr. Navarro decided to replicate the experiment. That is, she repeated the procedure as 

closely as possible: she could randomly sample 5 new people and measure their IQ. On a second simulation, the mean IQ in 

my sample is 101. When Dr. Navarro repeated the simulation 10 times, she obtained the results shown in Table . As you 

can see, the sample mean varies from one replication to the next. 

Table - 10 Replications of Simulated IQ Scores, Each with a Sample Size of N=5 (CC-BY-SA Danielle Navarro) 

Replication Time Person 1 Person 2 Person 3 Person 4 Person 5 Mean 

Replication 1 90 82 94 99 110 95.0 

Replication 2 78 88 111 111 117 101.0 

Replication 3 111 122 91 98 86 101.6 

Replication 4 98 96 119 99 107 103.8 

Replication 5 105 113 103 103 98 104.4 

Replication 6 81 89 93 85 114 92.4 

Replication 7 100 93 108 98 133 106.4 

Replication 8 107 100 105 117 85 102.8 

Replication 9 86 119 108 73 116 100.4 

Replication 10 95 126 112 120 76 105.8 

Now suppose that Dr. Navarro decided to keep going in this fashion, replicating this “five IQ scores” simulation over and over 

again. Every time she replicate the simulation, she write down the sample mean. Over time, she would be amassing a new data 

set, in which every experiment generates a single data point, a single mean. What if Dr. Navarro continued like this for 10,000 

replications, and then drew a histogram? Using the magical powers of statistical software, that’s exactly what Dr. Navarro did, 

and you can see the results in Figure . As this picture illustrates, the average of 5 IQ scores is usually between 90 


110. If you sample 5 people at random and calculate their average IQ, you’ll almost certainly get a number between 80 and 

120, even though there are quite a lot of individuals who have IQs above 120 or below 80. For comparison, the black line plots 

the population distribution of IQ scores. But more importantly, what Figure highlights is that if we replicate an 

 


experiment over and over again, what we end up with is a distribution of sample means! This distribution has a special name 

in statistics: it’s called the sampling distribution of the mean. 

Sampling distributions are another important theoretical idea in statistics, and they’re crucial for understanding the behavior of 

small samples. For instance, when Dr. Navarro ran the very first “five IQ scores” experiment, the sample mean turned out to 

be 95. If she repeats the experiment, the sampling distribution tells me that we can expect to see a sample mean anywhere 

between 80 and 120. 

Figure - Distribution of Means from 10,000 Samples of 5 IQ Scores (CC-BY-SA Danielle Navarro from 

Learning 


Not Just Distribution of Sample Means 

One thing to keep in mind when thinking about sampling distributions is that any sample statistic you might care to calculate 

has a sampling distribution. For example, suppose that each time Dr. Navarro replicated the “five IQ scores” experiment but 

wrote down the largest IQ score in the experiment. This would give her a data set that started out like in Figure : 

 

Figure - Distribution of Highest IQ from 10,000 Samples of 5 IQ Scores (CC-BY-SA Danielle Navarro 

from Learning 


Doing this over and over again would give Dr. Navarro a very different sampling distribution, namely the sampling 

distribution of the maximum. The sampling distribution of the maximum of 5 IQ scores is shown in Figure . Not 

surprisingly, if you pick 5 people at random and then find the person with the highest IQ score, they’re going to have an above 

average IQ. As shown in Figure , most of the time you’ll end up with someone whose IQ is measured in the 100 

to 140 

range. 

Central Limit Theorem 


An illustration of the how sampling distribution of the mean depends on sample size. In each panel, Dr. Navarro generated 

10,000 samples of IQ data, and calculated the mean IQ observed within each of these data sets. The histograms in these plots 

show the distribution of these means (i.e., the sampling distribution of the mean). Each individual IQ score was drawn from a 

normal distribution with mean 100 and standard deviation 15, which is shown as the solid black line. 

Figure - Distribution of Mean IQ from 10,000 Samples of 1 IQ Score (CC-BY-SA Danielle Navarro from 

Learning 


Figure - Distribution of Mean IQ from 10,000 Samples of 2 IQ Scores (CC-BY-SA Danielle Navarro from 

Learning 



Figure - Distribution of Mean IQ from 10,000 Samples of 10 IQ Scores (CC-BY-SA Danielle Navarro from 

Learning 


In Figure , each data set contained only a single observation, so the mean of each sample is just one person’s 

IQ score. As 

a consequence, the sampling distribution of the mean is of course identical to the population distribution of IQ scores because 

there are 10,000 individual scores. When we raise the sample size to 2, as in Figure , the mean of any one 

sample tends 

to be closer to the population mean than any one person’s IQ score, and so the histogram (i.e., the sampling distribution) is a 

bit narrower than the population distribution. By the time we raise the sample size to 10 (Figure ), we can see that 


distribution of sample means tend to be fairly tightly clustered around the true population mean. 

Sample Size Matters 

At this point I hope you have a pretty good sense of what sampling distributions are, and in particular what the sampling 

distribution of the mean is. This section talks about how the sampling distribution of the mean changes as a function of sample 

size. Intuitively, you already know part of the answer: if you only have a few observations, the sample mean is likely to be 

quite inaccurate: if you replicate a small experiment and recalculate the mean you’ll get a very different answer. In other 

words, the sampling distribution is quite wide. If you replicate a large experiment and recalculate the sample mean you’ll 

probably get the same answer you got last time, so the sampling distribution will be very narrow. You can see this visually in 

the Figures in this section: the bigger the sample size, the narrower the sampling distribution gets. We can quantify this effect 

by calculating the standard deviation of the sampling distribution, which is referred to as the standard error. The standard 

error of a statistic is often denoted SE, and since we’re usually interested in the standard error of the sample mean, we often 

use the acronym SEM. As you can see just by looking at the picture, as the sample size N increases, the SEM decreases. 

Okay, so that’s one part of the story. However, there’s something I’ve been glossing over so far. All my examples up to this 

point have been based on the “IQ scores” simlulations, and because IQ scores are usually from a normal distribution, I’ve 

assumed that the population distribution is shaped like that, a symmetrical, medium bell-shaped curve. What if the population 

distribution isn’t normally distributed? What happens to the sampling distribution of the mean? The remarkable thing is this: 

no matter what shape your population distribution is, as N increases the sampling distribution of the mean starts to look more 

like a normal distribution. 

To give you a sense of this, Dr. Navarro ran some simulations . To do this, she started with the “ramped” distribution shown in 

the histogram in Figure . As you can see by comparing the triangular shaped histogram to the 

bell curve plotted by the 

black line, the population distribution doesn’t look very much like a normal distribution at all. Next, Dr. Navarro simulated 

the results of a large number of experiments. In each experiment she took N=2 samples from this "ramp" distribution, and then 

calculated the sample mean. Figure plots the histogram of these sample means (i.e., the 

sampling distribution of the 

mean for N=2). This time, the histogram produces a ∩-shaped distribution (reverse-U shaped distribution): it’s still not normal, 

but it’s a lot closer to the black line than the population distribution in Figure . When Dr. Navarro 

increases the sample 

size to N=4, the sampling distribution of the mean is very close to normal (Figure , and by the time we reach 

a sample 

size of N=8 (Figure ) it’s almost perfectly symmetrical and bell-shaped!. As you can see, even though the 

original 

population distribution is non-normal, the sampling distribution of the mean becomes pretty close to normal by the time you 


have a sample of even 4 observations. In other words, as long as your sample size isn’t tiny, the sampling distribution of the 

mean will be approximately normal no matter what your population distribution looks like! 

Figure 

with R) 

 

- Ramp-Shaped Distribution 10,000 Samples of 1 Person (CC-BY-SA Danielle Navarro from Learning Statistics 

Figure - Distribution 10,000 Samples of 2 People (CC-BY-SA Danielle Navarro from Learning Statistics with R) 

 

Figure - Distribution 10,000 Samples of 4 People (CC-BY-SA Danielle Navarro from Learning Statistics with R) 

 


7 Ã Ê 

Figure - Distribution 10,000 Samples of 8 People (CC-BY-SA Danielle Navarro from Learning Statistics with 

R) 

On the basis of these figures, it seems like we have evidence for all of the following claims about the sampling distribution of 

the mean: 

The mean of the sampling distribution will be VERY similar to the mean of the population. 

The standard deviation of the sampling distribution (i.e., the standard error) gets smaller (taller and narrower distribution) 

as the sample size increases. 

The shape of the sampling distribution becomes more like a normal distribution as the sample size increases. 

As it happens, not only are all of these statements true, there is a very famous theorem in statistics that proves all three of 

them, known as the central limit theorem. Among other things, the central limit theorem tells us that if the population 

distribution has mean μ and standard deviation σ, then the sampling distribution of the mean also has mean μ, and the standard 

error of the mean is 

4&. 

Because we divide the population standard deviation σ by the square root of the sample size N, the SEM gets smaller as the 

sample size increases. It also tells us that the shape of the sampling distribution becomes normal. 

This result is useful for all sorts of things. It tells us why large experiments are more reliable than small ones, and because it 

gives us an explicit formula for the standard error it tells us how much more reliable a large experiment is. Pay attention to this 

formula for the standard error, it will come back! It tells us why the normal distribution is, well, normal. In real experiments, 

many of the things that we want to measure are actually averages of lots of different quantities (e.g., arguably, “general” 

intelligence as measured by IQ is an average of a large number of “specific” skills and abilities), and when that happens, the 

averaged quantity should follow a normal distribution. Because of this mathematical law, the normal distribution pops up over 

and over again in real data. 





4.7: Putting it All Together 

So, we've spend a lot of time going over different distributions. Why? Because they can answer research questions! 

Standard Normal Curve 

This is where it all comes together! What we can start doing with these distributions is comparing them. This might not sound 

like much, but it's the foundation of statistics, and allows us to answer research questions and test hypotheses. 

Probability Distributions: I know that I can make predictions from probability distributions. I can use probability 

distributions to understand the likelihood of specific events happening. 

Law of Large Numbers: I know through the Law of Large Numbers that enough of samples, their scores will be normally 

distributed (with all of the important characteristics that includes). 

Central Limit Theorem: I know through the Central Limit Theorem that if I get the means of enough samples, that the 

sampling distribution of means will be normally distributed. 

Standard Normal Distributions: I know that the mean is converted to always be zero, and the standard deviation is 

standardized to always be 1 in Standard Normal Distributions. 

In Figure , the x-axis corresponds to the value of some variable, and the y-axis tells us something about how likely we 

are 

to observe that value. Notice that the y-axis is labeled “Probability Density” and not “Probability”. There is a subtle and 

somewhat frustrating characteristic of continuous distributions that makes the y-axis behave a bit oddly: the height of the curve 

here isn’t actually the probability of observing a particular x value. On the other hand, it is true that the heights of the curve 

tells you which x values are more likely (the higher ones!). This will be discussed in the Continuous Variable section. 

Figure - Standard Normal Curve (CC-BY-SA Danielle Navarro from Learning Statistics with R) 

 

You can see where the name “bell curve” comes from in Figure \(\PageIndex{1}\: it looks a bit like a bell. Notice that, unlike 

the plots that I drew to illustrate the binomial distribution, the picture of the normal distribution in Figure \(\PageIndex{1}\ 

shows a smooth curve instead of “histogram-like” bars. This isn’t an arbitrary choice: the normal distribution is continuous, 

whereas the binomial is discrete. For instance, in the die rolling example, it was possible to get 3 skulls or 4 skulls, but 

impossible to get 3.9 skulls. The figures that I drew in the previous section reflected this fact. Continuous quantities don’t have 

this constraint. For instance, suppose we’re talking about the weather. The temperature on a pleasant Spring day could be 23 

degrees, 24 degrees, 23.9 degrees, or anything in between since temperature is a continuous variable, and so a normal 

distribution might be quite appropriate for describing Spring temperatures. 

Normal Distributions: I know that there are predictable portions of scores between the mean and standard deviation. 

The last important piece is learning that a special feature of normal distributions is that we know the proportions (percentages) 

of cases that should (probability) fall within each standard deviation around the mean. Irrespective of what the actual mean 

and standard deviation are, 68.3% of the area falls within 1 standard deviation of the mean. Similarly, 95.4% of the distribution 

falls within 2 standard deviations of the mean, and 99.7% of the distribution is within 3 standard deviations. This idea is 

illustrated in the follow Figures. 


Figure - About 68% of Scores are One Standard Deviation Below through One Standard Deviation Above the Mean 

(CC- 

BY-SA Danielle Navarro from Learning Statistics with R) 

Figure - About 95% of Scores are Two Standard Deviations Below through Two Standard Deviations Above 


Mean (CC-BY-SA Danielle Navarro from Learning Statistics with R) 

Figure - About 16% of Scores Should Be Less Than Two Standard Deviations Below the Mean (CC-BY-SA 

Danielle 

Navarro from Learning Statistics with R) 


Figure - About 34% of Scores Should Be One Standard Deviations Below the Mean (CC-BY-SA Danielle Navarro 

from 

Learning Statistics with R) 

There is a 68.3% chance that a score from any sample will fall within the shaded area of Figure , and a 95.4% chance that 

an observation will fall within two standard deviations of the mean (the shaded area in Figure ). Similarly, there 

is a 

15.9% chance that an observation will be below one standard deviation below the mean. There is a 34.1% chance that the 

observation is greater than one standard deviation below the mean but still below the mean (Figure ). Notice that if 

you 

add these two numbers together you get 15.9+34.1=50. For normally distributed data, there is a 50% chance that an 

observation falls below the mean. And of course that also implies that there is a 50% chance that it falls above the mean. 

Continuous Variables 

There’s something Dr. Navarro was trying to hide throughout this discussion of the normal distribution, but she just couldn't do 

it. Many introductory textbooks omit this completely; they might be right to do so: this “thing” that I’m hiding is weird and 

counterintuitive even by the admittedly distorted standards that apply in statistics. Fortunately, it’s not something that you need 

to understand at a deep level in order to do basic statistics: rather, it’s something that starts to become important later on when 

you move beyond the basics. So, if it doesn’t make complete sense, don’t worry: I'm mostly going over it to help think about 

the differences between qualitative and quantitative variables, and how quantitative variables can be discrete or continuous 

(even though we act like they are continuous). 

Throughout the discussion of the normal distribution, there’s been one or two things that don’t quite make sense. Perhaps you 

noticed that the y-axis in these figures is labeled “Probability Density” rather than density. Let's spend a little time thinking 

about what it really means to say that X is a continuous variable. Let’s say we’re talking about the temperature outside. The 

thermometer tells me it’s 23 degrees, but I know that’s not really true. It’s not exactly 23 degrees. Maybe it’s 23.1 degrees, I 

think to myself. But I know that that’s not really true either, because it might actually be 23.09 degrees. But, I know that… 

well, you get the idea. The tricky thing with genuinely continuous quantities is that you never really know exactly what they 

are. 

Now think about what this implies when we talk about probabilities. Suppose that tomorrow’s maximum temperature is 

sampled from a normal distribution with mean 23 and standard deviation 1. What’s the probability that the temperature will be 

exactly 23 degrees? The answer is “zero”, or possibly, “a number so close to zero that it might as well be zero”. Why is this? 

It’s like trying to throw a dart at an infinitely small dart board: no matter how good your aim, you’ll never hit it. In real life 

you’ll never get a value of exactly 23. It’ll always be something like 23.1 or 22.99998 or something. In other words, it’s 

completely meaningless to talk about the probability that the temperature is exactly 23 degrees. However, in everyday 

language, if I told you that it was 23 degrees outside and it turned out to be 22.9998 degrees, you probably wouldn’t call me a 

liar. Because in everyday language, “23 degrees” usually means something like “somewhere between 22.5 and 23.5 degrees”. 

And while it doesn’t feel very meaningful to ask about the probability that the temperature is exactly 23 degrees, it does seem 

sensible to ask about the probability that the temperature lies between 22.5 and 23.5, or between 20 and 30, or any other range 

of temperatures. 

The point of this discussion is to make clear that, when we’re talking about continuous distributions, it’s not meaningful to talk 

about the probability of a specific value. However, what we can talk about is the probability that the value lies within a 


particular range of values. To find out the probability associated with a particular range, what you need to do is calculate the 

“area under the curve”. We’ve seen this concept already: in Figure , the shaded areas shown depict 

genuine probabilities 

(e.g., in Figure it shows the probability of observing a value that falls within 1 standard deviation of the 

mean). 

Okay, so that explains part of the story. I’ve explained a little bit about how continuous probability distributions should be 

interpreted (i.e., area under the curve is the key thing). In terms of the plots we’ve been drawing, probability density 

corresponds to the height of the curve. The densities themselves aren’t meaningful in and of themselves: but they’re “rigged” 

to ensure that the area under the curve is always interpretable as genuine probabilities. To be honest, that’s about as much as 

you really need to know for now. 

We are about to finish up learning abstract theory about distributions, and moving on to actually using them! 





4.8: Summary- The Bigger Picture 

The concepts and ideas presented in this chapter are likely not intuitive at first. Probability is a tough topic for everyone, but 

the tools it gives us are incredibly powerful and enable us to do amazing things with data analysis. They are the heart of how 

inferential statistics work. 

To summarize, the probability that an event happens is the number of outcomes that qualify as that event (i.e. the number of 

ways the event could happen) compared to the total number of outcomes (i.e. how many things are possible). This extends to 

graphs like a pie chart, where the biggest slices take up more of the area and are therefore more likely to be chosen at random. 

This idea then brings us back around to our normal distribution, which can also be broken up into regions or areas, each of 

which are bounded by one or two standard deviations around the mean. The probability of randomly getting a score in the 

specified region can then be found. Thus, the larger the region, the more likely an event is, and vice versa. Because the tails of 

the distribution are, by definition, smaller and we go farther out into the tail, the likelihood or probability of finding a result 

out in the extremes becomes small. 







5: USING Z 

Our first statistical analysis to answer a research question! 


What's a z-score and why do I care? 


Can you calculate a z-score? Yes! 


Practice! 


Our first critical value table! 


Practice! 


Here's a table of z-scores and their probabilities. 


Can you do more with z-scores than find probabilities? 


If you're still not sure what a z-score is, or why it might matter, check out this summary. 


You've done all of this math, now what? 

1 8/27/2021

5.1: Introduction to z-scores 

-scores and the standard normal distribution go hand-in-hand. A -score will tell you exactly where in the standard 6 

normal 

distribution a value is located, and any normal distribution can be converted into a standard normal distribution by converting 

all of the scores in the distribution into -scores, a process known as standardization. 

We saw in the previous chapter that standard deviations can be used to divide the normal distribution: 68% of the distribution 

falls within 1 standard deviation of the mean, 95% within (roughly) 2 standard deviations, and 99.7% within 3 standard 

deviations. Because -scores are in units of standard deviations, this means that 68% of scores fall between 6 = 6 -1.0 6 and = 

1.0 and so on. We call this 68% (or any percentage we have based on our -scores) the proportion of the area under the 6 

curve. 

Any area under the curve is bounded by (defined by, delineated by, etc.) by a single -score or pair of -scores. 

6 

An important property to point out here is that, by virtue of the fact that the total area under the curve of a distribution is 

always equal to 1.0 (see section on Normal Distributions at the beginning of this chapter), these areas under the curve can be 

added together or subtracted from 1 to find the proportion in other areas. For example, we know that 6 

the area between = -1.0 

and = 1.0 (i.e. within one standard deviation of the mean) contains 68% of the area under the 6 

curve, which can be 

represented in decimal form at 0.6800 (to change a percentage to a decimal, simply move the decimal point 2 places to the 

left). Because the total area under the curve is equal to 1.0, that means that the proportion of the area outside = 6 -1.0 6 and = 

1.0 is equal to 1.0 – 0.6800 = 0.3200 or 32% (see Figure below). This area is called the area in the tails of 


distribution. Because this area is split between two tails and because the normal distribution is symmetrical, each tail has 

exactly one-half, or 16%, of the area under the curve. 

6 6 

Figure 

: Shaded areas represent the area under the curve in the tails 

We will have much more to say about this concept in the coming chapters. As it turns out, this is a quite powerful idea that 

enables us to make statements about how likely an outcome is and what that means for research questions we would like to 

answer and hypotheses we would like to test. But first, we need to make a brief foray into some ideas about probability. 


 




Ã cc 

4 

5.2: Calculating z-scores 

A -score is a standardized version of a raw score ( ) that gives information about the relative location of that score 6 within 4 

its 

distribution. The formula for converting a raw score from a sample into 6 a -score is: 

4 Ã c c cc 

6 

/ 

As you can see, -scores combine information about where the distribution is located (the mean/center) with 6 

how wide the 

distribution is (the standard deviation/spread) to interpret a raw score ( ). Specifically, -scores will tell us how far the 4 score 6 

is 

away from the mean in units of standard deviations and in what direction. 

The value of a -score has two parts: the sign (positive or negative) and the 6 magnitude 6 

(the actual number). The sign of the - 

score tells you in which half of the distribution the z-score falls: a positive sign (or no sign) indicates that the score is above 

the mean and on the right hand-side or upper end of the distribution, and a negative sign tells you the score is below the mean 

and on the left-hand side or lower end of the distribution. The magnitude of the number tells you, in units of standard 

deviations, how far away the score is from the center or mean. The magnitude can take on any value between negative and 

positive infinity, but for reasons we will see soon, they generally fall between -3 and 3. 

A -score value of -1.0 tells us that this z-score is 1 standard deviation (because of 6 

the magnitude 1.0) below (because of the 

negative sign) the mean. Similarly, a -score value of 1.0 tells us that this -score is 6 1 6 standard deviation above the mean. 

Thus, these two scores are the same distance away from the mean but in opposite directions. A -score 6 of -2.5 is two-and-ahalf 

standard deviations below the mean and is therefore farther from the center than both of the previous scores, 6 and a -score 

of 0.25 is closer than all of the ones before. For now, we will use a rough cut-off of 1.5 standard deviations in either direction 

as the difference between close scores (those within 1.5 standard deviations or between = -1.5 and = 1.5) 6 and 6 

extreme 

scores (those farther than 1.5 6 6 

standard deviations – below = -1.5 or above = 1.5). 

Practice 

Raw Score to z-score 

We can convert raw scores into -scores to get a better idea of where in the distribution those scores fall. Let’s say we get 6 

a 

score of 68 on an exam. We may be disappointed to have scored so low, but perhaps it was just a very hard exam. Having 

information about the distribution of all scores in the class would be helpful to put some perspective on ours. We find out that 

the class got an average score of 54 with a standard deviation of 8. To find out our relative location within this distribution, we 

simply convert our test score into 6 a -score. 

Ã 

6 

We find that we are 1.75 standard deviations above the average, above our rough cut off for close and far. Suddenly our 68 is 

looking pretty good! 

/ 

 


Figure : Raw and standardized versions of a single score (CC-BY-NC-SA Foster et al. from An Introduction to 


Figure shows both the raw score and the -score on their respective distributions. Notice that the red line indicating 

where each score lies is in the same relative spot for both. This is because transforming a raw score into a -score does not 

change its relative location, it only makes it easier to know precisely where it is. 

-scores are also useful for comparing scores from different distributions. Let’s say we take the SAT and score 501 on both 

the math and critical reading sections. Does that mean we did equally well on both? Scores on the math portion are distributed 

normally with a mean of 511 and standard deviation of 120, so our -score on the math section is 

which is just slightly below average (note that use of “math” as a subscript; subscripts are used when presenting multiple 

versions of the same statistic in order to know which one is which and have no bearing on the actual calculation). The critical 

reading section has a mean of 495 and standard deviation of 116, so 

So even though we were almost exactly average on both tests, we did a little bit better on the critical reading portion relative to 

other people. 

Finally, -scores are incredibly useful if we need to combine information from different measures that are on different scales. 

Let’s say we give a set of employees a series of tests on things like job knowledge, personality, and leadership. We may want 

to combine these into a single score we can use to rate employees for development or promotion, but look what happens when 

we take the average of raw scores from different scales, as shown in Table : 

Table : Raw test scores on different scales (ranges in parentheses). 

Raw Scores Job Knowledge (0 – 100) Personality (1 –5) Leadership (1 – 5) Average 

Employee 1 98 4.2 1.1 34.43 

Employee 2 96 3.1 4.5 34.53 

Employee 3 97 2.9 3.6 34.50 

Because the job knowledge scores were so big and the scores were so similar, they overpowered the other scores and removed 

almost all variability in the average. However, if we standardize these scores into -scores, our averages retain more variability 

and it is easier to assess differences between employees, as shown in Table . 

Table : Standardized scores. 


-Scores 

Job Knowledge (0 

– 100) 

6 

Personality (1 –5) Leadership (1 – 5) 

Average 

Employee 1 1.00 1.14 -1.12 0.34 

Employee 2 -1.00 -0.43 0.81 -0.20 

Employee 3 0.00 -0.71 0.30 -0.14 

z-score to Raw Score 

Another convenient characteristic of -scores is that they can be converted into any “scale” that we would like. Here, the 6 

term 

scale means how far apart the scores are (their spread) and where they are located (their central tendency). This can be very 

useful if we don’t want to work with negative numbers or if we have a specific range we would like to present. The 

formula for transforming to for a 6 4 sample is: 

4 6/ cc 

Notice that this is just a rearrangement of the original formulas for calculating from raw scores. Dr. MO prefers to just 6 

use 

the original z-score formula and do algebra to figure out the raw score when she has a z-score because she doesn't want to have 

to remember more formulas, but others prefer to do less algebra and have more formulas; it's your choice! 

Let’s say we create a new measure of intelligence, and initial calibration finds that our scores have a mean of 40 and standard 

deviation of 7. Three people who have scores of 52, 43, and 34 want to know how well they did on the measure. We can 

convert their raw scores into 6-scores: 

Ã 

6 

 

 

Ã 

6 

 

 

Ã 

A problem is that these new -scores aren’t exactly intuitive for many people. We can give people information 6 

about their 

relative location in the distribution (for instance, the first person scored well above average), or we can translate these 6 scores 

into the more familiar metric of IQ scores, which have a mean of 100 and standard deviation of 16: 

6 

Ã 

 

*2 Ç 

*2 Ç 

*2 Ã Ç 

We would also likely round these values to 127, 107, and 87, respectively, for convenience. 





Example 

Exercise 

Exercise 

5.2.1: Practice Calculating z-scores 

Assume the following scores represent a sample of statistics classes taken by five psychology professors: 2, 3, 5, 5, 6. If 

the standard deviation is 1.64, what is the z-score for each of these professor's? 

Solution 

The mean is 4.2 stats classes taken 

 

 

( ). 

c 

 

 

 

4 Ã cc 

6 

Ã 

Ã 

Ã 

/ 

 

 

4 Ã cc 

6 

Ã 

Ã 

Ã 

/ 

 

 

4 Ã cc 

6 

Ã 

 

+0$ 

/ 

 

 

4 Ã cc 

6 

Ã 

 

 

PS You might want to practice calculating the standard deviation yourself to make sure that you haven't forgotten how! 

/ 

 

 

Your turn! 

Calculate -scores for the three IQ scores provided, which were taken from a population with a mean of 100 and 6 

standard 

deviation of 16: 112, 109, 88. 

Answer 

= 0.75 

6 

= 0.56 

6 

6 

= -0.75 

This time, you'll get the z-score and will need to find the IQ scores. Remember, you can do this with the z-score formula that 

you used above and to algebra to find \(x|), or you an use the other z-score formula. 

Use the -scores provided to find two IQ scores taken from a population with a mean of 100 and standard deviation of 16: 

6 

6 

= 2.19 

= -0.06 

Answer 

6 

The IQ scores are 135 (for = 2.19) and 99 ( = -0.06) . 

6 6 



Michelle Oja 7/30/2021 5.2.1.1 CC-BY-NC-SA https://stats.libretexts.org/@go/page/22049



☰ 

5.3: Introduction to the z table 

To introduce the table of critical z-scores, we'll first refresh and add to what you learned last chapter about distributions 

Probability Distributions and Normal Distributions 

Recall that the normal distribution has an area under its curve that is equal to 1 and that it can be split into sections by drawing 

a line through it that corresponds to standard deviations from the mean. These lines marked specific z-scores. These sections 

between the marked lines have specific probabilities of scores falling in these areas under the normal curve. 

First, let’s look back at the area between = -1.00 and = 1.00 presented in Figure . We were told earlier that this region 

contains 68% of the area under the curve. Thus, if we randomly chose a -score from all possible z-scores, there is a 68% 

chance that it will be between and (within one standard deviation below and one standard deviation 

above the mean) because those are the -scores that satisfy our criteria. 

Figure : There is a 68% chance of selection a -score from the blue-shaded region (CC-BY-NC-SA Foster et al. from An 


Take a look at the normal distribution in Figure which has a line drawn through it as = 1.25. This line creates two 

sections of the distribution: the smaller section called the tail and the larger section called the body. Differentiating between the 

body and the tail does not depend on which side of the distribution the line is drawn. All that matters is the relative size of the 

pieces: bigger is always body. 

Figure 


: Body and tail of the normal distribution (CC-BY-NC-SA Foster et al. from An Introduction to Psychological 

We can then find the proportion of the area in the body and tail based on where the line was drawn (i.e. at what -score). 

Mathematically this is done using calculus, but we don't need to know how to do all that! The exact proportions for are given 

you to you in the Standard Normal Distribution Table, also known at the -table. Using the values in this table, we can find the 

area under the normal curve in any body, tail, or combination of tails no matter which -scores are used to define them. 

Let’s look at an example: let’s find the area in the tails of the distribution for values less than = -1.96 (farther negative and 

therefore more extreme) and greater than = 1.96 (farther positive and therefore more extreme). Dr. Foster didn't just pick this 

z-score out of nowhere, but we'll get to that later. Let’s find the area corresponding to the region illustrated in Figure , 

which corresponds to the area more extreme than = -1.96 and = 1.96. 


☰ 

Figure : Area in the tails beyond = -1.96 and = 1.96 (CC-BY-NC-SA Foster et al. from An Introduction to 


If we go to the -table shown in the Critical Values of z Table page (which can also be found from the Common Critical Value 

Tables at the end of this book in the Back Matter with the glossary and index), we will see one column header that has a , 

bidirectional arrows, and then . This means that, for the entire table (all 14ish columns), there are really two columns (or 

sub-columns. The numbers on the left (starting with -3.00 and ending with 3.00) are z-scores. The numbers on the right 

(starting with .00135 and ending with .99865) are probabilities (p-values). So, if you multiply the p-values by 100, you get a 

percentage. 

Let’s start with the tail for = 1.96. What p-value corresponds to 1.96 from the z-table in Table ? 

Example 

What p-value corresponds to 1.96 from the z-table in Table ? 

Solution 

For z = 1.96, p = .97500 

If we multiply that by 100, that means that 97.50% of the scores in this distribution will be below this score. Look at Figure 

again. This is saying that 97.5 % of scores are outside of the shaded area on the right. That means that 2.5% of scores in 

a normal distribution will be higher than this score (100% - 97.50% = 2.50%). In other words, the probability of a raw score 

being higher than a z-score is p=.025. 

If do the same thing with |(z = -1.96|), we find that the p-value for is .025. That means that of raw scores 

should be below a z-score of ; according to Figure , that is the shaded area on the left side. What did we just 

learn? That the shaded areas for the same z-score (negative or positive) are the same p-value, the same probability. We can 

also find the total probabilities of a score being in the two shaded regions by simply adding the areas together to get 0.0500. 

Thus, there is a 5% chance of randomly getting a value more extreme than or (this particular value and 

region will become incredibly important later). And, because we know that z-scores are really just standard deviations, this 

means that it is very unlikely (probability of ) to get a score that is almost two standard deviations away from the mean ( 

below the mean or 1.96 above the mean). 

Attributions & Contributors 





Example 

Exercise 

Example 

4 Ã cc 

5.3.1: Practice Using the z Table 

It's time to practice with the z-table! 

Find the -score that bounds the top 9% of the 6 

distribution. 

Ã 

 

 

 

 

Solution 

Because we are looking for top 9%, we need to look for the p-value closest to p = .91000 ( 

) because 

the p-values (probabilities) in the z Table show the probability of score being lower, but this question is asking for top 

9%, not the portion lower than . There should be of scores lower than the top . 

The closest p-value to p = .91000 ( ) is 0.90988. The z-score for p = 0.90988 iz z=1.34. 

The z-score for the top $9\%$ of the distribution is z=1.34 (for p=0.90988, the closest probability to , which marks 

everyone lower than the top ). 

Your turn! 

Find the -score that bounds (p=0.25000) of the lower tail of the 6 

distribution. 

Hint: You don't have to subtract anything for this one because the question is asking about the scores that are lower. 

Answer 

The z-score for 25% of the lower tail of the distribution is (for p=0.25143, the closes probability to 6 0.25000 

Ã 

( )). 

Now, let's try some scenarios... 

The heights of women in the United States are normally distributed with a mean of 63.7 inches and a standard deviation 

of 2.7 inches. If you randomly select a woman in the United States, what is the probability that she will be between taller 

than 64 inches? 

Solution 

X (raw score) = 64 inches 

= 63.7 inches 

c 

s=2.7 inches 

 

6 

cc 

Ã 

 

Finding z=0.11 on the z Table, we see that p = 0.543860. This is the probability that a score will be lower than our raw 

score, but the question asked the proportion who would be taller. 

/ 

 

 

Ã 

, g g 

Final Answer (in words): The probability that a woman in the U.S. would be 

64 inches or taller is 0.4562, or . 


Exercise 

Exercise 

Your turn! 

The heights of men in the United States are normally distributed with a mean of 69.1 inches and a standard deviation of 

2.9 inches. What proportion of men are taller than 6 feet (72 inches)? 

Answer 

Final Answer: The probability that a man in the U.S. would be 72 inches or taller is 0.15866, or . 

Last one, on something that you might find relevant! 

Imagine that you scored 82 points on a final exam. After the final, you find out that the average score on the exam was 78 

with a standard deviation of 7. What proportion (in a percentage) did worse than you (earned a lower score)? 

Answer 

The proportion of students in the class who did worse than you (earned a lower score) should be . 






☰ 

5.3.2: Table of Critical Values of z 

Table shows z-scores, their probability (p-value), and percentage. If this table is too unwieldy, here is a PDF of a z-score table with only three columns (z-score, p-value, percent) 

with 

more than 600 rows of z-scores (instead of Table ). 

 

Michelle Oja 8/6/2021 5.3.2.1 CC-BY https://stats.libretexts.org/@go/page/22052

☰ Table - Table of z-scores, Probabilities, and Percentages 

Scores from -1.99 to -1.00 Scores from -0.99 to zero Scores from 0.01 to 1.00 Scores from 1.01 to 2.00 

 

z-Score p-value % z-Score p-value % z-Score p-value % z-Score p-value % 

-1.99 0.0233 2.33% -0.99 0.1611 16.11% 0.01 0.5040 50.40% 1.01 0.8438 84.38% 

-1.98 0.0239 2.39% -0 98 0.1635 16.35% 0.02 0.5080 50.80% 1.02 0.8461 84.61% 

-1.97 0.0244 2.44% -0.97 0.1660 16.60% 0.03 0.5120 51.20% 1.03 0.8485 84.85% 

-1.96 0.0250 2.50% -0.96 0.1685 16.85% 0.04 0.5160 51.60% 1.04 0.8508 85.08% 

-1.95 0.0256 2.56% -0.95 0.1711 17.11% 0.05 0.5199 51.99% 1.05 0.8531 85.31% 

-1.94 0.0262 2.62% -0.94 0.1736 17.36% 0.06 0.5239 52.39% 1.06 0.8554 85.54% 

-1.93 0.0268 2.68% -0.93 0.1762 17.62% 0.07 0.5279 52.79% 1.07 0.8577 85.77% 

-1.92 0.0274 2.74% -0.92 0.1788 17.88% 0.08 0.5319 53.19% 1.08 0.8599 85.99% 

-1.91 0.0281 2.81% -0.91 0.1814 18.14% 0.09 0.5359 53.59% 1.09 0.8621 86.21% 

-1.90 0.0287 2.87% -0.90 0.1841 18.41% 0.10 0.5398 53.98% 1.10 0.8643 86.43% 

-1.89 0.0294 2.94% -0.89 0.1867 18.67% 0.11 0.5438 54.38% 1.11 0.8665 86.65% 

-1.88 0.0301 3.01% -0.88 0.1894 18.94% 0.12 0.5478 54.78% 1.12 0.8686 86.86% 

-1.87 0.0307 3.07% -0.87 0.1922 19.22% 0.13 0.5517 55.17% 1.13 0.8708 87.08% 

-1.86 0.0314 3.14% -0.86 0.1949 19.49% 0.14 0.5557 55.57% 1.14 0.8729 87.29% 

-1.85 0.0322 3.22% -0.85 0.1977 19.77% 0.15 0.5596 55.96% 1.15 0.8749 87.49% 

-1.84 0.0329 3.29% -0.84 0.2005 20.05% 0.16 0.5636 56.36% 1.16 0.8770 87.70% 

-1.83 0.0336 3.36% -0.83 0.2033 20.33% 0.17 0.5675 56.75% 1.17 0.8790 87.90% 

-1.82 0.0344 3.44% -0.82 0.2061 20.61% 0.18 0.5714 57.14% 1.18 0.8810 88.10% 

-1.81 0.0352 3.52% -0.81 0.2090 20.90% 0.19 0.5754 57.54% 1.19 0.8830 88.30% 

-1.80 0.0359 3.59% -0.80 0.2119 21.19% 0.20 0.5793 57.93% 1.20 0.8849 88.49% 

-1.79 0.0367 3.67% -0.79 0.2148 21.48% 0.21 0.5832 58.32% 1.21 0.8869 88.69% 

-1.78 0.0375 3.75% -0.78 0.2177 21.77% 0.22 0.5871 58.71% 1.22 0.8838 88.38% 

-1.77 0.0384 3.84% -0.77 0.2207 22.07% 0.23 0.5910 59.10% 1.23 0.8907 89.07% 

-1.76 0.0392 3.92% -0.76 0.2236 22.36% 0.24 0.5948 59.48% 1.24 0.8925 89.25% 

-1.75 0.0401 4.01% -0.75 0.2266 22.66% 0.25 0.5987 59.87% 1.25 0.8944 89.44% 

-1.74 0.0409 4.09% -0.74 0.2297 22.97% 0.26 0.6026 60.26% 1.26 0.8562 85.62% 

-1.73 0.0418 4.18% -0.73 0.2327 23.27% 0.27 0.6064 60.64% 1.27 0.8980 89.80% 

-1.72 0.0427 4.27% -0.72 0.2358 23.58% 0.28 0.6103 61.03% 1.28 0.8997 89.97% 

-1.71 0.0436 4.36% -0.71 0.2389 23.89% 0.29 0.6141 61.41% 1.29 0.9015 90.15% 

-1.70 0.0446 4.46% -0.70 0.2420 24.20% 0.30 0.6179 61.79% 1.30 0.9032 90.32% 

-1.69 0.0455 4.55% -0.69 0.2451 24.51% 0.31 0.6217 62.17% 1.31 0.9349 93.49% 

-1.68 0.0465 4.65% -0.68 0.2483 24.83% 0.32 0.6255 62.55% 1.32 0.9366 93.66% 

-1.67 0.0475 4.75% -0.67 0.2514 25.14% 0.33 0.6293 62.93% 1.33 0.9382 93.82% 

-1.66 0.0485 4.85% -0.66 0.2546 25.46% 0.34 0.6331 63.31% 1.34 0.9099 90.99% 

-1.65 0.0495 4.95% -0.65 0.2579 25.79% 0.35 0.6368 63.68% 1.35 0.9115 91.15% 


-1.64 ☰ 0.0505 5.05% -0.64 0.2611 26.11% 0.36 0.6406 64.06% 1.36 0.9131 91.31% 

-1.63 0.0516 5.16% -0.63 0.2644 26.44% 0.37 0.6443 64.43% 1.37 0.9147 91.47% 

-1.62 0.0526 5.26% -0.62 0.2676 26.76% 0.38 0.6480 64.80% 1.38 0.9162 91.62% 

-1.61 0.0537 5.37% -0.61 0.2709 27.09% 0.39 0.6517 65.17% 1.39 0.9177 91.77% 

-1.60 0.0548 5.48% -0.60 0.2743 27.43% 0.40 0.6554 65.54% 1.40 0.9192 91.92% 

-1.59 0.0559 5.59% -0.59 0.2776 27.76% 0.41 0.6591 65.91% 1.41 0.9207 92.07% 

-1.58 0.0571 5.71% -0.58 0.2810 28.10% 0.42 0.6628 66.28% 1.42 0.9222 92.22% 

-1.57 0.0582 5.82% -0.57 0.2843 28.43% 0.43 0.6664 66.64% 1.43 0.9236 92.36% 

-1.56 0.0594 5.94% -0.56 0.2877 28.77% 0.44 0.6700 67.00% 1.44 0.9251 92.51% 

-1.55 0.0606 6.06% -0.55 0.2912 29.12% 0.45 0.6736 67.36% 1.45 0.9265 92.65% 

-1.54 0.0618 6.18% -0.54 0.2946 29.46% 0.46 0.6772 67.72% 1.46 0.9279 92.79% 

-1.53 0.0630 6.30% -0.53 0.2981 29.81% 0.47 0.6808 68.08% 1.47 0.9292 92.92% 

-1.52 0.0643 6.43% -0.52 0.3015 30.15% 0.48 0.6844 68.44% 1.48 0.9306 93.06% 

-1.51 0.0655 6.55% -0.51 0.3150 31.50% 0.49 0.6879 68.79% 1.49 0.9319 93.19% 

-1.50 0.0668 6.68% -0.50 0.3085 30.85% 0.50 0.6915 69.15% 1.50 0.9332 93.32% 

-1.49 0.0681 6.81% -0.49 0.3121 31.21% 0.51 0.6950 69.50% 1.51 0.9345 93.45% 

-1.48 0.0694 6.94% -0.48 0.3156 31.56% 0.52 0.6985 69.85% 1.52 0.9357 93.57% 

-1.47 0.0708 7.08% -0.47 0.3192 31.92% 0.53 0.7019 70.19% 1.53 0.9370 93.70% 

-1.46 0.0722 7.22% -0.46 0.3228 32.28% 0.54 0.7054 70.54% 1.54 0.9382 93.82% 

-1.45 0.0735 7.35% -0.45 0.3264 32.64% 0.55 0.7088 70.88% 1.55 0.9394 93.94% 

-1.44 0.0749 7.49% -0.44 0.3300 33.00% 0.56 0.7123 71.23% 1.56 0.9406 94.06% 

-1.43 0.0764 7.64% -0.43 0.3336 33.36% 0.67 0.7157 71.57% 1.57 0.9418 94.18% 

-1.42 0.0778 7.78% -0.42 0.3372 33.72% 0.58 0.7190 71.90% 1.58 0.9430 94.30% 

-1.41 0.0793 7.93% -0.41 0.3409 34.09% 0.59 0.7224 72.24% 1.59 0.9441 94.41% 

-1.40 0.0808 8.08% -0.40 0.3446 34.46% 0.60 0.7258 72.58% 1.60 0.9452 94.52% 

-1.39 0.0823 8.23% -0.39 0.3483 34.83% 0.61 0.7291 72.91% 1.61 0.9463 94.63% 

-1.38 0.0838 8.38% -0.38 0.3520 35.20% 0.62 0.7324 73.24% 1.62 0.9474 94.74% 

-1.37 0.0853 8.53% -0.37 0.3557 35.57% 0.63 0.7357 73.57% 1.63 0.9485 94.85% 

-1.36 0.0869 8.69% -0.36 0.3594 35.94% 0.64 0.7389 73.89% 1.64 0.9495 94.95% 

-1.35 0.0885 8.85% -0.35 0.3632 36.32% 0.65 0.7422 74.22% 1.65 0.9505 95.05% 

-1.34 0.0901 9.01% -0.34 0.3669 36.69% 0.66 0.7454 74.54% 1.66 0.9515 95.15% 

-1.33 0.0918 9.18% -0.33 0.3707 37.07% 0.67 0.7486 74.86% 1.67 0.9525 95.25% 

-1.32 0.0934 9.34% -0.32 0.3745 37.45% 0.68 0.7518 75.18% 1.68 0.9535 95.35% 

-1.31 0.0951 9.51% -0.31 0.3783 37.83% 0.69 0.7549 75.49% 1.69 0.9545 95.45% 

-1.30 0.0968 9.68% -0.30 0.3821 38.21% 0.70 0.7580 75.80% 1.70 0.9554 95.54% 

-1.29 0.0985 9.85% -0.29 0.3859 38.59% 0.71 0.7612 76.12% 1.71 0.9564 95.64% 

-1.28 0.1003 10.03% -0.28 0.3897 38.97% 0.72 0.7642 76.42% 1.72 0.9573 95.73% 

-1.27 0.1020 10.20% -0.27 0.3936 39.36% 0.73 0.7673 76.73% 1.73 0.9582 95.82% 


-1.26 ☰ 0.1038 10.38% -0.26 0.3974 39.74% 0.74 0.7704 77.04% 1.74 0.9591 95.91% 

-1.25 0.1057 10.57% -0.25 0.4013 40.13% 0.75 0.7734 77.34% 1.75 0.9599 95.99% 

-1.24 0.1075 10.75% -0.24 0.4052 40.52% 0.76 0.7764 77.64% 1.76 0.9608 96.08% 

-1.23 0.1094 10.94% -0.23 0.4091 40.91% 0.77 0.7794 77.94% 1.77 0.9616 96.16% 

-1.22 0.1112 11.12% -0.22 0.4129 41.29% 0.78 0.7823 78.23% 1.78 0.9625 96.25% 

-1.21 0.1131 11.31% -0.21 0.4168 41.68% 0.79 0.7852 78.52% 1.79 0.9633 96.33% 

-1.20 0.1151 11.51% -0.20 0.4207 42.07% 0.80 0.7881 78.81% 1.80 0.9641 96.41% 

-1.19 0.1170 11.70% -0.19 0.4247 42.47% 0.81 0.7910 79.10% 1.81 0.9649 96.49% 

-1.18 0.1190 11.90% -0.18 0.4286 42.86% 0.82 0.7939 79.39% 1.82 0.9656 96.56% 

-1.17 0.1210 12.10% -0.17 0.4325 43.25% 0.83 0.7967 79.67% 1.83 0.9664 96.64% 

-1.16 0.1230 12.30% -0.16 0.4364 43.64% 0.84 0.7996 79.96% 1.84 0.9671 96.71% 

-1.15 0.1251 12.51% -0.15 0.4404 44.04% 0.85 0.8023 80.23% 1.85 0.9678 96.78% 

-1.14 0.1271 12.71% -0.14 0.4443 44.43% 0.86 0.8051 80.51% 1.86 0.9686 96.86% 

-1.13 0.1292 12.92% -0.13 0.4483 44.83% 0.87 0.8079 80.79% 1.87 0.9693 96.93% 

-1.12 0.1314 13.14% -0.12 0.4522 45.22% 0.88 0.8106 81.06% 1.88 0.9700 97.00% 

-1.11 0.1335 13.35% -0.11 0.4562 45.62% 0.89 0.8133 81.33% 1.89 0.9706 97.06% 

-1.10 0.1357 13.57% -0.10 0.4602 46.02% 0.90 0.8159 81.59% 1.90 0.9713 97.13% 

-1.09 0.1379 13.79% -0.09 0.4641 46.41% 0.91 0.8186 81.86% 1.91 0.9719 97.19% 

-1.08 0.1401 14.01% -0.08 0.4681 46.81% 0.92 0.8212 82.12% 1.92 0.9726 97.26% 

-1 07 0.1423 14.23% -0.07 0.4721 47.21% 0.93 0.8238 82.38% 1.93 0.9732 97.32% 

-1.06 0.1446 14.46% -0.06 0.4761 47.61% 0.94 0.8264 82.64% 1.94 0.9738 97.38% 

-1.05 0.1469 14.69% -0.05 0.4801 48.01% 0.95 0.8289 82.89% 1.95 0.9744 97.44% 

-1.04 0.1492 14.92% -0.04 0.4841 48.41% 0.96 0.8315 83.15% 1.96 0.9750 97.50% 

-1.03 0.1515 15.15% -0.03 0.4880 48.80% 0.97 0.8340 83.40% 1.97 0.9756 97.56% 

-1.02 0.1539 15.39% -0.02 0.4920 49.20% 0.98 0.8365 83.65% 1.98 0.9762 97.62% 

-1.01 0.1563 15.63% -0.01 0.4960 49.60% 0.99 0.8389 83.89% 1.99 0.9767 97.67% 

-1.00 0.1587 15.87% 0.00 0.5000 50.00% 1.00 0.8413 84.13% 2.00 0.9773 97.73% 

Attributions & Contributors 


Table 1 

 

: z Table. (CC-BY-SA; modified by Michelle Oja from Jsmura via Wikimedia Commons) 

Tables modified by Michelle Oja, originally provided by Jsmura via Wikimedia Commons) 


Example 

Example 

5.4: Predicting Amounts 

You might see how z-scores are useful for determining probabilities of scores, but you can use the z-score, probability, and the 

characteristics of a standard normal distribution so predict how many people could earn lower (or higher) than scores. To 

show how, we’ll use Final Exam scores (scores on a 100-point final exam from 20 students from an unknown college provided 

by OpenIntro.org). The average Final Exam score 77.7 points, with a standard deviation of 8.44 points. 

Let’s start with some context about who and what we are looking at. 

1. For the Final Exam scores, who is the sample? Your answer should include the number of people in the sample. 

2. Based on this sample, who could be the population? 


4. What information was provided? 

a. N: _____ 

b. Mean: _____ points 

c. Standard deviation: _____ 

Solution 

Add text here. 

1. 20 students from an unknown college 

2. Anything reasonable (probably college students). 

3. Final Exam scores (on a 100-point exam) 

4. What information was provided? 

a. N: 20 

b. Mean: 77.70 points 

c. Standard deviation: 8.44 

The Research Question will be: How many students passed the Final Exam (earned a score of 70 points or higher)? And 

we’ll use z-scores and the z-table to figure that out! Let’s walk through each step. 

1. What’s the formula for computing a z-score? 

2. Compute the z-score for X (70 points, from the Research Question). 

3. Find the p-value from the z Table for that z-score. 

4. To answer the Research Question, do we need to subtract this from 1 (or 18.141% from 100%)? Why or why not? 

If so, do it. 

5. You’re not done yet! The Research Question asks how many students, not what proportion (percentage). 

6. Write a concluding sentence for the Research Question. Round to the nearest whole person. 

Solution 

1. What’s the formula for computing a z-score? 


Exercise 

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3. z = -0.91 so p = 0.18141 

4. Yes, we do need to subtract because the p-value is telling us that 18.141% of the Final Exam scores are lower than 

our 70 points (z=-0.91), but the Research Question asked how many students scored higher then 70 points. 

 

 

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5. We have 20 students (N = 20). To find the amount of students, we multiply the N by the proportion (before it is a 

percentage): 

)+1*0 g 

6. Based on the mean, standard deviation, and size of this sample, 16 of the 20 students should pass the Final Exam 

(earned a score of 70 points or higher). 

Your turn! We’ll use the same sample of Final Exam scores (scores on a 100-point final exam from 20 students from an 

unknown college, with the average Final Exam score of 77.7 points and the standard deviation of 8.44 points. The Research 

Question will be: How many students earned 90 points or higher on the Final Exam? 


2. Find the p-value from the z Table for that z-score. 

3. To answer the Research Question, do we need to subtract this from 1 (or 100%)? Why or why not? If so, do 

it. 

4. You’re not done yet! The Research Question asks how many students, not what proportion (percentage). 

5. Write a concluding sentence for the Research Question. Round to the nearest whole person. 

Answer 


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2. z=1.46 so p = 0.92785 

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3. Yes, we do need to subtract because the p-value is telling us that 92.79% of the Final Exam scores are lower 

than 90 points (z = 1.46), but the Research Question asked how many students scored higher then 90 points. 

’ +. Ã 

4. We have 20 students (N = 20). To find the amount of students, we multiply the N by the proportion (before it is 

a percentage): 

)+1*0 g 

5. Based on the mean, standard deviation, and size of this sample, 1 of the 20 students should earn 90 points or 

higher on the Final Exam. 


We're Getting There! 

The Standard Normal Curve and z-scores are a small piece of the puzzle to get to where we’re heading, to statistically 

comparing sample means. It’s okay if you’re not quite on board with z-scores yet, we’ll keep working on the concept! We use 

z-scores to compare one sample to compare with another sample. It’s okay if you’re still confused about the Standard Normal 

Curve, too. These are building blocks to get us to the exciting statistical comparisons that will be coming up. 




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6 

5.5: Summary of z Scores 

Although this section has some math and the z-score formula, it's really to hit the main points of z-scores. 

Summary of z-scores 

Suppose Dr. Navarro's friend is putting together a new questionnaire intended to measure “grumpiness”. The survey has 50 

questions, which you can answer in a grumpy way or not. Across a big sample (hypothetically, let’s imagine a million people 

or so!) the data are fairly normally distributed, with the mean grumpiness score being 17 out of 50 questions answered in a 

grumpy way, and the standard deviation is 5. In contrast, when Dr. Navarro takes the questionnaire, she answer 35 out of 50 

questions in a grumpy way. So, how grumpy is she? One way to think about would be to say that I have grumpiness of 35/50, 

so you might say that she's 70% grumpy. But that’s a bit weird, when you think about it. If Dr. Navarro's friend had phrased 

her questions a bit differently, people m

PSYC 2200 Elementary Statistics for the Behavioral and Social Sciences. 2021a

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