AP StatisticsSemester OneReviewPart 1Chapters 1-5

AP Statistics TopicsDescribing DataProducing DataProbabilityStatistical Inference

Describing DataCh 1: Describing Data: Graphically andNumericallyCh 2: The Normal DistributionsCh 3: Describing BiVariateRelationshipsCh 4: More BiVariate Relationships

Chapter 1:Describing DataOur Introductory Chaptertaught us how to describe a setof data graphically andnumerically.Our focus in this chapter wasdescribing the Shape, Outliers,Center, and Spread of a dataset.

Describing DataWhen starting any data analysis, you should firstPLOT your data and describe what you see...DotplotStemplotBox-n-Whisker PlotHistogram

Describe the SOCSAfter plotting the data, note the SOCS:Shape: Skewed, Mound, Uniform, BimodalOutliers: Any “extreme” observationsCenter: Typical “representative” valueSpread: Amount of variability

Numeric DescriptionsWhile a plot provides a nicevisual description of a dataset,we often want a more detailednumeric summary of thecenter and spread.

Measures of VariabilityWhen describing the “spread” of a set ofdata, we can use:Range: Max-MinInterQuartile Range: IQR=Q3-Q1Standard Deviation: ! =(x " x) 2#n " 1

Numeric DescriptionsWhen describing the center and spreadof a set of data, be sure to provide anumeric description of each:Mean and Standard Deviation5-Number Summary: Min, Q1, Med,Q3, Max {Box-n-Whisker Plot}

Determining OutliersWhen an observation appears to bean outlier, we will want to providenumeric evidence that it is or isn’t“extreme”We will consider observations outliers if:More than 3 standard deviations fromthe mean.OrMore than 1.5 IQR’s outside the “box”

Chapter 1 Summary

Chapter 2:NormalDistributionsMany distributions in statisticscan be described asapproximately Normal.In this chapter, we learned howto identify and describe normaldistributions and how to doStandard Normal Calculations.

Density CurvesA Density Curve isa smooth, idealizedmathematical modelof a distribution.The area underevery densitycurve is 1.

The Normal DistributionMany distributions of data and many statisticalapplications can be described by an approximatelynormal distribution.Symmetric, Bell-shaped CurveCentered at Mean µDescribed as N(µ,! )

Empirical RuleOne particularly useful factabout approximately Normaldistributions is that68% of observations fallwithin one standarddeviation of µ95% fall within 2 standarddeviations of µ99.7% fall within 3 standarddeviations of µ

Standard Normal CalculationsThe empirical rule is useful when an observationfalls exactly 1,2,or 3 standard deviations from µ.When it doesn’t, we must standardize the value {zscore}and use a table to calculate percentiles, etc.z = x ! µ"

Assessing NormalityTo assess the normality of a set of data, we can’trely on the naked eye alone - not all mound shapeddistributions are normal.Instead, we should make a Normal Quantile Plot andlook for linearity.LinearityNormality

Chapter 3DescribingBiVariateRelationshipsIn this chapter, we learnedhow to describe bivariaterelationships.We focused on quantitativedata and learned how toperform least squaresregression.

Bivariate RelationshipsLike describing univariate data, thefirst thing you should do withbivariate data is make a plot.ScatterplotNote Strength, Direction, Form

Correlation “r”We can describe the strengthof a linear relationship withthe Correlation Coefficient, r-1 ! r ! 1The closer r is to 1 or -1,the stronger the linearrelationship between xand y.

Least Squares RegressionWhen we observe a linearrelationship between x and y, weoften want to describe it with a “lineof best fit” y=a+bx.We can find this line byperforming least-squaresregression.We can use the resulting equationto predict y-values for given x-values.

Assessing the FitIf we hope to make useful predictions of y we mustassess whether or not the LSRL is indeed the bestfit. If not, we may need to find a different model.Residual Plot

Making PredictionsIf you are satisfied that the LSRL provides anappropriate model for predictions, you can use itto predict a y-hat for x’s within the observed rangeof x-values.ŷ = a + bxPredictions for observed x-values can beassessed by noting the residual.Residual = observed y - predicted y

Chapter 3 Summary

Chapter 4More BiVariateRelationshipsIn this chapter, we learnedhow to find models that fitsome nonlinearrelationships.We also explored how todescribe categoricalrelationships.

NonLinear RelationshipsIf data is not best described by a LSRL, we may beable to find a Power or Exponential model that canbe used for more accurate predictions.Power Model:Exponential Model:ˆ y =10 a x by ˆ =10 a 10 bx!!

Transforming DataIf (x,y) is non-linear, we can transform it to try toachieve a linear relationship.If transformed data appears linear, we can find aLSRL and then transform back to the originalterms of the data(x, log y) LSRL > Exponential Model(log x, log y) LSRL > Power Model

The Question of CausationJust because we observe a strong relationship orstrong correlation between x and y, we can notassume it is a causal relationship.

Relations in Categorical DataWhen categorical data is presented in a two-waytable, we can explore the marginal and conditionaldistributions to describe the relationship betweenthe variables.

Chapter 5Producing DataIn this chapter, we learnedmethods for collecting datathrough sampling andexperimental design.

Sampling DesignOur goal in statistics is often to answer a questionabout a population using information from asample.Observational Study vs. ExperimentThere are a number of ways to select a sample.We must be sure the sample is representativeof the population in question.

SamplingIf you are performing anobservational study, yoursample can be obtained in anumber of ways:Convenience - ClusterSystematicSimple Random SampleStratified Random Sample

Experimental DesignIn an experiment, we impose a treatment with thehopes of establishing a causal relationship.Experiments exhibit 3 PrinciplesRandomizationControlReplication

Experimental DesignsLike Observational Studies, Experiments can take anumber of different forms:Completely Controlled RandomizedComparative ExperimentBlockedMatched Pairs

Chapters 6-9 Tomorrow

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