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them. However, observe that the sum of the deviations is 0! This is true of any dataset, since the mean represents the “balance” of the dataset. Due to mathematical concerns that we won‟t state here, mathematicians decided to square these values, since squaring converts all signed numbers into positive values. Value ̅ ( ̅) 0 -17.875 319.5156 1 -16.875 284.7656 3 -14.875 221.2656 10 -7.875 62.01563 8 -9.875 97.51563 7 -10.875 118.2656 4 -13.875 192.5156 110 92.125 8487.016 Mean: 17.875 Sum: 9782.88 Great, now they can be summed up to give 9782.88! Thus, we have found the following: ∑( ̅) One would think that dividing by 8 would now be appropriate to find the average. Due to mathematical properties that are beyond the scope of this course, the division will be by 7, which is . Thus: ∑( ̅) This value that we have found is called the variance. NOTE: The division by has to do with the fact that we are often dealing with a sample in inferential statistics and hope to make conclusions above a population. Sample Variance The variance of a sample, an uninterpretable measure of variability denoted by by the following formula: , can be found ∑( ̅) To make all of these calculations more meaningful (to have a true average), we should probably “unsquare” the value that we have. When we do this, we get the sample standard deviation: Statistics for Decision-Making in Business © Milos Podmanik Page 68
√ ∑( ̅) √ √ This is what we can think of as the average deviation of each point from the mean. It is clearly high for this dataset. What is causing it The outlier of 110! Conclusion: On average, values in the dataset deviate from the mean by about 37 units. Sample Standard Deviation The standard deviation of a sample, denoted , is given by the following formula: ∑( ̅) √ Note that this is simply the square root of the variance. In Excel, the standard deviation can be calculated simply by using the function below: = stdev(cell range) Example 1: A river with mild current is known to have an average depth of 3 feet with a standard deviation of 3 feet. The bottom is not visible. Is the river safe to cross by foot Also, what is the variance SOLUTION: Since there is a standard deviation of 3 feet, we can conclude, that, on average, the river depth deviates by 3 feet from the mean. It would not be unusual to encounter a part of the river with a depth of 6 or more feet. Therefore, the river should not be crossed by foot. Since the standard deviation is the square root of the variance, the variance is the square of the standard deviation. That is, Thus, the variance is 9. The variance does not have a valuable interpretation. 2.4.2 How Do We Interpret the Value We Get Think about this: n is a fixed value for our sample, specifically 5. The only thing that could make s 2 large or small is the numerator. Thus, if the deviations are large (a bad thing!), then the squared deviations will be large, and so the sum of squares will be large. This implies a large standard deviation. Statistics for Decision-Making in Business © Milos Podmanik Page 69
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Statistics for Decision- Making in
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A Note to Students This book is far
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6.3 Confidence Interval for ̂ 202
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which has undergone no additional r
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of transporting such an animal. Not
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Although far from last, we will con
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Account Type Revenue ($) New $5,296
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For the present time, we‟ll proce
Page 17 and 18: Row Labels Average of Revenue ($) M
Page 19 and 20: (SOURCE: Essentials of Modern Busin
Page 21 and 22: 1.3Statistics in Excel When conduct
Page 23 and 24: We can make it a bit more presentab
Page 25 and 26: For measures such as the range, Exc
Page 27 and 28: We get: We can do the same for Old.
Page 29 and 30: Chapter 2 Visual Representations of
Page 31 and 32: We now have to enter a formula for
Page 33 and 34: We know from the data that this val
Page 35 and 36: Alternatively, it is possible to in
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Page 39 and 40: Selected the copied graph. In Chart
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Page 43 and 44: Frequency 2.2 Visualizing Quantitat
Page 45 and 46: You can either choose to have Excel
Page 47 and 48: If we had additional variables, the
Page 49 and 50: Select PivotChart and select the fi
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Page 53 and 54: Relative Frequency Histogram of Cal
Page 55 and 56: a. Using Excel, find the mean and r
Page 57 and 58: ̅ ̅ account for this discrepancy,
Page 59 and 60: 2.3.2 Percentile Another useful too
Page 61 and 62: 2.3.4 Rank What if, on the other ha
Page 63 and 64: Check the “Analysis ToolPak” an
Page 65 and 66: You‟ll immediately notice that a
Page 67: ̅ 2.4 Descriptive Statistics - Var
Page 71 and 72: This gives us a nice measure of how
Page 73 and 74: We can immediately see the mean and
Page 75 and 76: Lastly, a normal curve is the most
Page 77 and 78: Frequency ̅ Histogram of TV's Owne
Page 79 and 80: 58.3 34.6 35.5 45.4 38.6 63.8 53.9
Page 81 and 82: Chapter 3 Probability and Decision
Page 83 and 84: Proportion of Rainy Days 1.2 Propor
Page 85 and 86: This means that if samples of HFCS
Page 87 and 88: Are we confident in accusing a lung
Page 89 and 90: 3.2 Joint Probability In the previo
Page 91 and 92: Company 1 Choices Y Y Y Y Y Y Y Y N
Page 93 and 94: There is less than a 1% chance that
Page 95 and 96: We can now see that the situation i
Page 97 and 98: sequence, one ball is chose at rand
Page 99 and 100: 3.3 Probability of Unions Imagine t
Page 101 and 102: Physical .20 .10 .30 Mental .40 .30
Page 103 and 104: SOLUTION: We first arrange this inf
Page 105 and 106: Homework Problems - 3.3 1. A gaming
Page 107 and 108: 3.4 Conditional Probability In many
Page 109 and 110: ) : The Arizona Cardinals make it t
Page 111 and 112: We could just as well have written,
Page 113 and 114: That is, the probability that the c
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3.5 Combinations and Permutations R
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1 st Bushel 2 nd Bushel U1 U1 U2 U2
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Which, in its final state gives: Th
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Combination - Order Does NOT Matter
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( ) There is only a .9% chance that
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1. Are repeats/replacements allowed
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Object 4 Object 5 Also notice that
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e. 2. Your classmate was absent whe
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3.6 Expected Value Imagine that you
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An expected value is actually not s
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Where The expected value is: , - (
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SOLUTION: We first note that there
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Probability 0.7 0.6 0.5 0.4 0.3 0.2
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Statistics for Decision-Making in B
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For men, we see that the most frequ
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So, we had 10 trials and wanted to
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Example 1: A fair-two sided coin is
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( ) ( ) ( ) ( ) Example 2: A fair,
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Expected Value of a Binomial Random
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f. What is the probability that g.
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Probability For instance, we see th
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Density 0.25 Time Speng Waiting in
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Suppose we wish to find ( ), that i
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Density Resulting in a standard dev
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Density Without going into detail h
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a. What is the probability that the
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Statistics for Decision-Making in B
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As you can see, this is a difficult
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We wish to know the area of the sha
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We can easily find that the probabi
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In the second section, we can check
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Chapter 6 Sampling Distributions an
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Thus, the standard deviation would
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1.7 to 1.8 1.8 to 1.9 1.9 to 2 2 to
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2.4 to 2.5 2.5 to 2.6 2.6 to 2.7 2.
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The Central Limit Theorem (CLT) has
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can set this up in our applet by ha
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3) We cannot use this sample to cal
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Thus, we need to find the lower and
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We see that the standard deviation
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√ The lower limit is: √ And the
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following data was collected on the
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{ So, we have a set of twenty 1‟s
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DULY CAUTIONED: The assumptions her
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Homework Problems -6.2 1. In a samp
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Since this is a mathematical questi
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Example 3: Many older homes have el
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Hypothesis Test Conclusion Test Say
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Hypothesis Test Conclusion Since yo
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SOLUTION: d. A generic conclusion s
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9. Based on the “Structure of a H
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1.2 Descriptive VS. Inferential Sta
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Relative Frequency 35% 30% 25% 20%
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Symmetric: 35 30 25 20 15 10 5 0 10
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Repair Cost Mean 971 Standard Error
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el freq Nitrous Oxide (thous. Tons)
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Probability Pizza Size Distribution
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e. ( ) f. ; the average response ti
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2. a. The long-run proportion of al
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e. 20% of all children are expected
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e. The middle 50% score between abo
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5. b. This might indicate that the
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̅ We have that ̅ and √ . So our
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6. This is the probability that we
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