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The Practice of Statistics Third Edition Chapter 1: Exploring Data 1.2 Describing Distributions with Numbers Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

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Objectives for 1.2 Given a data set, How do you compute mean, median, quartiles, and the five-number summary? How do you construct a box plot using the five- number summary? How do you compute the inter-quartile range? How do you identify an outlier using the inter- quartile range rule? How do you compute the standard deviation and variance?

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Measure for The Center of a Distribution

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The Means of a Data Set So far, we know several measures of central tendency of a set of numbers: means, median, and mode. The means is the arithmetic average of the data set.

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The Mean of a Data Set “Average Value” Σ (sigma) means to add them all up. All the data values and get a total. Take the total and divide by the number of data.

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Example - Mean Joey’s first 14 quiz grades in a marking period were 86, 84, 91, 75, 78, 80, 74, 87, 76, 96, 82, 90, 98, 93 Find the mean. Answer 85. Use calculator – Stat edit, enter data in L1 Second Stat, Math, Mean( L1), Enter

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The Median of the Data Set Median is the center of the data set. Half of the data set is above and Half is below the median. The 50 th Percentile. The median may or may not be in the data set.

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Calculation for Median “Middle Value”

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Example - Median Joey’s first 14 quiz grades in a marking period were 86, 84, 91, 75, 78, 80, 74, 87, 76, 96, 82, 90, 98, 93 Find the median. Answer 85. Use calculator – Stat edit, enter data in L1 Second Stat, Math, Median( L1), Enter

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Terminology “A measure is resistant” A measure that does not respond strongly to the influence of outliers (extreme observations). Furthermore, a measure that is resistant does not respond strongly to changes in a few observations.

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Are mean and median resistant? Mean and Median Applet

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Mean vs Median Mean is not a resistant measure. –It is sensitive to the influence of a few extreme observations (outliers). –It is sensitive to skewed distributions. The mean is pulled towards the tail. Median is resistant. –It is resistant to extreme values and skewed distributions. For skewed distributions the median is the better measure for center.

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Measure for Spread Range Quartiles Five Number Summary The Standard Deviation

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Range The difference between the largest value and the smallest value. Gives the full spread of the data. But may be dependent on outliers.

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Quartiles We can describe the spread (variability of a distribution) by giving several percentiles (pth percentile of a distribution) Typically we use 25 th percentile, 50 th percentile, 75 th percentile. Q1, median, Q3.

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Example Joey’s first 14 quiz grades in a marking period were 86, 84, 91, 75, 78, 80, 74, 87, 76, 96, 82, 90, 98, 93 Find Q1, median, and Q3. Answer: Q1 = 78, Median = 85, Q3 = 91 Using the calculator – STAT, CALC, 1-Var Stats L1, ENTER

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Five Number Summary Using the calculator, we again use 1-Var Stats.

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Five Number Summary Computer Software Output

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Graphical Display of 5 Number Summary

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Example - Boxplot Joey’s first 14 quiz grades in a marking period were 86, 84, 91, 75, 78, 80, 74, 87, 76, 96, 82, 90, 98, 93 Answer 74 78 85 91 98 Calculator STAT PLOT, make appropriate selections on the menu, ZOOM, 9:Zoom Stat

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Interquartile Range

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Identifying Outliers

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Variance and Standard Deviation

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Example – Variance and Standard Deviation Joey’s first 14 quiz grades in a marking period were 86, 84, 91, 75, 78, 80, 74, 87, 76, 96, 82, 90, 98, 93 Calculate the variance and standard deviation. 8686-85=11 8484-85=-11 9191-85=636 7575-85=-10100 7878-85=-749 8080-85=-525 7474-85=-11121 8787-85=24 7676-85=-981 9696-85=11121 8282-85=-39 9090-85=525 9898-85=13169 9393-85=864 Total 1190 Tot 806 Standard Deviation Calculator – STAT EDIT, enter data in list 1, QUIT STAT CALC 1-Var Stat

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Standard Deviation The standard deviation is zero when there is no spread. The Standard deviation gets larger as the spread increases.

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Impact of adding a constant to all data in the set? Joey’s first 14 quiz grades in a marking period were –86, 84, 91, 75, 78, 80, 74, 87, 76, 96, 82, 90, 98, 93 Add 32 points to each score, then store in L2. Compute 1-Var Stat. What has changed? The five-number summary has changed but the standard deviation has not? The measure the spread remains the same?

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The impact of multiplying each data in the set by a constant? Using the data set in L1 multiply the 2. Compute 1-Var Stat. What has changed? The five-number summary has changed by 2 times and the standard deviation has changed by 2 times. The measure of the spread has increased.

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