94 CHAPTER 3 Describing, Exploring, and Comparing Data FIGURE 3-2 Dotplots of “Rock ‘n’ Roller Coaster” and “Tower of Terror” Wait Times TABLE 3-1 Wait Times (min) For Rock ‘n’ Roller Coaster and Tower of Terror Rock ‘n’ Roller Coaster 15 20 45 45 45 55 55 70 75 95 110 Tower of Terror 45 45 45 50 50 55 60 60 65 75 80 Table 3-2 summarizes the different measures of center for the “Rock ‘n’ Roller Coaster” and “Tower of Terror” wait times shown above. In the spirit of describing, exploring, and comparing data, let’s examine the data in Table 3-1 and their dotplots in Figure 3-2, which includes 10 AM wait times on different days for “Rock ‘n’ Roller Coaster” and “Tower of Terror.” These data are taken from Data Set 33 “Disney World Wait Times” in Appendix B. The wait times for each ride have been sorted from shortest to longest. Simpson’s Paradox Consider these data: When playing a basketball game, Bart makes 4 of 10 shots in the first half (40% success) and 3 of 4 shots in the second half (75% success), while Homer makes 1 of 4 shots in the first half (25% success) and 7 of 10 shots in the second half (70% success). See that Bart was better than Homer in each half, but for the entire game, Homer made 8 of 14 shots (57.1% success), while Bart made 7 of 14 shots (50% success). Bart was better than Homer in each half, but Homer was better overall. This is a good example of Simpson’s paradox, which states that groups can have means that show one trend, but the trend is reversed when the groups are combined. One classic and real situation occurred when the Berkeley graduate program admitted 44% of male applicants and 35% of female applicants (bias in favor of males), but when looking at individual departments, there appeared to be a bias in favor of female applicants. YOUR TURN. For Exercise 5 “Super Bowl Jersey Numbers,” determine why the mean and median are not meaningful. EXAMPLE 6 Critical Thinking and Measures of Center See each of the following illustrating situations in which the mean and median are not meaningful statistics. a. Zip codes of the Gateway Arch in St. Louis, White House, Air Force division of the Pentagon, Empire State Building, and Statue of Liberty: 63102, 20500, 20330, 10118, 10004. (The zip codes don’t measure or count anything. The numbers are just labels for geographic locations.) b. Ranks of selected medical schools of Harvard, Johns Hopkins, New York University, Stanford University, and Duke University: 1, 2, 3, 4 10. (The ranks reflect an ordering, but they don’t measure or count anything.) c. Numbers on the jerseys of the starting offense for the New England Patriots when they won Super Bowl LIII: 12, 26, 46, 15, 11, 87, 77, 62, 60, 69, 61. (The numbers on the football jerseys don’t measure or count anything; they are just substitutes for names.) d. Top 5 annual compensation of chief executive officers (in millions of dollars): 513.3, 256.0, 146.6, 141.7, 130.7. (Such “top 5” or “top 10” lists include data that are not at all representative of the larger population.) e. The 50 mean ages computed from the means in each of the 50 states. (If you calculate the mean of those 50 values, the result is not the mean age of people in the entire United States. The population sizes of the 50 different states must be taken into account, as described in the weighted mean introduced in Part 2 of this section.)
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