126 CHAPTER 3 Describing, Exploring, and Comparing Data Quartiles Just as there are 99 percentiles that divide the data into 100 groups, there are three quartiles that divide the data into four groups. DEFINITION Quartiles are measures of location, denoted Q1, Q2, and Q3, which divide a set of data into four groups with about 25% of the values in each group. CAUTION Just as there is not universal agreement on a procedure for finding percentiles, there is not universal agreement on a single procedure for calculating quartiles, and different technologies often yield different results. If you use a calculator or software for exercises involving quartiles, you may get results that differ somewhat from the answers obtained by using the procedures described here. Here are descriptions of quartiles that are more accurate than those given in the preceding definition: Q1 (First quartile): Same value as P25. It separates the bottom 25% of the sorted values from the top 75%. (To be more precise, at least 25% of the sorted values are less than or equal to Q1, and at least 75% of the values are greater than or equal to Q1.) Q2 (Second quartile): Same as P50 and same as the median. It separates the bottom 50% of the sorted values from the top 50%. Q3 (Third quartile): Same as P75. It separates the bottom 75% of the sorted values from the top 25%. (To be more precise, at least 75% of the sorted values are less than or equal to Q3, and at least 25% of the values are greater than or equal to Q3.) Finding values of quartiles can be accomplished with the same procedure used for finding percentiles. Simply use the relationships shown in the margin. In Example 4 we found that P25 = 25 minutes, so it follows that Q1 = 25 minutes. Q1 = P25 Q2 = P50 Q3 = P75 Nielsen Ratings for College Students The Nielsen ratings are one of the most important measures of television viewing, and they affect billions of dollars in television advertising. In the past, the television viewing habits of college students were ignored, with the result that a large segment of the important young viewing audience was ignored. Nielsen Media Research is now including college students who do not live at home. Some television shows have large appeal to viewers in the 18-24 age bracket, and the ratings of such shows have increased substantially with the inclusion of college students. For males, NBC’s Sunday Night Football broadcast had an increase of 20% after male college students were included. Higher ratings ultimately translate into greater profits from charges to commercial sponsors. These ratings also give college students recognition that affects the programming they receive. T r o m t o v CP EXAMPLE 5 Converting a Percentile to a Data Value Refer to the fifty sorted “Space Mountain” wait times in Table 3-6. Use Figure 3-7 to find the 90th percentile, denoted by P90. SOLUTION Referring to Figure 3-7, we see that the sample data are already sorted, so we can proceed to compute the value of the locator L. In this computation, we use k = 90 because we are attempting to find the value of the 90th percentile, and we use n = 50 because there are 50 data values. L = k 100 # n = 90 100 # 50 = 45 Since L = 45 is a whole number, we proceed to the box in Figure 3-7 located at the right. We now see that the value of the 90th percentile is midway between the Lth (45th) value and the next value in the original set of data. That is, the value of the 90th percentile is midway between the 45th value and the 46th value. The 45th value in Table 3-6 is 60 and the 46th value is 75, so the value midway between them is 67.5 minutes. We conclude that the 90th percentile is P90 = 67.5 minutes. YOUR TURN. Do Exercise 21 “Percentile.”
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