Section 3.3 Measures of Position 129 Check Your Understanding 1. Following are final exam scores, arranged in increasing order, for 28 students in an introductory statistics course. 58 59 62 64 67 68 69 71 73 74 74 75 76 76 76 77 78 78 78 82 82 84 86 87 87 88 91 97 a. Find the first quartile of the scores. b. Find the third quartile of the scores. c. Fred got a 73 on the exam. On what percentile is this score? d. Students whose scores are on the 80th percentile or above will get a grade of A. Louisa got an 88 on the exam. Will she get an A? 2. For the years 1869–2007, the 90th percentile of annual snowfall in Central Park in New York City was 50.1 inches. Approximately what percentage of years had snowfall less than 50.1 inches? Answers are on page 139. Objective 4 Compute the five-number summary for a data set The Five-Number Summary The five-number summary of a data set consists of the median, the first quartile, the third quartile, the minimum value, and the maximum value. These values are generally arranged in order. DEFINITION The five-number summary of a data set consists of the following quantities: Minimum First quartile Median Third quartile Maximum EXAMPLE 3.27 Constructing a five-number summary Table 3.11 presents the number of students absent in a middle school in northwestern Montana for each school day in January 2008. Construct the five-number summary. Table 3.11 Number of Absences in January 2008 Date Number Absent Date Number Absent Date Number Absent Jan. 2 65 Jan. 14 59 Jan. 23 42 Jan. 3 67 Jan. 15 49 Jan. 24 45 Jan. 4 71 Jan. 16 42 Jan. 25 46 Jan. 7 57 Jan. 17 56 Jan. 28 100 Jan. 8 51 Jan. 18 45 Jan. 29 59 Jan. 9 49 Jan. 21 77 Jan. 30 53 Jan. 10 44 Jan. 22 44 Jan. 31 51 Jan. 11 41 Solution Step 1: We arrange the numbers in increasing order. The ordered numbers are: 41 42 42 44 44 45 45 46 49 49 51 51 53 56 57 59 59 65 67 71 77 100 Step 2: The minimum is 41 and the maximum is 100. Step 3: We use the methods described in Example 3.25 to compute the first and third quartiles. The first quartile is Q1 = 45, and the third quartile is Q3 = 59. Step 4: We use the method described in Section 3.1 to compute the median. The median is 51. Step 5: The five-number summary is 41 45 51 59 100
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