12.4 Measures of Dispersion 797 Measures of central tendency by themselves do not always give sufficient information to analyze a situation and make decisions. For example, suppose Apple is considering two companies to produce batteries for its iPads. Testing shows that Company A batteries have a mean life of 10 hours. Company B batteries have a mean life of 9.5 hours. If both manufacturers’ batteries cost the same, which one should be purchased? The average battery life may not be the most important factor. If half of Company A batteries last only 5 hours, while half last 15 hours, there is a large variability in the life of the batteries. If all of Company B batteries last between 9.0 and 10.0 hours, the batteries are more consistent and reliable. This example illustrates the importance of knowing something about the spread, or variability, of the data. In this section we will discuss two measures of variability or dispersion. Amble Design/Shutterstock Measures of Dispersion SECTION 12.4 LEARNING GOALS Upon completion of this section, you will be able to: 7 Calculate the range of a set of data. 7 Calculate the standard deviation of a set of data. of data multiplied by its weight; and wΣ is the sum of the weights. For example, suppose that students in a class need to submit a report that counts for 20% of their grade, they need to take a midterm exam that counts for 30% of their grade, and they need to take a final exam that counts for 50% of their grade. Suppose that a student got a 72 on the report, an 85 on the midterm exam, and a 93 on the final exam. To determine this student’s weighted average, first determine xw xw : 72(0.20) 85(0.30) 93(0.50) 86.4. Σ Σ = + + = Next determine w, Σ the sum of the weights: w 0.20 0.30 0.50 1.00. Σ = + + = Now determine the weighted average as follows. Weighted average xw w 86.4 1.00 86.4 = Σ Σ = = Thus, the weighted average is 86.4. Note that wΣ does not always have to be 1.00. In Exercises 75 and 76, use the weighted average formula. 75. Course Average Suppose that your final grade for a course is determined by a midterm exam and a final exam. The midterm exam is worth 40% of your grade, and the final exam is worth 60%. If your midterm exam grade is 84 and your final exam grade is 94, calculate your final weighted average. 90 76. Grade Point Average In a four-point grade system, an A corresponds to 4.0 points, a B corresponds to 3.0 points, a C corresponds to 2.0 points, and a D corresponds to 1.0 points. No points are awarded for an F. Last semester, Tanya received a B in a four-credit hour course, an A in a three-credit hour course, a C in a three-credit hour course, and an A in another three-credit hour course. A grade point average (GPA) is calculated as a weighted average using the credit hours as weights and the number of points corresponding to the grade as pieces of data. Calculate Tanya’s GPA for the previous semester. (Round your answer to the nearest hundredth.) 3.23 Recreational Mathematics 77. Average of Your Exams a) Calculate the mean, median, mode, and midrange of your exam grades in your mathematics course. Answers will vary. b) Which measure of central tendency best represents your average grade? Answers will vary. c) Which measure of central tendency would you rather use as your average grade? Answers will vary. 78. Purchases Matthew purchased some items at Staples each day for five days. The mode of the number of items Matthew purchased is higher than the median of the number of items he purchased. The median of the number of items Matthew purchased is higher than the mean of the number of items he purchased. He purchased at least two items but no more than seven items each day. a) How many items did Matthew purchase each day? (Note: There is more than one correct answer.) Answers will vary. One example is 2, 3, 5, 7, 7. b) Determine the mean, median, and mode for your answer to part (a). Answers will vary. Research Activity 79. Stanines and Deciles Two other measures of location that we did not mention in this section are stanines and deciles. Use statistics books, books on educational testing and measurements, and Internet websites to write a report on what stanines and deciles are and when percentiles, quartiles, stanines, and deciles are used. *See Instructor Answer Appendix
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