96 CHAPTER 3 Describing, Exploring, and Comparing Data SOLUTION Remember, when working with data summarized in a frequency distribution, we make calculations possible by pretending that all sample values in each class are equal to the class midpoint. For example, consider the first class interval of 0–14 with a frequency of 6. We pretend that each of the 6 commute times is 7 minutes (the class midpoint). With the commute time of 7 minutes repeated 6 times, we have a total of 7.0# 6.0 = 42.0, as shown as the first entry in the rightmost column of Table 3-3. We can then add those results to find the sum of all sample values. The bottom row of Table 3-3 shows the two components we need for the calculation of the mean (as in Formula 3-2): Σf = 50 and Σ1f # x2 = 1730.0. We calculate the mean using Formula 3-2 as follows: x = Σ1f # x2 Σf = 1730.0 50 = 34.6 minutes The result of x = 34.6 minutes is an approximation because it is based on the use of class midpoint values instead of the original list of 50 commute times. The mean of 31.4 minutes (rounded from 31.42) found by using all of the original commute times is a more accurate result. YOUR TURN. Do Exercise 29 “Frequency Distribution.” FORMULA 3-3 Weighted mean: x = Σ1w# x2 Σw Formula 3-3 tells us to first multiply each weight w by the corresponding value x, then to add the products, and then finally to divide that total by the sum of the weights, Σw. Calculating a Weighted Mean When different x data values are assigned different weights w, we can compute a weighted mean. Formula 3-3 can be used to compute the weighted mean. Go Figure 4.77 hours: “Average” time adult Americans watch TV each day EXAMPLE 8 Computing Grade-Point Average In her first semester of college, a student of the author took five courses. Her final grades, along with the number of credits for each course, were A (3 credits), A (4 credits), B (3 credits), C (3 credits), and F (1 credit). The grading system assigns quality points to letter grades as follows: A = 4; B = 3; C = 2; D = 1; F = 0. Compute her grade-point average. SOLUTION Use the numbers of credits as weights: w = 3, 4, 3, 3, 1. Replace the letter grades of A, A, B, C, and F with the corresponding quality points: x = 4, 4, 3, 2, 0. We now use Formula 3-3 as shown below. The result is a first-semester grade-point average of 3.07. (In using the preceding round-off rule, the result should be rounded to 3.1, but it is common to round grade-point averages to two decimal places.) x = Σ1w# x2 Σw = 13 * 42 + 14 * 42 + 13 * 32 + 13 * 22 + 11 * 02 3 + 4 + 3 + 3 + 1 = 43 14 = 3.07 YOUR TURN. Do Exercise 33 “Weighted Mean.”
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