Section 3.1 Measures of Center 95 Objective 5 Approximate the mean with grouped data Approximating the Mean with Grouped Data Sometimes we don’t have access to the raw data in a data set, but we are given a frequency distribution. In these cases we can approximate the mean.We use Table 3.1 to illustrate the method. This table presents the number of text messages sent via cell phone by a sample of 50 high school students. We will approximate the mean number of messages sent. Table 3.1 Number of Text Messages Sent by High School Students Number of Text Messages Sent Frequency 0–49 10 50–99 5 100–149 13 150–199 11 200–249 7 250–299 4 We present the method for approximating the mean with grouped data. Procedure for Approximating the Mean with Grouped Data Step 1: Compute the midpoint of each class. The midpoint of a class is found by taking the average of the lower class limit and the lower limit of the next larger class. For the last class, there is no next larger class, but we use the lower limit that the next larger class would have. Step 2: For each class, multiply the class midpoint by the class frequency. Step 3: Add the products Midpoint×Frequency over all classes. Step 4: Divide the sum obtained in Step 3 by the sum of the frequencies. EXAMPLE 3.9 Approximating the mean with grouped data Compute the approximate mean number of messages sent, using Table 3.1. Solution The calculations are summarized in Table 3.2. Step 1: Compute the midpoints: For the first class, the lower class limit is 0. The lower limit of the next class is 50. The midpoint is therefore 0 + 50 2 = 25 We continue in this manner to compute the midpoint of each class. Note that for the last class, we average the lower limit of 250 with 300, which is the lower limit that the next class would have. Step 2: Multiply the midpoints by the frequencies as shown in the column in Table 3.2 labeled “Midpoint×Frequency.” Step 3: Add the products Midpoint×Frequency, to obtain 6850. Step 4: The sum of the frequencies is 50. The mean is approximated by 6850/50 = 137. Table 3.2 Calculating the Mean Number of Text Messages Sent by High School Students Class Midpoint Frequency Midpoint×Frequency 0–49 25 10 25 × 10 = 250 50–99 75 5 75 × 5 = 375 100–149 125 13 125 × 13 = 1625 150–199 175 11 175 × 11 = 1925 200–249 225 7 225 × 7 = 1575 250–299 275 4 275 × 4 = 1100 Sum = 50 Sum = 6850 Mean ≈ 6850 50 = 137
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