Section 3.2 Measures of Spread 111 We present the procedure for approximating the standard deviation from grouped data. Procedure for Approximating the Standard Deviation 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. Then compute the mean as described in Section 3.1. Step 2: For each class, subtract the mean from the class midpoint to obtain Midpoint−Mean. Step 3: For each class, square the difference obtained in Step 2 to obtain (Midpoint−Mean)2, and multiply by the frequency to obtain (Midpoint−Mean)2 ×Frequency. Step 4: Add the products (Midpoint−Mean)2 ×Frequency over all classes. Step 5: Compute the sum of the frequencies n. To compute the population variance, divide the sum obtained in Step 4 by n. To compute the sample variance, divide the sum obtained in Step 4 by n − 1. Step 6: Take the square root of the variance obtained in Step 5. The result is the standard deviation. EXAMPLE 3.16 Computing the standard deviation for grouped data Compute the approximate sample standard deviation of the number of messages sent, using the data given in Table 3.6. Solution The calculations are summarized in Table 3.7. Table 3.7 Calculating the Variance and Standard Deviation of the Number of Text Messages Class Midpoint Frequency Mean Midpoint−Mean (Midpoint−Mean)2×Frequency 0–49 25 10 137 −112 12544 × 10 = 125,440 50–99 75 5 137 −62 3844 × 5 = 19,220 100–149 125 13 137 −12 144 × 13 = 1,872 150–199 175 11 137 38 1444 × 11 = 15,884 200–249 225 7 137 88 7744 × 7 = 54,208 250–299 275 4 137 138 19044 × 4 = 76,176 Sum = 50 Sum = 292,800 Variance = 292,800 50 − 1 = 5975.51020 Standard deviation = √ 5975.51020 = 77.3014 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 midpoints of each of the classes. 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.We computed the sample mean in Example 3.9 in Section 3.1. The sample mean is ¯x = 137. Step 2: For each class, subtract the mean from the class midpoint as shown in the column labeled “Midpoint−Mean.” Step 3: For each class, square the difference obtained in Step 2 and multiply by the frequency as shown in the column labeled “(Midpoint−Mean)2 ×Frequency.” Step 4: Add the products (Midpoint − Mean)2 ×Frequency over all classes, to obtain the sum 292,800.
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