90 CHAPTER 2 Descriptive Statistics Standard Deviation for Grouped Data In Section 2.1, you learned that large data sets are usually best represented by frequency distributions. The formula for the sample standard deviation for a frequency distribution is Sample standard deviation = s = BΣ1x - x22f n - 1 where n = Σf is the number of entries in the data set. Finding the Standard Deviation for Grouped Data You collect a random sample of the number of children per household in a region. The results are listed below. Find the sample mean and the sample standard deviation of the data set. 1 3 1 1 1 1 2 2 1 0 1 1 0 0 0 1 5 0 3 6 3 0 3 1 1 1 1 6 0 1 3 6 6 1 2 2 3 0 1 1 4 1 1 2 2 0 3 0 2 4 SOLUTION These data could be treated as 50 individual entries, and you could use the formulas for mean and standard deviation. Because there are so many repeated numbers, however, it is easier to use a frequency distribution. x f xf 0 10 0 1 19 19 2 7 14 3 7 21 4 2 8 5 1 5 6 4 24 Σ = 50 Σ = 91 x − x 1x − x22 1x − x22 f -1.82 3.3124 33.1240 -0.82 0.6724 12.7756 0.18 0.0324 0.2268 1.18 1.3924 9.7468 2.18 4.7524 9.5048 3.18 10.1124 10.1124 4.18 17.4724 69.8896 Σ = 145.38 x = Σxf n = 91 50 = 1.82 ≈ 1.8 Sample mean Use the sum of squares to find the sample standard deviation. s = BΣ1x - x22f n - 1 = A145.38 49 ≈ 1.7 Sample standard deviation So, the sample mean is about 1.8 children, and the sample standard deviation is about 1.7 children. TRY IT YOURSELF 8 Change three of the 6’s in the data set to 4’s. How does this change affect the sample mean and sample standard deviation? Answer: Page A37 EXAMPLE 8 Study Tip Remember that formulas for grouped data require you to multiply by the frequencies.
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