3-2 Measures of Variation 113 SOLUTION The key to solving this problem is to recognize that 70 and 130 are each exactly 2 standard deviations away from the mean of 100, as shown below: 2 standard deviations = 2s = 2 1152 = 30 Therefore, 2 standard deviations from the mean is 100 - 30 = 70 or 100 + 30 = 130 The empirical rule tells us that about 95% of all values are within 2 standard deviations of the mean, so about 95% of all IQ scores are between 70 and 130. YOUR TURN. Do Exercise 41 “The Empirical Rule.” Another concept helpful in understanding or interpreting a value of a standard deviation is Chebyshev’s theorem. The empirical rule applies only to data sets with bell-shaped distributions, but Chebyshev’s theorem applies to any data set. Unfortunately, results from Chebyshev’s theorem are only approximate. Because the results are lower limits (“at least”), Chebyshev’s theorem has limited usefulness. Chebyshev’s Theorem The proportion of any set of data lying within K standard deviations of the mean is always at least 1 - 1>K2, where K is any positive number greater than 1. For K = 2 and K = 3, we get the following statements: ■ At least 3>4 (or 75%) of all values lie within 2 standard deviations of the mean. ■ At least 8>9 (or 89%) of all values lie within 3 standard deviations of the mean. Comparing Variation in Different Samples or Populations In Section 3-1 we included sample wait time data from “Rock ‘n’ Roller Coaster” and “Tower of Terror,” and those two data sets had the same means, medians, modes, and midranges. However, those two data sets have very different amounts of variation, as illustrated in the following example. EXAMPLE 7 Chebyshev’s Theorem IQ scores have a mean of 100 and a standard deviation of 15. What can we conclude from Chebyshev’s theorem? SOLUTION Applying Chebyshev’s theorem with a mean of 100 and a standard deviation of 15, we can reach the following conclusions: ■ At least 3>4 (or 75%) of IQ scores are within 2 standard deviations of the mean (between 70 and 130). ■ At least 8>9 (or 89%) of all IQ scores are within 3 standard deviations of the mean (between 55 and 145). YOUR TURN. Do Exercise 43 “Chebyshev’s Theorem.”
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