SECTION 2.4 Measures of Variation 89 The Empirical Rule applies only to (symmetric) bell-shaped distributions. What if the distribution is not bell-shaped, or what if the shape of the distribution is not known? The next theorem gives an inequality statement that applies to all distributions. It is named after the Russian statistician Pafnuti Chebychev (1821–1894). The portion of any data set lying within k standard deviations (k 7 1) of the mean is at least 1 - 1 k2 . • k = 2: In any data set, at least 1 - 1 22 = 3 4 , or 75%, of the data lie within 2 standard deviations of the mean. • k = 3: In any data set, at least 1 - 1 32 = 8 9 , or about 88.9%, of the data lie within 3 standard deviations of the mean. Chebychev’s Theorem Using Chebychev’s Theorem The age distributions for Georgia and Iowa are shown in the histograms. Apply Chebychev’s Theorem to the data for Georgia using k = 2. What can you conclude? Is an age of 90 unusual for a Georgia resident? Explain. (Source: Based on U.S. Census Bureau) Age (in years) Population (in thousands) 300 600 1200 1500 900 5 1525354555657585 ≈ 38.2 ≈ 22.6 μ σ Georgia Age (in years) Population (in thousands) 100 200 300 400 500 ≈ 39.8 ≈ 23.7 μ σ 5 1525354555657585 Iowa SOLUTION The histogram on the left shows Georgia’s age distribution. Moving two standard deviations to the left of the mean puts you below 0, because m - 2s ≈ 38.2 - 2(22.6) = -7.0. Moving two standard deviations to the right of the mean puts you at m + 2s ≈ 38.2 + 2(22.6) = 83.4. By Chebychev’s Theorem, you can say that at least 75% of the population of Georgia is between birth (0) and 83.4 years old. Also, because 90 7 83.4, an age of 90 lies more than two standard deviations from the mean. So, this age is unusual. TRY IT YOURSELF 7 Apply Chebychev’s Theorem to the data for Iowa using k = 2. What can you conclude? Is an age of 80 unusual for an Iowa resident? Explain. Answer: Page A37 EXAMPLE 7 Study Tip In Example 7, Chebychev’s Theorem gives you an inequality statement that says at least 75% of the population of Georgia is under the age of 83.4. This is a true statement, but it is not nearly as strong a statement as could be made from reading the histogram. In general, Chebychev’s Theorem gives the minimum percent of data entries that fall within the given number of standard deviations of the mean. Depending on the distribution, there is probably a higher percent of data falling in the given range.
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