88 CHAPTER 2 Descriptive Statistics Many real-life data sets have distributions that are approximately symmetric and bell-shaped (see figure below). For instance, the distributions of men’s and women’s heights in the United States are approximately symmetric and bell-shaped (see the figures at the left and bottom left). Later in the text, you will study bell-shaped distributions in greater detail. For now, however, the Empirical Rule can help you see how valuable the standard deviation can be as a measure of variation. 13.59% 13.59% 34.13% 34.13% About 95% within 2 standard deviations About 99.7% within 3 standard deviations 2.14% 2.14% 0.13% 0.13% x + s x − 2s x − 3s x −s x + 2s x + 3s x About 68% within 1 standard deviation Bell-Shaped Distribution For data sets with distributions that are approximately symmetric and bell-shaped (see figure above), the standard deviation has these characteristics. 1. About 68% of the data lie within one standard deviation of the mean. 2. About 95% of the data lie within two standard deviations of the mean. 3. About 99.7% of the data lie within three standard deviations of the mean. Empirical Rule (or 68–95–99.7 Rule) Using the Empirical Rule A survey was conducted by the National Center for Health Statistics to find the mean height of women in the United States. For a sample of women ages 20–29, the mean height was 64.1 inches and the standard deviation was 2.6 inches. Estimate the percent of women whose heights are between 58.9 and 64.1 inches. (Adapted from National Center for Health Statistics) SOLUTION The distribution of women’s heights is shown at the left. Because the distribution is bell-shaped, you can use the Empirical Rule. The mean height is 64.1, so when you subtract two standard deviations from the mean height, you get x - 2s = 64.1 - 2(2.6) = 58.9. Because 58.9 is two standard deviations below the mean height, the percent of the heights between 58.9 and 64.1 inches is about 13.59%+ 34.13%= 47.72%. Interpretation So, about 47.72% of women are between 58.9 and 64.1 inches tall. TRY IT YOURSELF 6 Estimate the percent of women ages 20–29 whose heights are between 64.1 inches and 66.7 inches. Answer: Page A37 EXAMPLE 6 Picturing the World A survey was conducted by the National Center for Health Statistics to find the mean height of men in the United States.The histogram shows the distribution of heights for a sample of men ages 20–29. In this group, the mean was 69.2 inches and the standard deviation was 2.8 inches. (Adapted from National Center for Health Statistics) Height (in inches) Relative frequency (in percent) 63 2 4 6 8 10 12 14 16 18 65 67 69 71 73 75 Heights of Men in the U.S. Ages 20–29 Estimate which two heights contain the middle 95% of the data. The height of a twenty-fiveyear-old man is 74 inches. Is this height unusual? Why or why not? x x + s x −s x + 3s x − 3s x + 2s x − 2s 13.59% 34.13% 61.5 64.1 66.7 69.3 71.9 58.9 56.3 Height (in inches) Heights of Women in the U.S. Ages 20–29
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