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5.9 The 689599.7 Rule for Normal Distributions According to the 689599.7 rule, in any normal distribution: • About 68% of the observations fall within one standard deviation of the mean. • About 95% of the observations fall within two standard deviations of the mean. • About 99.7% of the observations fall within three standard deviations of the mean. By remembering these three numbers, you can think about normal distributions without making detailed calculations. The figure below illustrates the 689599.7 rule. Note that the Greek letter μ is used for the mean, and the Greek letter σ is used for the standard deviation. Example: Suppose the mean (μ) of a normal distribution is 70, and the standard deviation (σ) is 10. Determine the range of values that fall in the middle 68%, the middle 95%, and the middle 99.7%. 1 Example 2: The heights of American women 1824 years old fit the normal distribution with a mean (μ) of 64.5" and a standard deviation (σ) of 2.5". a) In what range of heights do the middle 68% of American women's heights fall? b) In what range do the middle 95% fall? c) What percent of American women are taller than 69.5"? d) What percent of American women are shorter than 62"? Example 3: The distribution of scores on the SAT college entrance exam is close to normal, with a mean μ = 500 and a standard deviation σ = 100. a) How high must a student score to fall in the top 25%? b) What percent of scores fall between 200 and 800? c) What percent of scores are above 700? Homework: p. 205: 45-48 (Use handout to help.) 2