12.5 The Normal Curve 809 Since the curve representing the normal distribution is symmetric, 50% of the data always fall above (to the right of) the mean and 50% of the data fall below (to the left of) the mean. In addition, every normal distribution has approximately 68% of the data between the value that is one standard deviation below the mean, and the value that is one standard deviation above the mean; see Fig. 12.25. Approximately 95% of the data fall between the value that is two standard deviations below the mean and the value that is two standard deviations above the mean. Approximately 99.7% of the data fall between the value that is three standard deviations below the mean and the value that is three standard deviations above the mean. These three percentages, 68%, 95%, and 99.7%, are used in what is referred to as the empirical rule . 68% Mean Three standard deviations below the mean Three standard deviations above the mean Two standard deviations below the mean Two standard deviations above the mean One standard deviation below the mean One standard deviation above the mean Figure 12.25 Thus, if a normal distribution has a mean of 100 and a standard deviation of 10, then approximately 68% of all the data fall between − 100 10 and + 100 10, or between 90 and 110. Approximately 95% of all the data fall between − 100 20 and + 100 20, or between 80 and 120, and approximately 99.7% of all the data fall between − 100 30, and + 100 30, or between 70 and 130. The empirical rule is summarized as follows. Profile in Mathematics David Blackwell (1919–2010) David H. Blackwell (1919–2010), professor of statistics, was the author of more than 90 publications on statistics, probability, game theory, set theory, dynamic programming, and information theory. Blackwell, past president of the American Statistical Society, was the first African American elected to the National Academy of Sciences. When he received his Ph.D. in mathematics from the University of Illinois in 1941, he was only the sixth African American to receive a doctorate in mathematics in the United States. Blackwell taught both at the Institute for Advanced Study at Princeton University and at Howard University. In 1954, he joined the Department of Statistics at the University of California, Berkeley. Blackwell, who taught a wide variety of mathematics courses, said, “Basically, I’m not interested in doing research and I never have been. I’m interested in understanding, which is quite a different thing.” Empirical Rule In any normal distribution, • Approximately 68% of all the data lie within one standard deviation of the mean (in both directions). • Approximately 95% of all the data lie within two standard deviations of the mean (in both directions). • Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions). Suppose we know that the heights of males that are the same age are normally distributed. If we have a random sample of 200 males of the same age, by the empirical rule, we can conclude that approximately 95% of the males in the sample are expected to have a height within two standard deviations of the mean. Since × = × = 95% 200 0.95 200 190, we can expect that approximately 190 males in the sample will have a height within two standard deviations of the mean. Learning Catalytics Keyword: Angel-SOM-12.5 (See Preface for additional details.) z-Scores Now we turn our attention to z-scores. We use z -scores (or standard scores ) to determine how far, in terms of standard deviations, a given data value is from the mean of the distribution. For example, a data value that has a z-score of 1.5 indicates the data value is 1.5 standard deviations above the mean. The z-score or standard score is calculated as follows. Jon Brenneis/The Chronicle Collection/Getty Images
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