Normal Distribution for AP Statistics (page 2)

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By — McGraw-Hill Professional
Updated on Feb 5, 2011

Chebyshev's Rule

The empirical rule works fine as long as the distribution is approximately normal. But what do you do if the shape of the distribution is unknown or distinctly nonnormal (as, say, skewed strongly to the right)? Remember that the empirical rule told you that, in a normal distribution, approximately 68% of the data are within one standard deviation of the mean, approximately 95% are within two standard deviations, and approximately 99.7% are within three standard deviations. Chebyshev's rule isn't as strong as the empirical rule, but it does provide information about the percent of terms contained in an interval about the mean for any distribution.

Let k be a number of standard deviations. Then, according to Chebyshev's rule, for k > 1, at least % of the data lie within k standard deviations of the mean. For example, if k = 2.5, then Chebyshev's rule says that at least % = 84% of the data lie with 2.5 standard deviations of the mean. If k = 3, note the difference between the empirical rule and Chebyshev's rule. The empirical rule says that approximately 99.7% of the data are within three standard deviations of . Chebyshev's says that at least % ≈ 89% of the data are within three standard deviations of . This also illustrates what was said in the previous paragraph about the empirical rule being stronger than Chebyshev's. Note that, if at least % of the data are within k standard deviations of , it follows (algebraically) that at most % lie more than k standard deviations from .

Knowledge of Chebyshev's rule is not required in the AP Exam, but its use is certainly OK and is common enough that it will be recognized by AP readers.

Practice problems for these concepts can be found at:

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