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For any normal distribution, about 68% of results will fall between +1 and -1 standard deviations from the mean, and about 95% will fall between +2 and -2 standard deviations. Chebyshev’s Inequality, sometimes called Chebyshev’s Theorem, allows you to extend this idea to any distribution: even if that distribution isn’t normal. The theorem states that for a population or sample, the proportion of observations is no less than (1 – (1 / k2 )), as long as the z score’s absolute value is less than or equal to k. You can only use Chebyshev’s Theorem to get results for standard deviations over 1.
Sample problem: a left-skewed distribution has a mean of 4.99 and a standard deviation of 3.13. Use Chebyshev’s Theorem to calculate the proportion of observations you would expect to find within two standard deviations (in other words, between -2 and +2 standard deviations) from the mean:
Step 1: Square the number of standard deviations:
22 = 4
Step 2: Divide 1 by your answer to Step 1:
1 / 4 = 0.25
Step 3: Subtract Step 2 from 1:
1 – 0.25 = 0.75
At least 75% of the observations fall between -2 and +2 standard deviations from the mean.
Warning: As you may be able to tell, the mean of your distribution has no effect of Chebyshev’s theorem! That fact can cause some wide variations in data, and some inaccurate results.