Aug 13

Chebyshev’s inequality

by David Harper, CFA, FRM, CIPM

Learning objective: Define, calculate and interpret Chebyshev’s inequality…

Chebyshev’s inequality is useful because, as Gujarati says, “we do not need to know the PDF or PMF [the density function]” to use it. In the forum, blueturtle noticed a typo in the assigned reading. Chebyshev’s inequality should be expressed as either of the following:

For example,

The probability that the random variable X falls at least 3 standard deviations from its mean (expected value) is less than or equal to (1/3)^2 = 1/9. (This is just the upper bound, the probability is likely less than 1/9).
The probability that the random variable X falls at least 4 standard deviations from its mean is less than or equal to (1/4)^2 = 1/16.
If we do not require normality, statements like “we saw 25-standard deviation moves” can be partially resolved; i.e., 25-standard deviation moves are impossible under normality but the upper bound, according to Chebyshev’s (under a non-normal distribution of course), is fully (1/25)^2 or about 0.16%.