What is Chebyshev's inequality?

1 Answer
Jan 21, 2016

Chebyshev’s inequality says that at least #1-1/K^2# of data from a sample must fall within K standard deviations from the mean, where K is any positive real number greater than one.

Explanation:

Let play with a few value of K:

  1. #K = 2# we have #1-1/K^2 = 1 - 1/4 = 3/4 = 75%#. So Chebyshev’s would tell us that 75% of the data values of any distribution must be within two standard deviations of the mean.
  2. #K = 3# we have #1 – 1/K^2 = 1 - 1/9 = 8/9 = 89%#. This time we have 89% of the data values within three standard deviations of the mean.
  3. #K = 4# we have #1 – 1/K^2 = 1 - 1/16 = 15/16 = 93.75%#. Now we have 93.75% of the data within four standard deviations of the mean.

This is consistent to saying that in Normal distribution 68% of the data is one standard deviation from the mean, 95% is two standard deviations from the mean, and approximately 99% is within three standard deviations from the mean. The difference is Chebyshev's theorem extends this principle to any distribution.