# What is Chebyshev's inequality?

Jan 21, 2016

Chebyshev’s inequality says that at least $1 - \frac{1}{K} ^ 2$ 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.