“How Deviant Can You Be?”, Paul A. Samuelson1968 (order statistics, probability theory)⁠:

For a finite universe of N items, it is proved no one can lie more than √(N − 1) standard deviations away from the mean.

This is an improvement over the result given by Chebyshev’s inequality: and a similar improvement is possible when speaking of how far from the mean any odd-number r out of N observations can lie. However, the relative inefficiency of Chebyshev’s inequality as applied to a finite universe does go to 0 as N goes to infinity.

See Also:

• The Limits of a Measure of Skewness