Gauss's inequality

In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode.

Let X be a unimodal random variable with mode m, and let τ ² be the expected value of (X − m)². (τ ² can also be expressed as (μ − m)² + σ ², where μ and σ are the mean and standard deviation of X.) Then for any positive value of k,

${\displaystyle \Pr(|X-m|>k)\leq {\begin{cases}\left({\frac {2\tau }{3k}}\right)^{2}&{\text{if }}k\geq {\frac {2\tau }{\sqrt {3}}}\\[6pt]1-{\frac {k}{\tau {\sqrt {3}}}}&{\text{if }}0\leq k\leq {\frac {2\tau }{\sqrt {3}}}.\end{cases}}}$

The theorem was first proved by Carl Friedrich Gauss in 1823.