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