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.

See also

• Vysochanskiï–Petunin inequality, a similar result for the distance from the mean rather than the mode
• Chebyshev's inequality, concerns distance from the mean without requiring unimodality
• Concentration inequality – a summary of tail-bounds on random variables.

References

• Gauss, C. F. (1823). "Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 5.
• Upton, Graham; Cook, Ian (2008). "Gauss inequality". A Dictionary of Statistics. Oxford University Press.
• Sellke, T.M.; Sellke, S.H. (1997). "Chebyshev inequalities for unimodal distributions". American Statistician. American Statistical Association. 51 (1): 34–40. doi:10.2307/2684690. JSTOR 2684690.
• Pukelsheim, F. (1994). "The Three Sigma Rule". American Statistician. American Statistical Association. 48 (2): 88–91. doi:10.2307/2684253. JSTOR 2684253.