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,

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

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

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.