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}](../I/m/c65749fbe42da2061054c3aecfcc5c7a.png)
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.