Vysochanskij–Petunin inequality

Let X be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ². Then, for any ${\textstyle \lambda >{\sqrt {\frac {8}{3}}}=1.63299...,}$

${\displaystyle P(\left|X-\mu \right|\geq \lambda \sigma )\leq {\frac {4}{9\lambda ^{2}}}.}$

(For a relatively elementary proof see e.g. [1]). Furthermore, the equality is attained for a random variable having a probability 1 − 4/(3 λ²) of being exactly equal to the mean, and which, when it is not equal to the mean, is distributed uniformly in an interval centered on the mean. When ${\textstyle \lambda <{\sqrt {\frac {8}{3}}},}$ there exist non-symmetric distributions for which the ${\textstyle {\frac {4}{9\lambda ^{2}}}}$ bound is exceeded.

The theorem refines Chebyshev's inequality by including the factor of 4/9, made possible by the condition that the distribution be unimodal.

It is common, in the construction of control charts and other statistical heuristics, to set λ = 3, corresponding to an upper probability bound of 4/81= 0.04938..., and to construct 3-sigma limits to bound nearly all (i.e. 95%) of the values of a process output. Without unimodality Chebyshev's inequality would give a looser bound of 1/9 = 0.11111....

An improved version of the Vysochanskij-Petunin inequality for one-sided tail bounds exists. For a unimodal random variable ${\displaystyle X}$ with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$, and ${\displaystyle r\geq 0}$, the one-sided Vysochanskij-Petunin inequality[2] holds as follows:

${\displaystyle \mathbb {P} (X-\mu \geq r)\leq {\begin{cases}{\dfrac {4}{9}}{\dfrac {\sigma ^{2}}{r^{2}+\sigma ^{2}}}&{\mbox{for }}r^{2}\geq {\dfrac {5}{3}}\sigma ^{2},\\{\dfrac {4}{3}}{\dfrac {\sigma ^{2}}{r^{2}+\sigma ^{2}}}-{\dfrac {1}{3}}&{\mbox{otherwise.}}\end{cases}}}$

The one-sided Vysochanskij-Petunin inequality, as well as the related Cantelli inequality, can for instance be relevant in the financial area, in the sense of "how bad can losses get".

• Gauss's inequality, a similar result for the distance from the mode rather than the mean
• Rule of three (statistics), a similar result for the Bernoulli distribution

• D. F. Vysochanskij, Y. I. Petunin (1980). "Justification of the 3σ rule for unimodal distributions". Theory of Probability and Mathematical Statistics. 21: 25–36.
• Report (on cancer diagnosis) by Petunin and others stating theorem in English