Vysochanskiï-Petunin inequality

In probability theory, the  Vysochanskij-Petunin inequality  gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean. The sole restriction the distribution is that it be unimodal and have finite variance. (This implies that it is a continuous probability distribution except at the mode, which may have a non-zero probability.) The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle".

Theorem. Let X be a random variable with unimodal distribution, mean &mu; and finite, non-zero variance &sigma;2. Then, for any &lambda; > &radic;(8/3) = 1.63299...


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

Furthermore, the limit is attained (that is, the probability is equal to 4/(9 λ2)) for a random variable having a probability 1 &minus; 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 centred on the mean. When &lambda; is less than &radic;(8/3), there exist unsymmetric distributions for which the 4/(9 λ2) limit is exceeded.

The theorem refines Chebyshëv'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 &lambda; = 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.