Vysochanskiï-Petunin inequality

In probability theory, the  Vysochanski&iuml;-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 on the random variable is that the distribution be unimodal (and the random variable continuous). 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 variance &sigma;2. Then, for any &lambda; > &radic;(8/3) = 1.63299...
 * $$P(\left|X-\mu\right|\geq \lambda\sigma)\leq\frac{4}{9\lambda^2}.$$

The theorem refines Chebyshev's inequality by imposing 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 and a continuous random variable, Chebyshev's inequality would give a looser bound of 1/9=0.11111...

Reference

 * Vysochanski&iuml;, D F & Petunin, Y I (1980) Justification of the 3&sigma; rule for unimodal distributions, Theory of Probability and Mathematical Statistics vol. 21 pp25-36