Vysochanskiï-Petunin inequality
In probability theory, the Vysochanskiï-Petunin inequality gives a lower bound for the probability that arandom 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. 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 μ and finite variance σ2. Then, for any sufficiently large positive number k:
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 k = 3, corresponding to an upper probability bound of 0.062, and to construct 3-sigma limits to bound nearly all of the values of a process output.
Reference
- Vysochanskiï, D F & Petunin, Y I (1980) Justification of the 3σ rule for unimodal distributions, Theory of Probability and Mathematical Statistics vol. 21 pp25-36
Referenced By
List of probability topics | List of statistical topics | ProbabilityApplications | Probability Applications
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