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| In [[probability theory]], the ''' Vysochanskij–[[Yuri Petunin|Petunin]] inequality ''' gives a lower bound for the [[probability]] that a [[random variable]] with finite [[variance]] lies within a certain number of [[standard deviation]]s of the variable's [[expected value|mean]], or equivalently an upper bound for the probability that it lies further away. The sole restriction on the [[probability distribution|distribution]] is that it be [[unimodal function|unimodal]] and have finite [[variance]]. (This implies that it is a [[continuous probability distribution]] except at the [[mode (statistics)|mode]], which may have a non-zero probability.)
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| The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle."
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| == Theorem ==
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| Let ''X'' be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ<sup>2</sup>. Then, for any λ > √(8/3) = 1.63299…,
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| :<math>P(\left|X-\mu\right|\geq \lambda\sigma)\leq\frac{4}{9\lambda^2}.</math>
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| Furthermore, the equality is attained for a random variable having a probability 1 − 4/(3 λ<sup>2</sup>) 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 λ is less than √(8/3), there exist non-symmetric distributions for which the 4/(9 λ<sup>2</sup>) bound is exceeded.
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| == Properties ==
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| The theorem refines [[Chebyshev's inequality]] by including the factor of 4/9, made possible by the condition that the distribution be unimodal.
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| It is common, in the construction of [[control chart]]s 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. 99.73%) of the values of a process output. Without unimodality Chebyshev's inequality would give a looser bound of 1/9 = 0.11111….
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| ==See also==
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| *[[Gauss's inequality]], a similar result for the distance from the mode rather than the mean
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| ==References==
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| * {{cite journal |author=D. F. Vysochanskij, Y. I. Petunin |year=1980 |title=Justification of the 3σ rule for unimodal distributions |journal=Theory of Probability and Mathematical Statistics |volume=21 |pages=25–36}}
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| * [http://m.njit.edu/CAMS/Technical_Reports/CAMS02_03/report4.pdf Report (on cancer diagnosis) by Petunin and others stating theorem in English]
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| {{DEFAULTSORT:Vysochanskij-Petunin inequality}}
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| [[Category:Probabilistic inequalities]]
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| [[Category:Statistical inequalities]]
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