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Added a note that the distribution symmetry must not be determined by using only the skewness of the distribution. |
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In [[probability theory]] and [[statistics]], '''skewness''' is a measure of the asymmetry of the [[probability distribution]] of a [[real number|real]]-valued [[random variable]] about its mean. The skewness value can be positive, zero, negative, or undefined.
For a [[unimodal]] distribution (a distribution with a single peak), negative skew commonly indicates that the ''tail'' is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value in skewness means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution
== Introduction ==
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== Relationship of mean and median ==
The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, and likewise for positive skew.<ref name="von Hippel 2005">{{cite journal |last=von Hippel |first=Paul T. |year=2005 |title=Mean, Median, and Skew: Correcting a Textbook Rule |url=http://www.amstat.org/publications/jse/v13n2/vonhippel.html |url-status=dead |journal=Journal of Statistics Education |volume=13 |issue=2 |archive-url=https://web.archive.org/web/20160220181456/http://www.amstat.org/publications/jse/v13n2/vonhippel.html |archive-date=2016-02-20}}</ref>
[[File:Relationship between mean and median under different skewness.png|thumb|434x434px|A general relationship of mean and median under differently skewed unimodal distribution.]]
In the older notion of [[nonparametric skew]], defined as <math>(\mu - \nu)/\sigma,</math> where <math>\mu</math> is the [[mean]], <math>\nu</math> is the [[median]], and <math>\sigma</math> is the [[standard deviation]], the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than (to the right of) the median, while negative/left nonparametric skew means the mean is less than (to the left of) the median. However, the modern definition of skewness and the traditional nonparametric definition do not always have the same sign: while they agree for some families of distributions, they differ in some of the cases, and conflating them is misleading.
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