Editing Mean (section)

===Statistical location===
{{See also|Average#Statistical location}}
[[File:Comparison mean median mode.svg|thumb|Comparison of the [[arithmetic mean]], [[median]], and [[mode (statistics)|mode]] of two skewed ([[log-normal distribution|log-normal]]) distributions]]
[[File:visualisation mode median mean.svg|thumb|upright|Geometric visualization of the mode, median and mean of an arbitrary probability density function<ref>{{cite web|title=AP Statistics Review - Density Curves and the Normal Distributions|url=http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions?action=purge|access-date=16 March 2015|archive-url=https://web.archive.org/web/20150402183703/http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions?action=purge|archive-date=2 April 2015|url-status=dead}}</ref>]]
In [[descriptive statistics]], the mean may be confused with the [[median]], [[Mode (statistics)|mode]] or [[mid-range]], as any of these may incorrectly be called an "average" (more formally, a measure of [[central tendency]]). The mean of a set of observations is the arithmetic average of the values; however, for [[skewness|skewed distributions]], the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the [[Exponential distribution|exponential]] and [[Poisson distribution|Poisson]] distributions.

====Mean of a probability distribution====
{{Main|Expected value}}
{{See also|Population mean}}
The mean of a [[probability distribution]] is the long-run arithmetic average value of a [[random variable]] having that distribution. If the random variable is denoted by <math>X</math>, then the mean is also known as the [[expected value]] of <math>X</math> (denoted <math>E(X)</math>). For a [[discrete probability distribution]], the mean is given by <math>\textstyle \sum xP(x)</math>, where the sum is taken over all possible values of the random variable and <math>P(x)</math> is the [[probability mass function]]. For a [[continuous probability distribution|continuous distribution]], the mean is <math>\textstyle \int_{-\infty}^{\infty} xf(x)\,dx</math>, where <math>f(x)</math> is the [[probability density function]].<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Population Mean|url=https://mathworld.wolfram.com/PopulationMean.html|access-date=2020-08-21|website=mathworld.wolfram.com|language=en}}</ref> In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the [[Lebesgue integration|Lebesgue integral]] of the random variable with respect to its [[probability measure]]. The mean need not exist or be finite; for some probability distributions the mean is infinite ({{math|+&infin;}} or {{math|−&infin;}}), while for others the mean is [[Undefined (mathematics)|undefined]].