Editing Median (section)

==Probability distributions==
For any [[real number|real]]-valued [[probability distribution]] with [[cumulative distribution function]]&nbsp;''F'', a median is defined as any real number&nbsp;''m'' that satisfies the inequalities
<math display="block">
\lim_{x\to m-} F(x) \leq \frac{1}{2} \leq F(m)
</math>
(cf. the [[Expected value#Uhl2023Bild1|drawing]] in the [[Expected value#Arbitrary real-valued random variables|definition of expected value for arbitrary real-valued random variables]]). An equivalent phrasing uses a random variable ''X'' distributed according to ''F'':
<math display="block">
\operatorname{P}(X\leq m) \geq \frac{1}{2}\text{ and } \operatorname{P}(X\geq m) \geq \frac{1}{2}\,.
</math>

[[File:visualisation mode median mean.svg|thumb|upright|[[Mode (statistics)|Mode]], median and mean ([[expected value]]) of a 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|access-date=16 March 2015| archive-url=https://web.archive.org/web/20150408230922/https://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions |archive-date=8 April 2015}}</ref>]]
Note that this definition does not require ''X'' to have an [[absolute continuity|absolutely continuous distribution]] (which has a [[probability density function]] ''f''), nor does it require a [[discrete distribution|discrete one]]. In the former case, the inequalities can be upgraded to equality: a median satisfies
<math display="block">\operatorname{P}(X \leq m) = \int_{-\infty}^m{f(x)\, dx} = \frac{1}{2}</math>
and
<math display="block">\operatorname{P}(X \geq m) = \int_m^{\infty}{f(x)\, dx} = \frac{1}{2}\,.</math>

Any [[probability distribution]] on the real number set <math>\R</math> has at least one median, but in pathological cases there may be more than one median: if ''F'' is constant 1/2 on an interval (so that ''f'' = 0 there), then any value of that interval is a median.

===Medians of particular distributions===	
The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking a well-defined mean, such as the [[Cauchy distribution]]:
* The median of a symmetric [[unimodal distribution]] coincides with the mode.
* The median of a [[symmetric distribution]] which possesses a mean ''μ'' also takes the value ''μ''.
** The median of a [[normal distribution]] with mean ''μ'' and variance ''σ''<sup>2</sup> is&nbsp;μ. In fact, for a normal distribution, mean = median = mode.
** The median of a [[uniform distribution (continuous)|uniform distribution]] in the interval [''a'',&nbsp;''b''] is (''a''&nbsp;+&nbsp;''b'')&nbsp;/&nbsp;2, which is also the mean.
* The median of a [[Cauchy distribution]] with location parameter ''x''<sub>0</sub> and scale parameter ''y'' is&nbsp;''x''<sub>0</sub>, the location parameter.
* The median of a [[Power law|power law distribution]] ''x''<sup>−''a''</sup>, with exponent ''a''&nbsp;>&nbsp;1 is 2<sup>1/(''a''&nbsp;−&nbsp;1)</sup>''x''<sub>min</sub>, where ''x''<sub>min</sub> is the minimum value for which the power law holds<ref>{{cite journal | arxiv=cond-mat/0412004 | doi=10.1080/00107510500052444 | title=Power laws, Pareto distributions and Zipf's law | year=2005 | last1=Newman | first1=M. E. J. | journal=Contemporary Physics | volume=46 | issue=5 | pages=323–351 | bibcode=2005ConPh..46..323N | s2cid=2871747 }}</ref>
* The median of an [[exponential distribution]] with [[rate parameter]] ''λ'' is the [[natural logarithm]] of 2 divided by the rate parameter: ''λ''<sup>−1</sup>ln&nbsp;2.
* The median of a [[Weibull distribution]] with shape parameter ''k'' and scale parameter ''λ'' is&nbsp;''λ''(ln&nbsp;2)<sup>1/''k''</sup>.