Editing Median (section)

===Optimality property===
The ''[[mean absolute error]]'' of a real variable ''c'' with respect to the [[random variable]]&nbsp;''X'' is
<math display="block">\operatorname{E}\left[\left|X-c\right|\right]</math>
Provided that the probability distribution of ''X'' is such that the above expectation exists, then ''m'' is a median of  ''X'' if and only if ''m'' is a minimizer of the mean absolute error with respect to ''X''.<ref>{{cite book |last=Stroock |first=Daniel |title=Probability Theory |url=https://archive.org/details/probabilitytheor00stro |url-access=limited |year=2011 |publisher=Cambridge University Press |isbn=978-0-521-13250-3 |pages=[https://archive.org/details/probabilitytheor00stro/page/n66 43] }}</ref> In particular, if ''m'' is a sample median, then it minimizes the arithmetic mean of the absolute deviations.<ref>{{cite book | last = DeGroot | first = Morris H. | mr = 0356303 | page = 232 | publisher = McGraw-Hill Book Co., New York-London-Sydney | title = Optimal Statistical Decisions | url = https://books.google.com/books?id=7rDY2_r4bmEC&pg=PA232 | year = 1970 | isbn = 9780471680291 }}</ref> Note, however, that in cases where the sample contains an even number of elements, this minimizer is not unique.

More generally, a median is defined as a minimum of
<math display="block">\operatorname{E}\left[\left|X - c\right| - \left|X\right|\right],</math>
as discussed below in the section on [[multivariate median]]s (specifically, the [[spatial median]]).

This optimization-based definition of the median is useful in statistical data-analysis, for example, in [[k-medians clustering|''k''-medians clustering]].