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{{About|the statistical concept}}
[[File:Finding the median.png|thumb|Finding the median in sets of data with an odd and even number of values]]
The '''median''' is the value separating the higher half from the lower half of a data [[Sample (statistics)|sample]] (a [[statistical population|population]] or a [[probability distribution]]). For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a [[continuous probability distribution]], the median is the value such that a number is equally likely to fall above or below it.

The median is a commonly used measure of the properties of a data set in [[statistics]] and [[probability theory]]. The basic advantage of the median in describing data compared to the [[Arithmetic mean|mean]] (often simply described as the "average") is that it is not [[Skewness|skewed]] so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. [[Median income]], for example, may be a better way to suggest what a "typical" income is.

Because of this, the median is of central importance<!-- pun not entirely unintended --> in [[robust statistics]], as it is the most [[resistant statistic]], having a [[breakdown point]] of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.

==Finite set of numbers==
The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest.

If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers

: 1, 3, 3, 6, 7, 8, 9

This list contains seven numbers. The median is the fourth of them, which is 6.

If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the [[arithmetic mean|mean]] of the two middle values.<ref name="StatisticalMedian">{{MathWorld |urlname=StatisticalMedian |title=Statistical Median }}</ref><ref>Simon, Laura J.; [http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm "Descriptive statistics"] {{webarchive|url=https://web.archive.org/web/20100730032416/http://www.stat.psu.edu/old_resources/ClassNotes/ljs_07/sld008.htm |date=2010-07-30 }}, ''Statistical Education Resource Kit'', Pennsylvania State Department of Statistics</ref> For example, in the data set

: 1, 2, 3, 4, 5, 6, 8, 9

the median is the mean of the middle two numbers: this is <math>(4 + 5)/2</math>, which is <math>4.5</math>. (In more technical terms, this interprets the median as the fully [[trimmed estimator|trimmed]] [[mid-range]]).

The formula used to find the index of the middle number of a data set of ''n'' numerically ordered numbers is <math>(n + 1)/2</math>. This either gives the middle number (for an odd number of values) or the halfway point between the two middle values. For example, with 14 values, the formula will give an index of 7.5, and the median will be taken by averaging the seventh (the floor of this index) and eighth (the ceiling of this index) values. So the median can be represented by the following formula:

: <math>\mathrm{median}(a) = \frac{a_{\lfloor(\#x+1) \div 2\rfloor} + a_{\lceil(\#x+1) \div 2\rceil}}{2}</math>

{| class="wikitable"
|+ Comparison of common [[average]]s of values { 1, 2, 2, 3, 4, 7, 9 }
! Type
! Description
! Example
! Result
|-
| align="center" | [[Arithmetic mean]]
| Sum of values of a data set divided by number of values: <math>\scriptstyle\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i</math>
| align="center" | (1&nbsp;+&nbsp;2&nbsp;+&nbsp;2&nbsp;+&nbsp;3&nbsp;+&nbsp;4&nbsp;+&nbsp;7&nbsp;+&nbsp;9)&nbsp;/&nbsp;7
| align="center" | '''4'''
|-
| align="center" | Median
| Middle value separating the greater and lesser halves of a data set
| align="center" | 1, 2, 2, '''3''', 4, 7, 9
| align="center" | '''3'''
|-
| align="center" | [[Mode (statistics)|Mode]]
| Most frequent value in a data set
| align="center" | 1, '''2''', '''2''', 3, 4, 7, 9
| align="center" | '''2'''
|}

One can find the median using the [[stem-and-leaf display|Stem-and-Leaf Plot]].

There is no widely accepted standard notation for the median, but some authors represent the median of a variable ''x'' either as ''x͂'' or as ''μ''<sub>1/2</sub><ref name="StatisticalMedian" /> sometimes also ''M''.<ref name="Sheskin2003">{{cite book|author=David J. Sheskin|title=Handbook of Parametric and Nonparametric Statistical Procedures: Third Edition|url=https://books.google.com/books?id=bmwhcJqq01cC&pg=PA7|accessdate=25 February 2013|date=27 August 2003|publisher=CRC Press|isbn=978-1-4200-3626-8|pages=7–}}</ref><ref name="Bissell1994">{{cite book|author=Derek Bissell|title=Statistical Methods for Spc and Tqm|url=https://books.google.com/books?id=cTwwtyBX7PAC&pg=PA26|accessdate=25 February 2013|year=1994|publisher=CRC Press|isbn=978-0-412-39440-9|pages=26–}}</ref> In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced.

The median is used primarily for [[skewness|skewed]] distributions, which it summarizes differently from the [[arithmetic mean]]. Consider the [[multiset]] { 1, 2, 2, 2, 3, 14 }. The median is 2 in this case, (as is the [[mode (statistics)|mode]]), and it might be seen as a better indication of [[central tendency]] (less susceptible to the exceptionally large value in data) than the [[arithmetic mean]] of 4.

The median is a popular [[summary statistic]] used in [[descriptive statistics]], since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of [[outlier]] values than is the [[mean]]. The widely cited empirical relationship between the relative locations of the mean and the median for skewed distributions is, however, not generally true.<ref>{{cite web|url=http://www.amstat.org/publications/jse/v13n2/vonhippel.html|title=Journal of Statistics Education, v13n2: Paul T. von Hippel|work=amstat.org}}</ref> There are, however, various relationships for the ''absolute'' difference between them; see below.

With an even number of observations (as shown above) no value need be exactly at the value of the median. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the [[medoid]].

In a population, at most half have values strictly less than the median and at most half have values strictly greater than it. If each set contains less than half the population, then some of the population is exactly equal to the median. For example, if ''a''&nbsp;<&nbsp;''b''&nbsp;<&nbsp;''c'', then the median of the list {''a'',&nbsp;''b'',&nbsp;''c''} is ''b'', and, if ''a''&nbsp;<&nbsp;''b''&nbsp;<&nbsp;''c''&nbsp;<&nbsp;''d'', then the median of the list {''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;''d''} is the mean of ''b'' and ''c''; i.e., it is (''b''&nbsp;+&nbsp;''c'')/2. As a median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.<ref name="Robson">{{cite book|last1=Robson|first1=Colin|title=Experiment, Design and Statistics in Psychology|date=1994|publisher=Penguin|isbn=0-14-017648-9|pages=42–45}}</ref>

The median can be used as a measure of [[location parameter|location]] when a distribution is [[skewness|skewed]], when end-values are not known, or when one requires reduced importance to be attached to [[outlier]]s, e.g., because they may be measurement errors.

A median is only defined on [[Weak ordering|ordered]] one-dimensional data, and is independent of any [[distance metric]]. A [[geometric median]], on the other hand, is defined in any number of dimensions.

The median is one of a number of ways of summarising the typical values associated with members of a statistical population; thus, it is a possible [[location parameter]]. The median is the 2nd [[quartile]], 5th [[decile]], and 50th [[percentile]]. Since the median is the same as the ''second quartile'', its calculation is illustrated in the article on [[quartile]]s. A median can be worked out for ranked but not numerical classes (e.g. working out a median grade when students are graded from A to F), although the result might be halfway between grades if there is an even number of cases.

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the [[Range (statistics)|range]], the [[interquartile range]], the [[mean absolute deviation]], and the [[median absolute deviation]].

For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the [[Efficiency (statistics)|efficiency]] of candidate estimators shows that the sample mean is more statistically efficient than the sample median when data are uncontaminated by data from heavy-tailed distributions or from mixtures of distributions, but less efficient otherwise, and that the efficiency of the sample median is higher than that for a wide range of distributions. More specifically, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean—see [[Efficiency (statistics)#Asymptotic efficiency|asymptotic efficiency]] and references therein.

==Probability distributions==
[[File:visualisation mode median mean.svg|thumb|100px|Geometric visualisation 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|accessdate=16 March 2015}}</ref>]]
For any [[probability distribution]] on the [[real number|real]] line '''R''' with [[cumulative distribution function]]&nbsp;''F'', regardless of whether it is any kind of continuous probability distribution, in particular an [[absolute continuity|absolutely continuous distribution]] (which has a [[probability density function]]), or a discrete probability distribution, a median is by definition any real number&nbsp;''m'' that satisfies the inequalities

:<math>\operatorname{P}(X\leq m) \geq \frac{1}{2}\text{ and }\operatorname{P}(X\geq m) \geq \frac{1}{2}\,\!</math>

or, equivalently, the inequalities

:<math>\int_{(-\infty,m]} dF(x) \geq \frac{1}{2}\text{ and }\int_{[m,\infty)} dF(x) \geq \frac{1}{2}\,\!</math>

in which a [[Lebesgue–Stieltjes integral]] is used. For an absolutely continuous probability distribution with [[probability density function]] ''ƒ'', the median satisfies

:<math>\operatorname{P}(X\geq m)=\operatorname{P}(X\leq m) = \int_{-\infty}^m f(x)\, dx=\frac{1}{2}.\,\!</math>

Any [[probability distribution]] on '''R''' has at least one median, but in specific cases there may be more than one median. Specifically, if a probability density is zero on an interval [''a'',&nbsp;''b''], and the [[cumulative distribution function]] at ''a'' is 1/2, any value between ''a'' and ''b'' will also be 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>[https://arxiv.org/pdf/cond-mat/0412004.pdf Newman, Mark EJ. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46.5 (2005): 323–351.]</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>.

==Populations==

===Optimality property===
The ''mean absolute error'' of a real variable ''c'' with respect to the [[random variable]]&nbsp;''X'' is
:<math>E(\left|X-c\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 |year=2011 |publisher=Cambridge University Press |isbn=978-0-521-13250-3 |pages=43 }}</ref> In particular, ''m'' is a sample median if and only if ''m'' minimizes the arithmetic mean of the absolute deviations.

More generally, a median is defined as a minimum of
:<math>E(|X-c| - |X| ),</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]]<!-- The "k" in that article title needs to be in LOWER CASE because it's mathematical notation. -->.

===Unimodal distributions===
[[File:Comparison mean median mode.svg|thumb|300px|Comparison of [[mean]], median and [[mode (statistics)|mode]] of two [[log-normal distribution]]s with different [[skewness]].]]

It can be shown for a [[Unimodality|unimodal]] distribution that the median <math>\tilde{X}</math> and the mean <math>\bar{X}</math> lie within (3/5)<sup>1/2</sup> ≈ 0.7746 standard deviations of each other.<ref name="unimodal">{{Cite journal|title=The Mean, Median, and Mode of Unimodal Distributions:A Characterization|journal=Theory of Probability and Its Applications|volume=41|issue=2|pages=210–223|doi=10.1137/S0040585X97975447|year = 1997|last1 = Basu|first1 = S.|last2=Dasgupta|first2=A.}}</ref> In symbols,

: <math>\frac{\left|\tilde{X} - \bar{X}\right|}{\sigma} \le \left(\frac{3}{5}\right)^{1/2}</math>

where |·| is the absolute value.

A similar relation holds between the median and the mode: they lie within 3<sup>1/2</sup> ≈ 1.732 standard deviations of each other:

: <math>\frac{|\tilde{X} - \mathrm{mode}|}{\sigma} \le 3^{1/2}.</math>

===Inequality relating means and medians===
If the distribution has finite variance, then the distance between the median and the mean is bounded by one [[standard deviation]].

This bound was proved by Mallows,<ref>{{cite journal |last=Mallows |first=Colin |title=Another comment on O'Cinneide |journal=The American Statistician |date=August 1991 |volume=45 |issue=3 |pages=257 | doi = 10.1080/00031305.1991.10475815}}</ref> who used [[Jensen's inequality]] twice, as follows. We have

: <math>\begin{align}
  |\mu - m| = |\operatorname{E}(X - m)| & \leq \operatorname{E}(|X - m|) \\
                                        & \leq \operatorname{E}(|X - \mu|) \\
                                        & \leq \sqrt{\operatorname{E}\left((X - \mu)^2\right)} = \sigma.
\end{align}</math>

The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex.  The second inequality comes from the fact that a median minimizes the [[absolute deviation]] function

:<math>a \mapsto \operatorname{E}(|X-a|).\,</math>

This proof also follows directly from [[Cantelli's inequality]].<ref>[http://www.montefiore.ulg.ac.be/~kvansteen/MATH0008-2/ac20112012/Class3/Chapter2_ac1112_vfinalPartII.pdf K.Van Steen ''Notes on probability and statistics'' ]</ref>
The result can be generalized to obtain a multivariate version of the inequality,<ref name=PicheRandomVectorsSequences>{{cite book|last=Piché|first=Robert|title=Random Vectors and Random Sequences|year=2012|publisher=Lambert Academic Publishing|isbn=978-3659211966}}</ref> as follows:
:  <math>\begin{align}
  \|\mu - m\| = \| \operatorname{E}(X - m)\|
    & \leq  \operatorname{E}\|X - m\|  \\
    & \leq  \operatorname{E}(\| X - \mu\| ) \\
    & \leq \sqrt{ \operatorname{E}\left(\|X - \mu\|^2\right) }
      = \sqrt{ \operatorname{trace}\left(\operatorname{var}(X)\right) }
\end{align}</math>

where ''m'' is a [[spatial median]], that is, a minimizer of the function
<math>a \mapsto \operatorname{E}(\|X-a\|).\,</math> The spatial median is unique when the data-set's dimension is two or more.<ref name="Kemperman">{{cite journal |first=Johannes H. B. |last=Kemperman |title=The median of a finite measure on a Banach space: Statistical data analysis based on the L1-norm and related methods |journal=Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987 |editor-first=Yadolah |editor-last=Dodge |publisher=North-Holland Publishing Co. |location=Amsterdam |pages=217–230 |mr=949228 |ref=harv |year=1987 }}</ref><ref name="MilasevicDucharme">{{cite journal |first=Philip |last=Milasevic |first2=Gilles R. |last2=Ducharme |title=Uniqueness of the spatial median |journal=[[Annals of Statistics]] |volume=15 |year=1987 |number=3 |pages=1332–1333 |mr=902264 |ref=harv |doi=10.1214/aos/1176350511}}</ref> An alternative proof uses the one-sided Chebyshev inequality; it appears in [[An inequality on location and scale parameters#An application: distance between the mean and the median|an inequality on location and scale parameters]].

==Jensen's inequality for medians==

Jensen's inequality states that for any random variable ''x'' with a finite expectation ''E''(''x'') and for any convex function ''f''

: <math> f[ E(x) ] \le E[ f(x) ] </math>

It has been shown<ref name=Merkle2005>{{cite journal |last=Merkle |first=M. |year=2005 |title=Jensen’s inequality for medians |journal=Statistics & Probability Letters |volume=71 |issue=3 |pages=277–281 |doi=10.1016/j.spl.2004.11.010 }}</ref> that if ''x'' is a real variable with a unique median ''m'' and ''f'' is a C function then

: <math> f(m) \le \operatorname{Median}[ f(x)] </math>

A C function is a real valued function, defined on the set of real numbers ''R'', with the property that for any real ''t''

: <math> f^{-1}\left( \,(-\infty, t]\, \right) = \{ x \in R \mid f(x) \le t \} </math>

is a closed [[interval (mathematics)|interval]], a [[singleton (mathematics)|singleton]] or an [[empty set]].

==Medians for samples==

===The sample median===

===={{anchor|Ninther}} {{anchor|Remedian}} Efficient computation of the sample median====
Even though [[sorting algorithm|comparison-sorting]] ''n'' items requires {{math|[[Big O notation|Ω]](''n'' log ''n'')}} operations, [[selection algorithm]]s can compute the [[order statistic|{{mvar|k}}'th-smallest of {{mvar|n}} items]] with only {{math|Θ(''n'')}} operations. This includes the median, which is the {{math|{{sfrac|''n''|2}}}}'th order statistic (or for an even number of samples, the [[arithmetic mean]] of the two middle order statistics).

Selection algorithms still have the downside of requiring {{math|Ω(''n'')}} memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates the median as the median of a three-element subsample; this is commonly used as a subroutine in the [[quicksort]] sorting algorithm, which uses an estimate of its input's median. A more [[robust estimator]] is [[John Tukey|Tukey]]'s ''ninther'', which is the median of three rule applied with limited recursion:<ref>{{cite journal |first1=Jon L. |last1=Bentley |first2=M. Douglas |last2=McIlroy |title=Engineering a sort function |journal=Software—Practice and Experience |volume=23 |issue=11 |pages=1249–1265 |year=1993 |url=http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.14.8162 |doi=10.1002/spe.4380231105}}</ref> if {{mvar|A}} is the sample laid out as an [[array (data structure)|array]], and

:{{math|med3(''A'') {{=}} median(''A''[1], ''A''[{{sfrac|''n''|2}}], ''A''[''n''])}},

then

:{{math|ninther(''A'') {{=}} med3(med3(''A''[1 ... {{sfrac|1|3}}''n'']), med3(''A''[{{sfrac|1|3}}''n'' ... {{sfrac|2|3}}''n'']), med3(''A''[{{sfrac|2|3}}''n'' ... ''n'']))}}

The ''remedian'' is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.<ref>{{cite journal |last1=Rousseeuw |first1=Peter J. |last2=Bassett |first2=Gilbert W. Jr.|title=The remedian: a robust averaging method for large data sets |journal=J. Amer. Stat. Assoc. |volume=85 |issue=409 |year=1990 |url=http://wis.kuleuven.be/stat/robust/papers/publications-1990/rousseeuwbassett-remedian-jasa-1990.pdf |pages=97–104 |doi=10.1080/01621459.1990.10475311}}</ref>

====Easy explanation of the sample median====
In individual series (if number of observation is very low) first one must arrange all the observations in order. Then count(''n'') is the total number of observation in given data.

If '''''n'' is odd''' then Median (''M'') = value of ((''n''&nbsp;+&nbsp;1)/2)th item term.

If '''''n'' is even''' then Median (''M'') = value of [(''n''/2)th item term + (''n''/2 + 1)th item term]/2

;For an odd number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7.

Start by sorting the values: 1, 2, 5, 7, 8.

In this case, the median is 5 since it is the middle observation in the ordered list.

The median is the ((''n''&nbsp;+&nbsp;1)/2)th item, where ''n'' is the number of values. For example, for the list {1,&nbsp;2,&nbsp;5,&nbsp;7,&nbsp;8}, we have ''n''&nbsp;=&nbsp;5, so the median is the ((5&nbsp;+&nbsp;1)/2)th item.
: median = (6/2)th item
: median = 3rd item
: median = 5

;For an even number of values

As an example, we will calculate the sample median for the following set of observations: 1, 6, 2, 8, 7, 2.

Start by sorting the values: 1, 2, 2, 6, 7, 8.

In this case, the arithmetic mean of the two middlemost terms is (2 + 6)/2 = 4. Therefore, the median is 4 since it is the arithmetic mean of the middle observations in the ordered list.

====Sampling distribution====

The distributions of both the sample mean and the sample median were determined by [[Pierre-Simon Laplace|Laplace]].<ref name=Stigler1973>{{cite journal
 | last = Stigler
 | first = Stephen
 | authorlink = Stephen Stigler
 |date= December 1973
 | title = Studies in the History of Probability and Statistics. XXXII: Laplace, Fisher and the Discovery of the Concept of Sufficiency
 | journal = Biometrika
 | volume = 60
 | issue = 3
 | pages = 439–445
 | doi = 10.1093/biomet/60.3.439
 | mr = 0326872 | jstor = 2334992
 }}</ref> The distribution of the sample median from a population with a density function <math>f(x)</math> is asymptotically normal with mean <math>m</math> and variance<ref name="Rider1960">{{cite journal |last=Rider |first=Paul R. |year=1960 |title=Variance of the median of small samples from several special populations |journal=[[Journal of the American Statistical Association|J. Amer. Statist. Assoc.]] |volume=55 |issue=289 |pages=148–150 |doi=10.1080/01621459.1960.10482056 }}</ref>

: <math> \frac{ 1 }{ 4n f( m )^2 }</math>

where <math>m</math> is the median of <math>f(x)</math> and <math>n</math> is the sample size.
For normal samples, the density is <math>f(m)=1/\sqrt{2\pi\sigma^2}</math>, thus for large samples the variance of the median equals <math>({\pi}/{2})\cdot(\sigma^2/n).</math><ref>{{cite book |last=Williams |first=D. |year=2001 |title=Weighing the Odds |publisher=Cambridge University Press |isbn=052100618X |page=165}}</ref>

These results have also been extended.<ref name="Stuart1994">{{cite book |title=Kendall's Advanced Theory of Statistics |first=Alan |last=Stuart |first2=Keith |last2=Ord |location=London |publisher=Arnold |year=1994 |isbn=0340614307 }}</ref> It is now known for the <math>p</math>-th quantile that the distribution of the sample <math>p</math>-th quantile is asymptotically normal around the <math>p</math>-th quantile with variance equal to
: <math> \frac{p( 1 - p )}{nf(x_p)^2}</math>
where <math>f(x_{p})</math> is the value of the distribution density at the <math>p</math>-th quantile.

; Numerical experimentation
In the case of a discrete variable, the sampling distribution of the median for small-samples can be investigated as follows.  We take the sample size to be an odd number <math> N = 2n + 1 </math>.  If a given value <math> v </math> is to be the median of the sample then two conditions must be satisfied.  The first is that at most <math> n </math> observations can have a value of <math> v - 1 </math> or less.  The second is that at most <math> n </math> observations can have a value of <math> v + 1 </math> or more. Let <math> i </math> be the number of observations that have a value of <math> v - 1 </math> or less and let <math> k </math> be the number of observations that have a value of <math> v + 1 </math> or more.  Then <math> i </math> and <math> k </math> both have a minimum value of 0 and a maximum of <math> n </math>.  If an observation has a value below <math> v </math>, it is not relevant how far below <math> v </math> it is and conversely, if an observation has a value above <math> v </math>, it is not relevant how far above <math> v </math> it is.  We can therefore represent the observations as following a trinomial distribution with probabilities  <math> F(v - 1) </math>, <math> f(v) </math> and  <math> 1 - F(v) </math>.  The probability that the median <math> m </math> will have a value <math> v </math> is then given by

: <math> \Pr(m = v) = \sum_{i=0}^n \sum_{k=0}^n \frac{N!}{i!(N-i-k)!k!} [F(v-1)]^i [f(v)]^{N-i-k} [1 - F(v)]^k. </math>

Summing this over all values of <math> v </math> defines a proper distribution and gives a unit sum.  In practice, the function <math> f(v) </math> will often not be known but it can be estimated from an observed frequency distribution.  An example is given in the following table where the actual distribution is not known but a sample of 3,800 observations allows a sufficiently accurate assessment of <math> f(v) </math>.

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;"
! v !! 0 !! 0.5 !! 1 !! 1.5 !! 2 !! 2.5 !! 3 !! 3.5 !! 4 !! 4.5 !! 5
|-
! f(v)
| 0.000 || 0.008 || 0.010 || 0.013 || 0.083 || 0.108 || 0.328 || 0.220 ||  0.202 || 0.023 || 0.005
|-
! F(v)
| 0.000 || 0.008 || 0.018 || 0.031 || 0.114 || 0.222 || 0.550 || 0.770 ||  0.972 || 0.995 || 1.000
|-
|}

Using these data it is possible to investigate the effect of sample size on the standard errors of the mean and median.  The observed mean is 3.16, the observed raw median is 3 and the observed interpolated median is 3.174.  The following table gives some comparison statistics.  The standard error of the median is given both from the above expression for <math> pr(m = v) </math> and from the asymptotic approximation given earlier.

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;"
! {{Diagonal split header|Statistic|Sample size}} !! 3 !! 9 !! 15 !! 21
|-
! Expected value of median
| 3.198 || 3.191 || 3.174 || 3.161
|-
! Standard error of median (above formula)
| 0.482 || 0.305 || 0.257 || 0.239
|-
! Standard error of median (asymptotic approximation)
| 0.879 || 0.508 || 0.393 || 0.332
|-
! Standard error of mean
| 0.421 || 0.243 || 0.188 || 0.159
|}
The expected value of the median falls slightly as sample size increases while, as would be expected, the standard errors of both the median and the mean are proportionate to the inverse square root of the sample size.  The asymptotic approximation errs on the side of caution by overestimating the standard error.

In the case of a continuous variable, the following argument can be used.  If a given value <math> v </math> is to be the median, then one observation must take the value <math> v </math>.  The elemental probability of this is <math> f(v)\,dv </math>.  Then, of the remaining <math> 2n </math> observations, exactly <math> n </math> of them must be above <math> v </math> and the remaining <math> n </math> below.  The probability of this is the <math> n </math>th term of a binomial distribution with parameters <math> F(v) </math> and <math> 2n </math>.  Finally we multiply by <math> 2n+1 </math> since any of the observations in the sample can be the median observation.  Hence the elemental probability of the median at the point <math> v </math>  is given by

: <math> f(v) \frac{(2n)!}{n!n!} [F(v)]^n [1 - F(v)]^n (2n + 1) \, dv. </math>

Now we introduce the beta function.  For integer arguments <math> \alpha </math> and <math> \beta </math>, this can be expressed as <math> \Beta(\alpha,\beta) = (\alpha - 1)! (\beta - 1)! / (\alpha + \beta - 1)! </math>.  Also, <math> f(v) = dF(v)/dv </math>.  Using these relationships and setting both <math> \alpha </math> and <math> \beta </math> equal to  <math> (n+1) </math> allows the last expression to be written as

: <math> \frac{[F(v)]^n[1 - F(v)]^n}{\Beta(n+1,n+1)} \, dF(v) </math>

Hence the density function of the median is a symmetric beta distribution over the unit interval which supports <math> F(v) </math>.  Its mean, as we would expect, is 0.5 and its variance is  <math> 1/(4(N+2)) </math>. The corresponding variance of the sample median is

: <math>\frac{ 1 }{ 4 (N + 2) f( m )^2 }.</math>

However this finding can only be used if the density function <math> f(v) </math> is known or can be assumed.  As this will not always be the case, the median variance has to be estimated sometimes from the sample data.

;Estimation of variance from sample data

The value of <math>(2 f(x))^{-2}</math>—the asymptotic value of <math>n^{-\frac{1}{2}} (\nu - m)</math> where <math>\nu</math> is the population median—has been studied by several authors. The standard "delete one" [[Resampling (statistics)#Jackknife|jackknife]] method produces [[consistent estimator|inconsistent]] results.<ref name=Efron1982>{{cite book |last=Efron |first=B. |year=1982 |title=The Jackknife, the Bootstrap and other Resampling Plans |publisher=SIAM |location=Philadelphia |isbn=0898711797 }}</ref> An alternative—the "delete k" method—where <math>k</math> grows with the sample size has been shown to be asymptotically consistent.<ref name=Shao1989>{{cite journal |last=Shao |first=J. |last2=Wu |first2=C. F. |year=1989 |title=A General Theory for Jackknife Variance Estimation |journal=[[Annals of Statistics|Ann. Stat.]] |volume=17 |issue=3 |pages=1176–1197 |jstor=2241717 |doi=10.1214/aos/1176347263}}</ref> This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,<ref name=Efron1979>{{cite journal |last=Efron |first=B. |year=1979 |title=Bootstrap Methods: Another Look at the Jackknife |journal=[[Annals of Statistics|Ann. Stat.]] |volume=7 |issue=1 |pages=1–26 |jstor=2958830 |doi=10.1214/aos/1176344552}}</ref> but converges very slowly ([[computational complexity theory|order]] of <math>n^{-\frac{1}{4}}</math>).<ref name=Hall1988>{{cite journal |last=Hall |first=P. |last2=Martin |first2=M. A. |year=1988 |title=Exact Convergence Rate of Bootstrap Quantile Variance Estimator |journal=Probab Theory Related Fields |volume=80 |issue=2 |pages=261–268 |doi=10.1007/BF00356105 }}</ref> Other methods have been proposed but their behavior may differ between large and small samples.<ref name=Jimenez-Gamero2004>{{cite journal |last=Jiménez-Gamero |first=M. D. |last2=Munoz-García |first2=J. |first3=R. |last3=Pino-Mejías |year=2004 |title=Reduced bootstrap for the median |journal=Statistica Sinica |volume=14 |issue=4 |pages=1179–1198 |doi= |url=http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=14&num=4&art=11 }}</ref>

;Efficiency{{anchor|Efficiency}}

The [[Efficiency (statistics)|efficiency]] of the sample median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size and on the underlying population distribution. For a sample of size <math>N = 2n + 1</math> from the [[normal distribution]], the efficiency for large N is

: <math> \frac{2}{\pi} \frac{N+2}{N} </math>

The efficiency tends to <math> \frac{2}{\pi} </math> as <math>N</math> tends to infinity.

In other words, the relative variance of the median will be <math>\pi/2 \approx 1.57</math>, or 57% greater than the variance of the mean – the [[standard error]] of the median will be 25% greater than that of the mean.<ref>[https://books.google.com.br/books?id=8bMj8m-4RDQC&lpg=PA130&dq=standard%20error%20of%20the%20median&pg=PA104#v=onepage&q&f=false]</ref>

===Other estimators===
For univariate distributions that are ''symmetric'' about one median, the [[Hodges–Lehmann estimator]] is a [[robust statistics|robust]] and highly [[Efficiency (statistics)|efficient estimator]] of the population median.<ref name="HM">{{cite book |last1=Hettmansperger |first1=Thomas P. |last2=McKean |first2=Joseph W. |title=Robust nonparametric statistical methods |series=Kendall's Library of Statistics |volume=5 |publisher=Edward Arnold |location=London |year=1998 |isbn=0-340-54937-8 |mr=1604954 |ref=harv }}</ref>

If data are represented by a [[statistical model]] specifying a particular family of [[probability distribution]]s, then estimates of the median can be obtained by fitting that family of probability distributions to the data and calculating the theoretical median of the fitted distribution.{{Citation needed|date=May 2012}} [[Pareto interpolation]] is an application of this when the population is assumed to have a [[Pareto distribution]].

===Coefficient of dispersion===

The coefficient of dispersion (CD) is defined as the ratio of the average absolute deviation from the median to the median of the data.<ref name=Bonett2006>{{cite journal | last1 = Bonett | first1 = DG | last2 = Seier | first2 = E | year = 2006 | title = Confidence interval for a coefficient of dispersion in non-normal distributions | url = | journal = Biometrical Journal | volume = 48 | issue = 1| pages = 144–148 | doi=10.1002/bimj.200410148}}</ref> It is a statistical measure used by the states of [[Iowa]], [[New York (state)|New York]] and [[South Dakota]] in estimating dues taxes.<ref>{{cite web |url=http://www.iowa.gov/tax/locgov/Statistical_Calculation_Definitions.pdf|title=Statistical Calculation Definitions for Mass Appraisal |work=Iowa.gov |quote=Median Ratio: The ratio located midway between the highest ratio and the lowest ratio when individual ratios for a class of realty are ranked in ascending or descending order. The median ratio is most frequently used to determine the level of assessment for a given class of real estate.|archive-url=https://web.archive.org/web/20101111214903/http://iowa.gov/tax/locgov/Statistical_Calculation_Definitions.pdf|archive-date=11 November 2010}}</ref><ref>{{cite web|url=http://www.tax.ny.gov/research/property/reports/cod/2010mvs/reporttext.htm|title=Assessment equity in New York: Results from the 2010 market value survey|archive-url=https://web.archive.org/web/20121106015231/http://www.tax.ny.gov/research/property/reports/cod/2010mvs/reporttext.htm|archive-date=6 November 2012}}</ref><ref>{{cite web|url=http://www.state.sd.us/drr2/publications/assess1199.pdf|title=Summary of the Assessment Process|work=state.sd.us|publisher=South Dakota Department of Revenue - Property/Special Taxes Division|archive-url=https://web.archive.org/web/20090510034115/http://www.state.sd.us/drr2/publications/assess1199.pdf|archive-date=10 May 2009}}</ref> In symbols

: <math> CD = \frac 1 n \frac{ \sum| m - x | }{ m } </math>

where ''n'' is the sample size, ''m'' is the sample median and ''x'' is a variate. The sum is taken over the whole sample.

Confidence intervals for a two-sample test in which the sample sizes are large have been derived by Bonett and Seier.<ref name="Bonett2006"/> This test assumes that both samples have the same median but differ in the dispersion around it. The confidence interval (CI) is bounded inferiorly by

: <math> \exp \left[ \log \left( \frac{t_a} {t_b} \right) - z_\alpha \left( \operatorname{var} \left[ \log \left( \frac{t_a} {t_b} \right) \right] \right)^{1/2} \right] </math>

where ''t''<sub>j</sub> is the mean absolute deviation of the ''j''<sup>th</sup> sample, var() is the variance and ''z<sub>α</sub>'' is the value from the normal distribution for the chosen value of ''α'': for ''α'' = 0.05, ''z<sub>α</sub>'' = 1.96. The following formulae are used in the derivation of these confidence intervals

: <math> \operatorname{var} [ \log ( t_a ) ] = \frac{1}{n} \left[ \frac{s_a^2} {t_a^2} + \left( \frac{x_a - \bar{x}} {t_a} \right)^2 - 1 \right] </math>

: <math> \operatorname{var} \left[ \log\left( \frac{t_a}{t_b} \right) \right] = \operatorname{var}[ \log( t_a ) ] + \operatorname{var}[ \log( t_b )] - 2r (\operatorname{var}[ \log(t_a) ] \operatorname{var}[ \log( t_b ) ] )^{1/2} </math>

where ''r'' is the Pearson correlation coefficient between the squared deviation scores

: <math> d_{ia} = | x_{ia} - \bar{x}_a | </math> and <math> d_{ib} = |x_{ib} - \bar{x}_b | </math>

''a'' and ''b'' here are constants equal to 1 and 2, ''x'' is a variate and ''s'' is the standard deviation of the sample.

==Multivariate median==
Previously, this article discussed the univariate median, when the sample or population had one-dimension. When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one.<ref name="HM" /><ref>Small, Christopher G. "A survey of multidimensional medians." International Statistical Review/Revue Internationale de Statistique (1990): 263–277. {{doi|10.2307/1403809}} {{JSTOR|1403809}}</ref><ref>Niinimaa, A., and H. Oja. "Multivariate median." Encyclopedia of statistical sciences (1999).</ref><ref>Mosler, Karl. Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach. Vol. 165. Springer Science & Business Media, 2012.</ref>

===Marginal median===
The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.<ref name="HM" /><ref>Puri, Madan L.; Sen, Pranab K.; ''Nonparametric Methods in Multivariate Analysis'', John Wiley & Sons, New York, NY, 197l. (Reprinted by Krieger Publishing)</ref>

===Centerpoint===

An alternative generalization of the median in higher dimensions is the [[Centerpoint (geometry)|centerpoint]].

==Other median-related concepts==

===Interpolated median===
When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median <math> m </math> is 3 since the median is the smallest value of <math> x </math> for which <math> F(x) </math> is greater than a half. But the interpolated median is somewhere between 2.50 and 3.50.  First we add half of the interval width <math> w </math> to the median to get the upper bound of the median interval. Then we subtract that proportion of the interval width which equals the proportion of the 33% which lies above the 50% mark.  In other words, we split up the interval width pro rata to the numbers of observations.  In this case, the 33% is split into 28% below the median and 5% above it so we subtract 5/33 of the interval width from the upper bound of 3.50 to give an interpolated median of 3.35. More formally, if the values <math> f(x) </math> are known, the interpolated median can be calculated from

: <math> m_\text{int} = m + w\left[\frac{1}{2} - \frac{F( m ) - \frac{1}{2} }{f( m )}\right]. </math>

Alternatively, if in an observed sample there are <math> k </math> scores above the median category, <math> j </math> scores in it and <math> i </math> scores below it then the interpolated median is given by

: <math> m_\text{int} = m + \frac{w}{2} \left[\frac{k - i} j\right]. </math>

===Pseudo-median===
{{Main|Pseudomedian}}
For univariate distributions that are ''symmetric'' about one median, the [[Hodges–Lehmann estimator]] is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population ''pseudo-median'', which is the median of a symmetrized distribution and which is close to the population median.<ref>{{Cite journal|last=Pratt|first=William K.|last2=Cooper|first2=Ted J.|last3=Kabir|first3=Ihtisham|date=1985-07-11|title=Pseudomedian Filter|journal=Architectures and Algorithms for Digital Image Processing II|volume=0534|pages=34|doi=10.1117/12.946562}}</ref> The Hodges–Lehmann estimator has been generalized to multivariate distributions.<ref name="Oja 2010 xiv+232">{{cite book|title=Multivariate nonparametric methods with&nbsp;''R'': An approach based on spatial signs and ranks|last=Oja|first=Hannu|publisher=Springer|year=2010|isbn=978-1-4419-0467-6|series=Lecture Notes in Statistics|volume=199|location=New York, NY|pages=xiv+232|doi=10.1007/978-1-4419-0468-3|mr=2598854|ref=harv}}</ref>

===Variants of regression===
The [[Theil–Sen estimator]] is a method for [[robust statistics|robust]] [[linear regression]] based on finding medians of [[slope]]s.<ref>{{citation
 | last = Wilcox | first = Rand R.
 | contribution = Theil–Sen estimator
 | isbn = 978-0-387-95157-7
 | pages = 207–210
 | publisher = Springer-Verlag
 | title = Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy
 | url = https://books.google.com/books?id=YSFb4QX2UIoC&pg=PA207
 | year = 2001}}.</ref>

===Median filter===
{{Main|Median filter}}
In the context of [[image processing]] of [[monochrome]] [[raster image]]s there is a type of noise, known as the [[salt and pepper noise]], when each pixel independently becomes black (with some small probability) or white (with some small probability), and is unchanged otherwise (with the probability close to 1). An image constructed of median values of neighborhoods (like 3×3 square) can effectively [[noise reduction|reduce noise]] in this case.{{Citation needed |date=May 2012 }}

===Cluster analysis===
{{main|k-medians clustering}}
In [[cluster analysis]], the [[k-medians clustering]] algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in [[k-means clustering]], is replaced by maximising the distance between cluster-medians.

===Median–median line===

This is a method of robust regression. The idea dates back to [[Abraham Wald|Wald]] in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent parameter <math>x</math>: a left half with values less than the median and a right half with values greater than the median.<ref name=Wald1940>{{cite journal |last=Wald |first=A. |year=1940 |title=The Fitting of Straight Lines if Both Variables are Subject to Error |journal=[[Annals of Mathematical Statistics]] |volume=11 |issue=3 |pages=282–300 |jstor=2235677 |doi=10.1214/aoms/1177731868 |url=http://dml.cz/bitstream/handle/10338.dmlcz/103573/AplMat_20-1975-2_3.pdf }}</ref> He suggested taking the means of the dependent <math>y</math> and independent <math>x</math> variables of the left and the right halves and estimating the slope of the line joining these two points. The line could then be adjusted to fit the majority of the points in the data set.

Nair and Shrivastava in 1942 suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples.<ref name=Nair1942>{{cite journal |title=On a Simple Method of Curve Fitting |first=K. R. |last=Nair |first2=M. P. |last2=Shrivastava |journal=Sankhyā: The Indian Journal of Statistics |volume=6 |issue=2 |year=1942 |pages=121–132 |jstor=25047749 }}</ref> Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.<ref name=Brown1951>{{cite book |last=Brown |first=G. W. |last2=Mood |first2=A. M. |year=1951 |chapter=On Median Tests for Linear Hypotheses |title=Proc Second Berkeley Symposium on Mathematical Statistics and Probability |location=Berkeley, CA |publisher=University of California Press |pages=159–166 |isbn= |zbl=0045.08606 }}</ref> Tukey combined these ideas and recommended dividing the sample into three equal size subsamples and estimating the line based on the medians of the subsamples.<ref name=Tukey1971>{{cite book |last=Tukey |first=J. W. |year=1977 |title=Exploratory Data Analysis |location=Reading, MA |publisher=Addison-Wesley |isbn=0201076160 |url=https://archive.org/details/exploratorydataa00tuke_0 }}</ref>

==Median-unbiased estimators==
{{main|Bias of an estimator#Median-unbiased estimators}}
Any [[Bias of an estimator|''mean''-unbiased estimator]] minimizes the [[risk]] ([[expected loss]]) with respect to the squared-error [[loss function]], as observed by [[Gauss]]. A [[Bias of an estimator#Median unbiased estimators, and bias with respect to other loss functions|''median''-unbiased estimator]] minimizes the risk with respect to the [[Absolute deviation|absolute-deviation]] loss function, as observed by [[Laplace]]. Other [[loss functions]] are used in [[statistical theory]], particularly in [[robust statistics]].

The theory of median-unbiased estimators was revived by [https://web.archive.org/web/20110310043642/http://www.universityofcalifornia.edu/senate/inmemoriam/georgewbrown.htm George W. Brown] in 1947:<ref name="Brown" />

{{quote|An estimate of a one-dimensional parameter θ will be said to be median-unbiased if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.|page 584}}

Further properties of median-unbiased estimators have been reported.<ref name="Lehmann" /><ref name="Birnbaum" /><ref name="vdW" /><ref name="Pf" />  Median-unbiased estimators are invariant under [[Injective function|one-to-one transformations]].

There are methods of constructing median-unbiased estimators that are optimal (in a sense analogous to the minimum-variance property for mean-unbiased estimators). Such constructions exist for probability distributions having [[monotone likelihood ratio|monotone likelihood-functions]].<ref>Pfanzagl, Johann. "On optimal median unbiased estimators in the presence of nuisance parameters." The Annals of Statistics (1979): 187–193.</ref><ref>Brown, L. D.; Cohen, Arthur; Strawderman, W. E. A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications. Ann. Statist. 4 (1976), no. 4, 712–722. {{doi|10.1214/aos/1176343543}}. http://projecteuclid.org/euclid.aos/1176343543.</ref> One such procedure is an analogue of the [[Rao–Blackwell theorem|Rao–Blackwell procedure]] for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of [[loss function]]s.<ref>Page 713: Brown, L. D.; Cohen, Arthur; Strawderman, W. E. A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications. Ann. Statist. 4 (1976), no. 4, 712–722. {{doi|10.1214/aos/1176343543}}. http://projecteuclid.org/euclid.aos/1176343543.</ref>

==History==

The idea of the median appeared in the 13th century in the [[Talmud]] <ref>[http://danadler.com/blog/2014/12/31/talmud-and-modern-economics/ Talmud and Modern Economics]</ref><ref>[http://www.wisdom.weizmann.ac.il/math/AABeyond12/presentations/Aumann.pdf Modern Economic Theory in the Talmud] by [[Yisrael Aumann]]</ref> (further {{citation needed|date=October 2015}} for possible older mentions)

The idea of the median also appeared later in [[Edward Wright (mathematician)|Edward Wright]]'s book on navigation (''Certaine Errors in Navigation'') in 1599 in a section concerning the determination of location with a [[compass]]. Wright felt that this value was the most likely to be the correct value in a series of observations.

In 1757, [[Roger Joseph Boscovich]] developed a regression method based on the [[L1 norm|''L''<sup>1</sup> norm]] and therefore implicitly on the median.<ref name=Stigler1986>{{cite book |last=Stigler |first=S. M. |year=1986 |title=The History of Statistics: The Measurement of Uncertainty Before 1900 |publisher=Harvard University Press |isbn=0674403401 |url=https://archive.org/details/historyofstatist00stig }}</ref>

In 1774, [[Pierre-Simon Laplace|Laplace]] suggested the median be used as the standard estimator of the value of a posterior pdf. The specific criterion was to minimize the expected magnitude of the error; <math>|\alpha - \alpha^{*}|</math> where <math>\alpha^{*}</math> is the estimate and <math>\alpha</math> is the true value. Laplaces's criterion was generally rejected for 150 years in favor of the [[least squares]] method of [[Carl Friedrich Gauss|Gauss]] and [[Adrien-Marie Legendre|Legendre]] which minimizes <math>(\alpha - \alpha^{*})^{2}</math> to obtain the mean.<ref>{{cite book|last1=Jaynes|first1=E.T.|title=Probability theory : the logic of science|date=2007|publisher=Cambridge Univ. Press|location=Cambridge [u.a.]|isbn=978-0-521-59271-0|page=172|edition=5. print.}}</ref> The distribution of both the sample mean and the sample median were determined by Laplace in the early 1800s.<ref name=Stigler1973 /><ref name=Laplace1818>Laplace PS de (1818) ''Deuxième supplément à la Théorie Analytique des Probabilités'', Paris, Courcier</ref>

[[Antoine Augustin Cournot]] in 1843 was the first<ref>{{Cite book|title=Dictionary of Mathematical Geosciences: With Historical Notes|last=Howarth|first=Richard|publisher=Springer|year=2017|isbn=|location=|pages=374}}</ref> to use the term ''median'' (''valeur médiane'') for the value that divides a probability distribution into two equal halves. [[Gustav Theodor Fechner]] used the median (''Centralwerth'') in sociological and psychological phenomena.<ref name=Keynes1921>Keynes, J.M. (1921) ''[[A Treatise on Probability]]''. Pt II Ch XVII §5 (p 201) (2006 reprint, Cosimo Classics, {{isbn|9781596055308}} : multiple other reprints)</ref> It had earlier been used only in astronomy and related fields. [[Gustav Theodor Fechner|Gustav Fechner]] popularized the median into the formal analysis of data, although it had been used previously by Laplace.<ref name=Keynes1921/>

[[Francis Galton]] used the English term ''median'' in 1881,<ref name=Galton1881>Galton F (1881) "Report of the Anthropometric Committee" pp 245–260. [https://www.biodiversitylibrary.org/item/94448 ''Report of the 51st Meeting of the British Association for the Advancement of Science'']</ref> having earlier used the terms ''middle-most value'' in 1869, and the ''medium'' in 1880.<ref>[https://www.encyclopediaofmath.org/index.php/Galton,_Francis ''encyclopediaofmath.org'']</ref><ref>[http://www.personal.psu.edu/users/e/c/ecb5/Courses/M475W/WeeklyReadings/Week%2013/DevelopmentOfModernStatistics.pdf  ''personal.psu.edu'']</ref>

==See also==
{{Portal|Mathematics}}
* [[Medoid]]s which are a generalisation of the median in higher dimensions
* [[Central tendency]]
** [[Mean]]
** [[Mode (statistics)|Mode]]
* [[Absolute deviation]]
* [[Bias of an estimator]]
* [[Concentration of measure]] for [[Lipschitz functions]]
* [[Median (geometry)]]
* [[Median graph]]
* [[Median search]]
* [[Median slope]]
* [[Median voter theory]]
* [[Weighted median]]

== References ==
{{Reflist
|refs =
<ref name="Brown">{{cite journal |last=Brown |first=George W. |year=1947 |title=On Small-Sample Estimation |journal=[[Annals of Mathematical Statistics]] |volume=18 |issue=4 |pages=582–585 |jstor=2236236 |doi=10.1214/aoms/1177730349 }}</ref>
<ref name="Lehmann">{{cite journal |author-link = Erich Leo Lehmann |last=Lehmann |first=Erich L. |year=1951 |title = A General Concept of Unbiasedness |journal=[[Annals of Mathematical Statistics]] |volume=22 |issue = 4 |pages=587–592 |jstor=2236928 |doi=10.1214/aoms/1177729549 }}</ref>
<ref name="Birnbaum">{{cite journal |author-link = Allan Birnbaum |last=Birnbaum |first=Allan |year=1961 |title = A Unified Theory of Estimation, I |journal=[[Annals of Mathematical Statistics]] |volume=32 |issue = 1 |pages=112–135 |jstor=2237612 |doi=10.1214/aoms/1177705145 }}</ref>
<ref name="vdW">{{cite journal |last=van der Vaart |first = H. Robert |year=1961 |title=Some Extensions of the Idea of Bias |journal=[[Annals of Mathematical Statistics]] |volume=32 |issue=2 |pages=436–447 |jstor=2237754 |doi=10.1214/aoms/1177705051 |mr=125674 }}</ref>
<ref name="Pf">{{cite book |title = Parametric Statistical Theory |last1=Pfanzagl |first1=Johann |author2 = with the assistance of R. Hamböker |year=1994 |publisher = Walter de Gruyter |isbn=3-11-013863-8 |mr=1291393 }}</ref>
}}

==External links==
* {{springer|title=Median (in statistics)|id=p/m063310}}
* [http://www.accessecon.com/pubs/EB/2004/Volume3/EB-04C10011A.pdf Median as a weighted arithmetic mean of all Sample Observations]
* [http://www.poorcity.richcity.org/cgi-bin/inequality.cgi On-line calculator]
* [http://www.statcan.gc.ca/edu/power-pouvoir/ch11/median-mediane/5214872-eng.htm Calculating the median]
* [http://mathschallenge.net/index.php?section=problems&show=true&titleid=average_problem A problem involving the mean, the median, and the mode.]
* {{MathWorld | urlname= StatisticalMedian | title= Statistical Median}}
* [http://www.poorcity.richcity.org/oei/#GiniHooverTheil Python script] for Median computations and [[income inequality metrics]]
* [https://arxiv.org/abs/0806.3301 Fast Computation of the Median by Successive Binning]
* [http://www.celiagreen.com/charlesmccreery/statistics/meanmedianmode.pdf 'Mean, median, mode and skewness'], A tutorial devised for first-year psychology students at Oxford University, based on a worked example.
* [https://www.popularmechanics.com/science/math/a28614640/complex-sat-math-problem/ The Complex SAT Math Problem Even the College Board Got Wrong]: Andrew Daniels in ''[[Popular Mechanics]]''

{{PlanetMath attribution|id=5900|title=Median of a distribution}}
{{Statistics|descriptive}}

[[Category:Means]]
[[Category:Robust statistics]]