![Finding the median](https://upload.wikimedia.org/wikipedia/commons/c/cf/Finding_the_median.png)
In
statistics and
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the median is the value separating the higher half from the lower half of a
data sample
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attem ...
, a
population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
, or a
probability distribution. For a
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
(often simply described as the "average") is that it is not
skewed
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimoda ...
by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value.
Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in
robust statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, su ...
, as it is the most
resistant statistic
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such ...
, having a
breakdown point
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such ...
of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result.
Finite data set of numbers
The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest.
If the data set has an odd number of observations, the middle one is selected. For example, the following list of seven numbers,
: 1, 3, 3, 6, 7, 8, 9
has the median of ''6'', which is the fourth value.
If the data set has an even number of observations, there is no distinct middle value and the median is usually defined to be the
arithmetic mean of the two middle values.
For example, this data set of 8 numbers
: 1, 2, 3, 4, 5, 6, 8, 9
has a median value of ''4.5'', that is
. (In more technical terms, this interprets the median as the fully
trimmed
''Trimmed'' is a 1922 American silent Western film directed by Harry A. Pollard and featuring Hoot Gibson. It is not known whether the film currently survives, and it may be a lost film.
Cast
* Hoot Gibson as Dale Garland
* Patsy Ruth Miller ...
mid-range
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
:M=\frac.
The mid-range is closely related to the range, a measure of ...
).
In general, with this convention, the median can be defined as follows: For a data set
of
elements, ordered from smallest to greatest,
: if
is odd,
: if
is even,
Formal definition
Formally, a median of a
population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
is any value such that at least half of the population is less than or equal to the proposed median and at least half is greater than or equal to the proposed median. As seen above, medians may not be unique. If each set contains less than half the population, then some of the population is exactly equal to the unique median.
The median is well-defined for any
ordered (one-dimensional) data, and is independent of any
distance metric
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
. The median can thus be applied to classes which are ranked but not numerical (e.g. working out a median grade when students are graded from A to F), although the result might be halfway between classes if there is an even number of cases.
A
geometric median
In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances ...
, on the other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the
medoid Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be ...
.
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''
1/2 sometimes also ''M''.
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 a special case of other
ways of summarizing the typical values associated with a statistical distribution: it is the 2nd
quartile, 5th
decile
In descriptive statistics, a decile is any of the nine values that divide the sorted data into ten equal parts, so that each part represents 1/10 of the sample or population. A decile is one possible form of a quantile; others include the quartile ...
, and 50th
percentile
In statistics, a ''k''-th percentile (percentile score or centile) is a score ''below which'' a given percentage ''k'' of scores in its frequency distribution falls (exclusive definition) or a score ''at or below which'' a given percentage fal ...
.
Uses
The median can be used as a measure of
location
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
when one attaches reduced importance to extreme values, typically because a distribution is
skewed
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimoda ...
, extreme values are not known, or
outliers are untrustworthy, i.e., may be measurement/transcription errors.
For example, consider the
multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
: 1, 2, 2, 2, 3, 14.
The median is 2 in this case, as is the
mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
, and it might be seen as a better indication of the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
than the
arithmetic mean of 4, which is larger than all but one of the values. However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true. At most, one can say that the two statistics cannot be "too far" apart; see below.
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.
Because the median is simple to understand and easy to calculate, while also a robust approximation to the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
, the median is a popular
summary statistic
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in
* a measure of ...
in
descriptive statistics
A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and an ...
. In this context, there are several choices for a measure of
variability: the
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
, the
interquartile range
In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the difference ...
, the
mean absolute deviation
The average absolute deviation (AAD) of a data set is the average of the Absolute value, absolute Deviation (statistics), deviations from a central tendency, central point. It is a summary statistics, summary statistic of statistical dispersion or ...
, and the
median absolute deviation
In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.
For a u ...
.
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 of candidate estimators shows that the sample mean is more statistically efficient
when—and only when— data is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions. Even then, 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.
Probability distributions
![visualisation mode median mean](https://upload.wikimedia.org/wikipedia/commons/3/33/Visualisation_mode_median_mean.svg)
For any
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued
probability distribution with
cumulative distribution function ''F'', a median is defined as any real number ''m'' that satisfies the inequalities
An equivalent phrasing uses a random variable ''X'' distributed according to ''F'':
Note that this definition does not require ''X'' to have an
absolutely continuous distribution (which has a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''f''), nor does it require a
discrete one. In the former case, the inequalities can be upgraded to equality: a median satisfies
Any
probability distribution on R 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 Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
:
* The median of a symmetric
unimodal distribution
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal ...
coincides with the mode.
* The median of a
symmetric distribution
In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) ...
which possesses a mean ''μ'' also takes the value ''μ''.
** The median of a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with mean ''μ'' and variance ''σ''
2 is μ. In fact, for a normal distribution, mean = median = mode.
** The median of a
uniform distribution in the interval
'a'', ''b''is (''a'' + ''b'') / 2, which is also the mean.
* The median of a
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
with location parameter ''x''
0 and scale parameter ''y'' is ''x''
0, the location parameter.
* The median of a
power law distribution
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
''x''
−''a'', with exponent ''a'' > 1 is 2
1/(''a'' − 1)''x''
min, where ''x''
min is the minimum value for which the power law holds
* The median of an
exponential distribution with
rate parameter ''λ'' is the natural logarithm of 2 divided by the rate parameter: ''λ''
−1ln 2.
* The median of a
Weibull distribution
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...
with shape parameter ''k'' and scale parameter ''λ'' is ''λ''(ln 2)
1/''k''.
Properties
Optimality property
The ''
mean absolute error
In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of ''Y'' versus ''X'' include comparisons of predicted versus observed, subsequent time versus initial time, and ...
'' of a real variable ''c'' with respect to the
random variable ''X'' is
:
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''. 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
:
as discussed below in the section on
multivariate median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
s (specifically, the
spatial median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
).
This optimization-based definition of the median is useful in statistical data-analysis, for example, in
''k''-medians clustering.
Inequality relating means and medians
![Comparison mean median mode](https://upload.wikimedia.org/wikipedia/commons/d/de/Comparison_mean_median_mode.svg)
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 Book and Sher in 1979 for discrete samples, and more generally by Page and Murty in 1982. In a comment on a subsequent proof by O'Cinneide, Mallows in 1991 presented a compact proof that uses
Jensen's inequality twice, as follows. Using , ·, for the
absolute value, we have
:
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 In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is posi ...
function
.
Mallows's proof can be generalized to obtain a multivariate version of the inequality
simply by replacing the absolute value with a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
:
:
where ''m'' is a
spatial median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
, that is, a minimizer of the function
The spatial median is unique when the data-set's dimension is two or more.
An alternative proof uses the one-sided Chebyshev inequality; it appears in
an inequality on location and scale parameters
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from th ...
. This formula also follows directly from
Cantelli's inequality.
Unimodal distributions
For the case of
unimodal
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal p ...
distributions, one can achieve a sharper bound on the distance between the median and the mean:
:
.
A similar relation holds between the median and the mode:
:
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''
:
This inequality generalizes to the median as well. We say a function is a C function if, for any ''t'',
:
is a
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
(allowing the degenerate cases of a
single point or an
empty set). Every convex function is a C function, but the reverse does not hold. If ''f'' is a C function, then
:
If the medians are not unique, the statement holds for the corresponding suprema.
Medians for samples
The sample median
Efficient computation of the sample median
Even though
comparison-sorting ''n'' items requires operations,
selection algorithm
In computer science, a selection algorithm is an algorithm for finding the ''k''th smallest number in a list or array; such a number is called the ''k''th ''order statistic''. This includes the cases of finding the minimum, maximum, and median e ...
s can compute the
th-smallest of items with only operations. This includes the median, which is the 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 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
Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961, it is still a commonly used algorithm for sorting. Overall, it is slightly faster than ...
sorting algorithm, which uses an estimate of its input's median. A more
robust estimator
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal distribution, normal. Robust Statistics, statistical methods have been developed ...
is
Tukey
John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the Cooley–Tukey FFT algorithm, fast Fourier Transform (FFT) algorithm and box plot. The Tukey's range test ...
's ''ninther'', which is the median of three rule applied with limited recursion: if is the sample laid out as an
array
An array is a systematic arrangement of similar objects, usually in rows and columns.
Things called an array include:
{{TOC right
Music
* In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
, and
:,
then
:
The ''remedian'' is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.
Sampling distribution
The distributions of both the sample mean and the sample median were determined by
Laplace.
The distribution of the sample median from a population with a density function
is asymptotically normal with mean
and variance
:
where
is the median of
and
is the sample size. A modern proof follows below. Laplace's result is now understood as a special case of
the asymptotic distribution of arbitrary quantiles.
For normal samples, the density is
, thus for large samples the variance of the median equals
(See also section
#Efficiency below.)
= Derivation of the asymptotic distribution
=
We take the sample size to be an odd number
and assume our variable continuous; the formula for the case of discrete variables is given below in . The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities
,
and
. For a continuous variable, the probability of multiple sample values being exactly equal to the median is 0, so one can calculate the density of at the point
directly from the trinomial distribution:
:
.
Now we introduce the beta function. For integer arguments
and
, this can be expressed as
. Also, recall that
. Using these relationships and setting both
and
equal to
allows the last expression to be written as
:
Hence the density function of the median is a symmetric beta distribution
pushed forward by
. Its mean, as we would expect, is 0.5 and its variance is
. By the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, the corresponding variance of the sample median is
:
.
The additional 2 is negligible
in the limit.
=Empirical local density
=
In practice, the functions
and
are often not known or assumed. However, they can be estimated from an observed frequency distribution. In this section, we give an example. Consider the following table, representing a sample of 3,800 (discrete-valued) observations:
Because the observations are discrete-valued, constructing the exact distribution of the median is not an immediate translation of the above expression for
; one may (and typically does) have multiple instances of the median in one's sample. So we must sum over all these possibilities:
:
Here, ''i'' is the number of points strictly less than the median and ''k'' the number strictly greater.
Using these preliminaries, 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 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.
Estimation of variance from sample data
The value of
—the asymptotic value of
where
is the population median—has been studied by several authors. The standard "delete one"
jackknife method produces
inconsistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
results.
An alternative—the "delete k" method—where
grows with the sample size has been shown to be asymptotically consistent.
This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,
but converges very slowly (
order of
).
Other methods have been proposed but their behavior may differ between large and small samples.
Efficiency
The
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
from the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, the efficiency for large N is
:
The efficiency tends to
as
tends to infinity.
In other words, the relative variance of the median will be
, or 57% greater than the variance of the mean – the relative
standard error
The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error o ...
of the median will be
, or 25% greater than the
standard error of the mean
The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of ...
,
(see also section
#Sampling distribution above.).
Other estimators
For univariate distributions that are ''symmetric'' about one median, the
Hodges–Lehmann estimator is a
robust
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
and highly
efficient estimator
In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, needs fewer input data or observations than a less efficient one to achie ...
of the population median.
If data is represented by a
statistical model specifying a particular family of
probability distributions, 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.
Pareto interpolation Pareto interpolation is a method of estimating the median and other properties of a population that follows a Pareto distribution. It is used in economics when analysing the distribution of incomes in a population, when one must base estimates on a ...
is an application of this when the population is assumed to have a
Pareto distribution.
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.
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.
Geometric median
The
geometric median
In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances ...
of a discrete set of sample points
in a Euclidean space is the point minimizing the sum of distances to the sample points.
:
In contrast to the marginal median, the geometric median is
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
with respect to Euclidean
similarity transformations such as
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
and
rotations.
Median in all directions
If the marginal medians for all coordinate systems coincide, then their common location may be termed the "median in all directions". This concept is relevant to voting theory on account of the
median voter theorem The median voter theorem is a proposition relating to ranked preference voting put forward by Duncan Black in 1948.Duncan Black, "On the Rationale of Group Decision-making" (1948). It states that if voters and policies are distributed along a one-d ...
. When it exists, the median in all directions coincides with the geometric median (at least for discrete distributions).
Centerpoint
An alternative generalization of the median in higher dimensions is the
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
is 3 since the median is the smallest value of
for which
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
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
are known, the interpolated median can be calculated from
:
Alternatively, if in an observed sample there are
scores above the median category,
scores in it and
scores below it then the interpolated median is given by
:
Pseudo-median
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. The Hodges–Lehmann estimator has been generalized to multivariate distributions.
Variants of regression
The
Theil–Sen estimator
In non-parametric statistics, the Theil–Sen estimator is a method for robustly fitting a line to sample points in the plane ( simple linear regression) by choosing the median of the slopes of all lines through pairs of points. It has also b ...
is a method for
robust
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
linear regression based on finding medians of
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
s.
Median filter
The
median filter
The median filter is a non-linear digital filtering technique, often used to remove noise from an image or signal. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an ...
is an important tool of
image processing, that can effectively remove any
salt and pepper noise from
grayscale
In digital photography, computer-generated imagery, and colorimetry, a grayscale image is one in which the value of each pixel is a single sample representing only an ''amount'' of light; that is, it carries only intensity information. Graysc ...
images.
Cluster analysis
In
cluster analysis, the
k-medians clustering
In statistics, ''k''-medians clusteringP. S. Bradley, O. L. Mangasarian, and W. N. Street, "Clustering via Concave Minimization," in Advances in Neural Information Processing Systems, vol. 9, M. C. Mozer, M. I. Jordan, and T. Petsche, Eds. Cambrid ...
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
Wald
WALD (1080 kHz) is an AM radio station licensed to Johnsonville, South Carolina. The station is part of the Worship and Word Network and is owned by Glory Communications, Inc., based in St. Stephen, South Carolina. It carries an Urban Gosp ...
in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent parameter
: a left half with values less than the median and a right half with values greater than the median.
He suggested taking the means of the dependent
and independent
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.
Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.
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.
Median-unbiased estimators
Any
''mean''-unbiased estimator minimizes the
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
(
expected loss Expected loss is the sum of the values of all possible losses, each multiplied by the probability of that loss occurring.
In bank lending (homes, autos, credit cards, commercial lending, etc.) the expected loss on a loan varies over time for a num ...
) with respect to the squared-error
loss function, as observed by
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. A
''median''-unbiased estimator minimizes the risk with respect to the
absolute-deviation loss function, as observed by
Laplace. Other
loss functions
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
are used in
statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics.
The theory covers approaches to statistical-decision problems and to statistica ...
, particularly in
robust statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, su ...
.
The theory of median-unbiased estimators was revived b
George W. Brownin 1947:
Further properties of median-unbiased estimators have been reported.
Median-unbiased estimators are invariant under
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-functions. One such procedure is an analogue of the
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 functions.
History
Scientific researchers in the ancient near east appear not to have used summary statistics altogether, instead choosing values that offered maximal consistency with a broader theory that integrated a wide variety of phenomena.
Within the Mediterranean (and, later, European) scholarly community, statistics like the mean are fundamentally a medieval and early modern development. (The history of the median outside Europe and its predecessors remains relatively unstudied.)
The idea of the median appeared in the 6th century in the
Talmud
The Talmud (; he, , Talmūḏ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law ('' halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the Talmud was the ce ...
, in order to fairly analyze divergent
appraisals. However, the concept did not spread to the broader scientific community.
Instead, the closest ancestor of the modern median is the
mid-range
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
:M=\frac.
The mid-range is closely related to the range, a measure of ...
, invented by
Al-Biruni
Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of Co ...
.
Transmission of Al-Biruni's work to later scholars is unclear. Al-Biruni applied his technique to
assaying metals, but, after he published his work, most assayers still adopted the most unfavorable value from their results, lest they appear to
cheat.
However, increased navigation at sea during the
Age of Discovery
The Age of Discovery (or the Age of Exploration), also known as the early modern period, was a period largely overlapping with the Age of Sail, approximately from the 15th century to the 17th century in European history, during which seafarin ...
meant that ship's navigators increasingly had to attempt to determine latitude in unfavorable weather against hostile shores, leading to renewed interest in summary statistics. Whether rediscovered or independently invented, the mid-range is recommended to nautical navigators in Harriot's "Instructions for Raleigh's Voyage to Guiana, 1595".
The idea of the median may have first appeared in
Edward Wright's 1599 book ''Certaine Errors in Navigation'' on a section about
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself wit ...
navigation. Wright was reluctant to discard measured values, and may have felt that the median — incorporating a greater proportion of the dataset than the
mid-range
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
:M=\frac.
The mid-range is closely related to the range, a measure of ...
— was more likely to be correct. However, Wright did not give examples of his technique's use, making it hard to verify that he described the modern notion of median.
The median (in the context of probability) certainly appeared in the correspondence of
Christiaan Huygens, but as an example of a statistic that was inappropriate for
actuarial practice.
The earliest recommendation of the median dates to 1757, when
Roger Joseph Boscovich
Roger Joseph Boscovich ( hr, Ruđer Josip Bošković; ; it, Ruggiero Giuseppe Boscovich; la, Rogerius (Iosephus) Boscovicius; sr, Руђер Јосип Бошковић; 18 May 1711 – 13 February 1787) was a physicist, astronomer, ...
developed a regression method based on the
''L''1 norm and therefore implicitly on the median.
In 1774,
Laplace made this desire explicit: he 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;
where
is the estimate and
is the true value. To this end, Laplace determined the distributions of both the sample mean and the sample median in the early 1800s.
[Laplace PS de (1818) ''Deuxième supplément à la Théorie Analytique des Probabilités'', Paris, Courcier] However, a decade later,
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and
Legendre developed the
least squares method, which minimizes
to obtain the mean. Within the context of regression, Gauss and Legendre's innovation offers vastly easier computation. Consequently, Laplaces' proposal was generally rejected until the rise of
computing devices 150 years later (and is still a relatively uncommon algorithm).
Antoine Augustin Cournot in 1843 was the first to use the term ''median'' (''valeur médiane'') for the value that divides a probability distribution into two equal halves.
Gustav Theodor Fechner
Gustav Theodor Fechner (; ; 19 April 1801 – 18 November 1887) was a German physicist, philosopher, and experimental psychologist. A pioneer in experimental psychology and founder of psychophysics (techniques for measuring the mind), he inspired ...
used the median (''Centralwerth'') in sociological and psychological phenomena.
[Keynes, J.M. (1921) '' A Treatise on Probability''. Pt II Ch XVII §5 (p 201) (2006 reprint, Cosimo Classics, : multiple other reprints)] It had earlier been used only in astronomy and related fields.
Gustav Fechner
Gustav Theodor Fechner (; ; 19 April 1801 – 18 November 1887) was a German physicist, philosopher, and experimental psychologist. A pioneer in experimental psychology and founder of psychophysics (techniques for measuring the mind), he ins ...
popularized the median into the formal analysis of data, although it had been used previously by Laplace,
and the median appeared in a textbook by
F. Y. Edgeworth.
Francis Galton used the English term ''median'' in 1881,
[Galton F (1881) "Report of the Anthropometric Committee" pp 245–260]
''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. ''personal.psu.edu''
/ref>
Statisticians encouraged the use of medians intensely throughout the 19th century for its intuitive clarity and ease of manual computation. However, the notion of median does not lend itself to the theory of higher moments as well as the arithmetic mean does, and is much harder to compute by computer. As a result, the median was steadily supplanted as a notion of generic average by the arithmetic mean during the 20th century.
See also
* Absolute deviation In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is posi ...
* Bias of an estimator
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In s ...
* Central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
* Concentration of measure
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random v ...
for Lipschitz functions
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ...
* Median graph
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices ''a'', ''b'', and ''c'' have a unique ''median'': a vertex ''m''(''a'',''b'',''c'') that belongs to shortest paths between each pair o ...
* Median of medians
In computer science, the median of medians is an approximate (median) selection algorithm, frequently used to supply a good pivot for an exact selection algorithm, mainly the quickselect, that selects the ''k''th smallest element of an initially u ...
– Algorithm to calculate the approximate median in linear time
* Median search
* Median slope
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
* Median voter theory
* Medoid Medoids are representative objects of a data set or a cluster within a data set whose sum of dissimilarities to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be ...
s – Generalization of the median in higher dimensions
Notes
References
External links
*
Median as a weighted arithmetic mean of all Sample Observations
On-line calculator
A problem involving the mean, the median, and the mode.
*
Python script
for Median computations and income inequality metrics
Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income and economic inequality among the participants in a particular economy, such as that of a specific country or of the world ...
Fast Computation of the Median by Successive Binning
'Mean, median, mode and skewness'
A tutorial devised for first-year psychology students at Oxford University, based on a worked example.
The Complex SAT Math Problem Even the College Board Got Wrong
Andrew Daniels in ''Popular Mechanics
''Popular Mechanics'' (sometimes PM or PopMech) is a magazine of popular science and technology, featuring automotive, home, outdoor, electronics, science, do-it-yourself, and technology topics. Military topics, aviation and transportation o ...
''
{{Statistics, descriptive
Means
Robust statistics