In
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
,
mean has two related meanings:
 the arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
(and is distinguished from the geometric meanThe geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
or harmonic meanIn mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
).
 the expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of a random variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
, which is also called the population mean.
There are other statistical measures that should not be confused with averages  including '
medianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
' and '
modeIn statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
'. Other simple statistical analyses use measures of spread, such as
rangeIn mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
,
interquartile rangeIn descriptive statistics, the interquartile range , also called the midspread or middle fifty, is a measure of statistical dispersion, being equal to the difference between the upper and lower quartiles...
, or
standard deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
.
For a realvalued random variable
X, the mean is the expectation of
X.
Note that not every
probability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
has a defined mean (or
varianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
); see the
Cauchy distributionThe Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...
for an example.
For a
data setA data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...
, the mean is the sum of the values divided by the number of values. The mean of a set of numbers
x_{1},
x_{2}, ...,
x_{n} is typically denoted by
, pronounced "
x bar". This mean is a type of arithmetic mean. If the data set were based on a series of observations obtained by
samplingIn statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....
a
statistical populationA statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...
, this mean is termed the "sample mean" (
) to distinguish it from the "population mean" (
or
_{x}). The mean is often quoted along with the standard deviation: the mean describes the central location of the data, and the standard deviation describes the spread. An alternative measure of dispersion is the mean deviation, equivalent to the average
absolute deviationIn statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median or sometimes the mean of the data set.D_i = x_im...
from the mean. It is less sensitive to outliers, but less mathematically tractable.
If a series of observations is sampled from a larger population (measuring the heights of a sample of adults drawn from the entire world population, for example), or from a
probability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
which gives the probabilities of each possible result, then the larger population or probability distribution can be used to construct a "population mean", which is also the
expected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
for a sample drawn from this population or probability distribution. For a finite population, this would simply be the arithmetic mean of the given property for every member of the population. For a probability distribution, this would be a sum or integral over every possible value weighted by the probability of that value. It is a universal convention to represent the population mean by the symbol
. In the case of a discrete probability distribution, the mean of a discrete random variable x is given by taking the product of each possible value of x and its probability P(x), and then adding all these products together, giving
.
The sample mean may differ from the population mean, especially for small samples, but the
law of large numbersIn probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.
As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below.
Arithmetic mean (AM)
The
arithmetic mean is the "standard" average, often simply called the "mean".
The
mean may often be confused with the
medianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
,
modeIn statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
or range. The mean is the arithmetic average of a set of values, or distribution; however, for
skewedIn probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a realvalued random variable. The skewness value can be positive or negative, or even undefined...
distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.
Nevertheless, many skewed distributions are best described by their mean – such as the
exponentialIn probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
and
Poisson distributionIn probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
s.
For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is
Geometric mean (GM)
The
geometric meanThe geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.
For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:
Harmonic mean (HM)
The
harmonic meanIn mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
is an average which is useful for sets of numbers which are defined in relation to some unit, for example
speedIn kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
(distance per unit of time).
For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is
Relationship between AM, GM, and HM
AM, GM, and HM satisfy these inequalities:
Equality holds only when all the elements of the given sample are equal.
Power mean
The
generalized meanIn mathematics, a generalized mean, also known as power mean or Hölder mean , is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.Definition:...
, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means. It is defined for a set of
n positive numbers
x_{i} by
By choosing the appropriate value for the parameter
m we get all means:

maximum 

quadratic mean 

arithmetic mean In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...


geometric mean The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...


harmonic meanIn mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....


minimum 
ƒmean
This can be generalized further as the generalized fmean
and again a suitable choice of an invertible
ƒ will give
>
arithmetic mean In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space... , 

harmonic meanIn mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.... , 

power mean, 

geometric mean The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the... . 
Weighted arithmetic mean
The
weighted arithmetic meanThe weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
is used, if one wants to combine average values from samples of the same population with different sample sizes:
The weights
represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.
Truncated mean
Sometimes a set of numbers might contain outliers, i.e. a datum which is much lower or much higher than the others.
Often, outliers are erroneous data caused by artifacts. In this case one can use a
truncated meanA truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both.For...
. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values.
Interquartile mean
The
interquartile meanThe interquartile mean is a statistical measure of central tendency, much like the mean , the median, and the mode....
is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.
Mean of a function
In
calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, and especially
multivariable calculusMultivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
, the mean of a function is loosely defined as the average value of the function over its
domainIn mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
. In one variable, the mean of a function
f(
x) over the interval (
a,b) is defined by

Recall that a defining property of the average value
of finitely many numbers
is that
. In other words,
is the
constant value which when
added to itself
times equals the result of adding the
terms of
. By analogy, a
defining property of the average value
of a function over the interval
is that

In other words,
is the
constant value which when
integrated over
equals the result of
integrating
over
. But by the second
fundamental theorem of calculusThe first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
, the integral of a constant
is just

See also the first mean value theorem for integration, which guarantees
that if
is continuous then there exists a point
such that

The point
is called the mean value of
on
. So we write
and rearrange the preceding equation to get the above definition.
In several variables, the mean over a relatively compact domain
U in a
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is defined by
This generalizes the
arithmetic mean. On the other hand, it is also possible to generalize the
geometric mean to functions by defining the geometric mean of
f to be
More generally, in measure theory and
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of nondeterministic events or measured quantities that may either be single...
either sort of mean plays an important role. In this context,
Jensen's inequalityIn mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context,...
places sharp estimates on the relationship between these two different notions of the mean of a function.
There is also a
harmonic average of functions and a
quadratic average (or
root mean square) of functions.
Mean of angles
Most of the usual means fail on circular quantities, like
angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s, daytimes,
fractional partAll real numbers can be written in the form n + r where n is an integer and the remaining fractional part r is a nonnegative real number less than one...
s of
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s. For those quantities you need a
mean of circular quantitiesIn mathematics, a mean of circular quantities is a mean which is suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means fail on circular quantities...
.
Fréchet mean
The
Fréchet meanThe Fréchet mean , is the point, x, that minimizes the Fréchet function, in cases where such a unique minimizer exists. The value at a point p, of the Fréchet function associated to a random point X on a complete metric space is the expected squared distance from p to X...
gives a manner for determining the "center" of a mass distribution on a
surfaceIn mathematics, specifically in topology, a surface is a twodimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary threedimensional Euclidean space R3 — for example, the surface of a ball...
or, more generally,
Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars.
It is sometimes also known as the
Karcher mean (named after Hermann Karcher).
Properties
All means share some properties and additional properties are shared by the most common means.
Some of these properties are collected here.
Weighted mean
A
weighted mean M is a function which maps tuples of positive numbers to a positive number
such that the following properties hold:
 "Fixed point
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
": M(1,1,...,1) = 1
 Homogeneity
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
: M(λ x_{1}, ..., λ x_{n}) = λ M(x_{1}, ..., x_{n}) for all λ and x_{i}. In vectorIn linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
notation: M(λ x) = λ Mx for all nvectors x.
 Monotonicity
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
: If x_{i} ≤ y_{i} for each i, then Mx ≤ My
It follows
 Boundedness
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
: min x ≤ Mx ≤ max x
 Continuity
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
:
 There are means which are not differentiable
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the power mean, or as a special case of a medianIn probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
), but is not differentiable.
 All means listed above, with the exception of most of the Generalized fmeans, satisfy the presented properties.
 If f is bijective, then the generalized fmean satisfies the fixed point property.
 If f is strictly monotonic, then the generalized fmean satisfy also the monotony property.
 In general a generalized fmean will miss homogeneity.
The above properties imply techniques to construct more complex means:
If
C,
M_{1}, ...,
M_{m} are weighted means and
p is a positive
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
,
then
A and
B defined by


are also weighted means.
Unweighted mean
Intuitively spoken, an
unweighted mean is a weighted mean with equal weights.
Since our definition of
weighted mean above does not expose particular weights,
equal weights must be asserted by a different way.
A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.
Thus we define
M to be an unweighted mean if it is a weighted mean
and for each
permutationIn mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
π of inputs, the result is the same.
 Symmetry
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
: Mx = M(πx) for all ntuples π and permutations π on ntuples.
Analogously to the weighted means,
if
C is a weighted mean and
M_{1}, ...,
M_{m} are unweighted means and
p is a positive
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
,
then
A and
B defined by


are also unweighted means.
Convert unweighted mean to weighted mean
An unweighted mean can be turned into a weighted mean by repeating elements.
This connection can also be used to state that a mean is the weighted version of an unweighted mean.
Say you have the unweighted mean
M and
weight the numbers by natural numbers
.
(If the numbers are
rationalIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
, then multiply them with the least common denominator.)
Then the corresponding weighted mean
A is obtained by
Means of tuples of different sizes
If a mean
M is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely
 Given an arbitrary tuple x, which is partitioned
In mathematics, a partition of a set X is a division of X into nonoverlapping and nonempty "parts" or "blocks" or "cells" that cover all of X...
into y_{1}, ..., y_{k}, then


 (See Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
.)
Population and sample means
The mean of a
populationA statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...
has an expected value of μ, known as the population mean. The sample mean makes a good
estimatorIn statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a
random variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
, not a constant, and consequently it will have its own distribution. For a random sample of
n observations from a normally distributed population, the sample mean distribution is

Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares, which changes the distribution of the sample mean from a normal distribution to a Student's t distribution with
n − 1
degrees of freedomIn statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
.
See also
 Algorithms for calculating mean and variance
Algorithms for calculating variance play a major role in statistical computing. A key problem in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with...
 Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
, same as central tendency
 Descriptive statistics
Descriptive statistics quantitatively describe the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics , in that descriptive statistics aim to summarize a data set, rather than use the data to learn about the population that the data are...
 For an independent identical distribution from the reals, the mean of a sample is an unbiased
In statistics, bias of an estimator 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. Otherwise the estimator is said to be biased.In ordinary English, the term bias is...
estimator for the mean of the population.
 Kurtosis
In probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a realvalued random variable...
 Law of averages
The law of averages is a lay term used to express a belief that outcomes of a random event will "even out" within a small sample.As invoked in everyday life, the "law" usually reflects bad statistics or wishful thinking rather than any mathematical principle...
 Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
 Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
 Spherical mean
In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.Definition:...
 Summary statistics
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount as simply as possible...
 Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
External links