Mean
Encyclopedia
In statistics
, mean has two related meanings:
There are other statistical measures that should not be confused with averages  including 'median
' and 'mode
'. Other simple statistical analyses use measures of spread, such as range
, interquartile range
, or standard deviation
.
For a realvalued random variable X, the mean is the expectation of X.
Note that not every probability distribution
has a defined mean (or variance
); see the Cauchy distribution
for an example.
For a data set
, 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 sampling
a statistical population
, 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 deviation
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 distribution
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 value
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 numbers
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.
The mean may often be confused with the median
, mode
or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed
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 exponential
and Poisson distribution
s.
For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is
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:
The harmonic mean
is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed
(distance per unit of time).
For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is
Equality holds only when all the elements of the given sample are equal.
, 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:
and again a suitable choice of an invertible ƒ will give >
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.
Often, outliers are erroneous data caused by artifacts. In this case one can use a truncated mean
. 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.
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.
, and especially multivariable calculus
, the mean of a function is loosely defined as the average value of the function over its domain
. 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 calculus
, 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 space
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 theory
either sort of mean plays an important role. In this context, Jensen's inequality
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.
s, daytimes, fractional part
s of real number
s. For those quantities you need a mean of circular quantities
.
gives a manner for determining the "center" of a mass distribution on a surface
or, more generally, Riemannian manifold
. 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).
Some of these properties are collected here.
such that the following properties hold:
It follows
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 number
,
then A and B defined by
are also weighted means.
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 permutation
π of inputs, the result is the same.
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 number
,
then A and B defined by
are also unweighted means.
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 rational
, then multiply them with the least common denominator.)
Then the corresponding weighted mean A is obtained by
has an expected value of μ, known as the population mean. The sample mean makes a good estimator
of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a random variable
, 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 freedom
.
Statistics
Statistics 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 meanArithmetic meanIn 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 meanGeometric 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 meanHarmonic 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 valueExpected 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...
of a random variableRandom 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 'median
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...
' and 'mode
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....
'. Other simple statistical analyses use measures of spread, such as range
Range (mathematics)
In 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 range
Interquartile range
In 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 deviation
Standard deviation
Standard 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 distribution
Probability distribution
In 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 variance
Variance
In 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 distribution
Cauchy distribution
The 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 set
Data set
A 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 sampling
Sampling (statistics)
In 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 population
Statistical population
A 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 deviation
Absolute deviation
In 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 distribution
Probability distribution
In 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 value
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...
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 numbers
Law of large numbers
In 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 median
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
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....
or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed
Skewness
In 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 exponential
Exponential distribution
In 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 distribution
Poisson distribution
In 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 meanGeometric 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...
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 mean
Harmonic mean
In 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 speed
Speed
In 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 meanGeneralized mean
In 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 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 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 mean Harmonic mean In 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 fmeanand again a suitable choice of an invertible ƒ will give >
arithmetic 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... , 

harmonic mean Harmonic mean In 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 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 meanWeighted 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...
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 mean
Truncated mean
A 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 meanInterquartile mean
The 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
Calculus 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 calculus
Multivariable calculus
Multivariable 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 domain
Domain (mathematics)
In 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 calculus
Fundamental theorem of calculus
The 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 space
Euclidean space
In 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 theory
Probability theory
Probability 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 inequality
Jensen's inequality
In 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 angleAngle
In 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 part
Fractional part
All 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 number
Real number
In 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 quantities
Mean of circular quantities
In 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 meanFréchet mean
The 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 surface
Surface
In 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 manifold
Riemannian manifold
In 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 numbersuch that the following properties hold:
 "Fixed pointFixed point (mathematics)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  HomogeneityHomogeneous functionIn 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 vectorCoordinate 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.  MonotonicityMonotonic functionIn 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
 BoundednessUpper boundIn 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  ContinuityContinuous functionIn 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 differentiableDerivativeIn 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 medianMedianIn 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 number
Real number
In 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 permutation
Permutation
In 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.
 SymmetrySymmetric functionIn 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 number
Real number
In 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 rational
Rational number
In 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 partitionedPartition of a setIn 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 hullConvex hullIn 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 populationStatistical population
A 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 estimator
Estimator
In 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 variable
Random variable
In 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 freedom
Degrees of freedom (statistics)
In 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 varianceAlgorithms for calculating varianceAlgorithms 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...
 AverageAverageIn 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 statisticsDescriptive statisticsDescriptive 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 unbiasedBias of an estimatorIn 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.  KurtosisKurtosisIn probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a realvalued random variable...
 Law of averagesLaw of averagesThe 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...
 MedianMedianIn 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)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 meanSpherical meanIn 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 statisticsSummary statisticsIn descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount as simply as possible...
 Weighted meanWeighted 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...