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In mathematics, particularly statistics, a measure of location is an operation that, given a collection or a probability distribution, produces an actual or hypothetical member of it that summarizes it by representing its overall position.

A central value, a mean,^[1][2] or an average is a measure of location whose value is generally in the middle of the data, rather than at or towards one end. (The word "average" is also used outside of mathematics to mean common, typical, or normal.)

A "(measure of) central tendency" may refer either to a central value, or, more rarely, to how close the data is to that central value (i.e. the opposite of statistical dispersion or variability). Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."^[3][4][5]

The phrases "the average" and "the mean" almost always refer specifically to the arithmetic mean, though other meanings are occasionally used depending on the context. For example, in education, "average" sometimes refers to "the three Ms" of (arithmetic) mean, median, and mode; additionally, the harmonic mean is implied in many situations involving rates or ratios.

The arithmetic mean or arithmetic average is the sum of the values divided by the number of values, i.e. ${\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$. In statistics, this sample mean is denoted using an overline (e.g. ${\displaystyle {\bar {x}}}$, pronounced "x bar"), and this population mean (the expected value) is denoted with the Greek letter mu (${\displaystyle \mu }$, pronounced /'mjuː/).^[6]

Any measure of location, given a collection within its domain, produces a value somewhere on or within the collection's bounding box (and so for real numbers, between its maximum and minimum). Therefore, if a collection consists entirely of the same value, any measure of location of it is that value.^[7]

Most measures of location^[a] are monotonic, i.e. moving a member of the input in one direction causes the measure to move in the same direction, or equivalently, if two collections of numbers A and B have the same number of elements, and they can be arranged such that each entry in A ≥ the corresponding entry in B, then the measure of A ≥ the measure of B.

All commonly-used measures of location are linearly homogeneous, i.e. multiplying every value by the same scale factor multiplies the measure of location by that same scale factor.

Most measures of location^[b] remain identical when the list of items is permuted, i.e. the ordering does not matter.

The arithmetic mean (L² center) and mid-range (L^∞ center) are unique (when they exist), while the median (L¹ center) and mode (L⁰ center) are not in general unique. This can be understood in terms of convexity of the associated functions (coercive functions).

The 2-norm and ∞-norm are strictly convex, and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the arithmetic mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point.

The 1-norm is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. Correspondingly, the median (in this sense of minimizing) is not in general unique, and in fact any point between the two central points of a discrete distribution with an even number of points minimizes average absolute deviation.

The 0-"norm" is not convex (hence not a norm). Correspondingly, the mode is not unique – for example, in a uniform distribution (whether discrete or continuous) any point is the mode.

There are many kinds of measures of location, most of which are considered averages or means, since they are in the center of the data in some sense. Most averages can be seen as minimizing their variation (statistical dispersion) from the data points by some measure, i.e. solving some optimization problem.

Some of the measures of variation are p-norms (though the function corresponding to the L⁰ space is not a norm, and is thus often referred to in quotes, as the 0-"norm"). For a given (finite) data set X, thought of as a vector x = (x₁,…,x_n), the dispersion about a point c is the "distance" from x to the constant vector c = (c,…,c) in the p-norm (normalized by the number of points n):

${\displaystyle f_{p}(c)=\left\|\mathbf {x} -\mathbf {c} \right\|_{p}:={\bigg (}{\frac {1}{n}}\sum _{i=1}^{n}\left|x_{i}-c\right|^{p}{\bigg )}^{1/p}}$

For p = 0 and p = ∞ these functions are defined by taking limits, respectively as p → 0 and p → ∞. For p = 0 the limiting values are 0⁰ = 0 and a⁰ = 1 for a ≠ 0, so the difference becomes simply equality, so the 0-norm counts the number of unequal points, which is minimized by the mode. For p = ∞ the largest number dominates, and thus the ∞-norm is the maximum difference, which is minimized by the mid-range. Using p = 1 results in the median, and p = 2 the arithmetic mean.

Some of these measures are robust against outliers, such as the median and the trimean.

The table of mathematical symbols explains the symbols used below. Herein, "amn" means argmin and "mx" means maximum.

Name	Equation or description	As solution to optimization problem					Minimum level of measurement as a scale in Stevens's typology	Result for [1,2,2,3,4,7,9]	Notes
Medoid	A representative object of a set ${\displaystyle {\mathcal {X}}}$ of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function ${\displaystyle d}$	${\displaystyle {\underset {y\in {\mathcal {X}}}{\operatorname {amn} }}\sum _{i=1}^{n}d(y,x_{i})}$	d(y, x)				(depends on d)	(depends on d)
Quasi-arithmetic mean, generalized f-mean, or Kolmogorov-Nagumo-de Finetti mean	Applies a function before computing the arithmetic mean, and then applies the inverse function to that mean: ${\displaystyle f^{-1}\left({\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\right)}$	${\displaystyle {\underset {x\in \operatorname {dom} (f)}{\operatorname {amn} }}\,\sum _{i=1}^{n}(f(x)-f(x_{i}))^{2}}$	${\displaystyle (f(y)-f(x))^{2}}$	f(x)	f⁻¹(y)		(depends on f)	(depends on f)	The optimization problem is valid if f is monotonic.
Generalized mean, power mean, or Hölder mean	The p-th root of, the sum of p-th powers divided by the count: ${\displaystyle {\sqrt[{p}]{{\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}}}}$	${\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {amn} }}\,\sum _{i=1}^{n}(x^{p}-x_{i}^{p})^{2}}$	${\displaystyle (y^{p}-x^{p})^{2}}$	xᵖ	${\displaystyle y^{\frac {1}{p}}}$	p	(depends on p)	(depends on p)	May have problems with negative values.
Arithmetic mean or centroid	The sum of the values divided by their count: ${\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}=}$${\displaystyle {\frac {1}{n}}(x_{1}+\cdots +x_{n})}$	Minimizing the standard deviation, or: ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}(x-x_{i})^{2}}$	(y - x)²	x	y	1	Interval	(1+...+9)/7 = 28/7 = 4
Quadratic mean or root mean square (RMS)	The square root of, the sum of squares divided by the count:${\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}=}$${\displaystyle {\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+\cdots +x_{n}^{2}\right)}}}$	${\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {amn} }}\,\sum _{i=1}^{n}(x^{2}-x_{i}^{2})^{2}}$	(y² - x²)²	x²	√y	2	Ratio	√((1²+...+9²)/7) ≈ √23.43 ≈ 4.84	Has problems with negative values.
Cubic mean	The cube root of, the sum of cubes divided by the count:${\displaystyle {\sqrt[{3}]{{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{3}}}=}$${\displaystyle {\sqrt[{3}]{{\frac {1}{n}}\left(x_{1}^{3}+\cdots +x_{n}^{3}\right)}}}$	${\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {amn} }}\,\sum _{i=1}^{n}(x^{3}-x_{i}^{3})^{2}}$	(y³ - x³)²	x³	∛y	3	Ratio	∛((1³+...+9³)/7) ≈ ∛168.57 ≈ 5.524	The optimization problem is valid for non-negative numbers (≥0).
Harmonic mean	The count divided by the sum of reciprocals: ${\displaystyle {\frac {n}{{\frac {1}{x_{1}}}++\cdots +{\frac {1}{x_{n}}}}}}$	${\displaystyle {\underset {x\in \mathbb {R} _{\neq 0}}{\operatorname {amn} }}\,\sum _{i=1}^{n}\left({\frac {1}{x}}-{\frac {1}{x_{i}}}\right)^{2}}$	(1/y - 1/x)²	1/x	1/y	-1	Ratio	7/(⅟₁+...+⅟₉) ≈ 7/2.837 ≈ 2.467	The values should all have the same sign.
Geometric mean	The nth root of the product, where n is the count: ${\displaystyle {\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}=}$${\displaystyle {\sqrt[{n}]{x_{1}\times \dotsb \times x_{n}}}}$	${\displaystyle {\underset {x\in \mathbb {R} _{>0}}{\operatorname {amn} }}\,\sum _{i=1}^{n}(\log(x)-\log(x_{i}))^{2}}$	(log(y) - log(x))²	ln(x)	exp(y)	lim →0	Ratio	${\displaystyle {\sqrt[{7}]{1{\times }\cdots {\times }9}}\approx }$${\displaystyle {\sqrt[{7}]{3024}}\approx 3.1421}$	The mean is well-defined for all non-negative numbers, but the logarithm is not defined at zero.
${\displaystyle \tau }$th quantile	𝜏 is a number between 0 and 1, proportional to the 0-indexing position in the sorted list of values. See Quantile regression § Sample quantile	Minimizing the total tilted absolute value loss (AKA quantile loss or pinball loss):${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\!\sum _{i=1}^{n}\!\operatorname {mx} \!{\begin{cases}\!\!(1-\tau )(x-x_{i})\\\!\!\tau (x_{i}-x)\end{cases}}}$ = ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\sum _{i=1}^{n}{\begin{aligned}&|x-x_{i}|+{}\\&(1-2\tau )\,x\end{aligned}}}$	mx{(1-𝜏)(y-x),𝜏(x-y)} = |y-x| + (1-2𝜏)y	𝜏			Ordinal	(depends on τ)	Generally not considered an average. The optimization problem is only unique for 0<𝜏<1.
Minimum	The smallest value in the data set See sample maximum and minimum			0			Ordinal	1	Not an average.
Maximum	The largest value in the data set			1			Ordinal	9	Not an average.
Midrange	The midpoint (arithmetic mean) of the maximum and minimum: ${\displaystyle {\frac {1}{2}}\left(\max x+\min x\right)}$	Minimizing the maximum deviation: ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,{\underset {i\in \{1,\dots ,n\}}{\operatorname {mx} }}\,|x-x_{i}|}$					Ordinal	(1+9)/2 = 5
Lower quartile (Q1)	The value such that one-quarter of the data is less than (or equal to?) it, and three-quarters is greater than (or equal to?) it	${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}\operatorname {mx} {\begin{cases}{\frac {3}{4}}(x-x_{i})\\{\frac {1}{4}}(x_{i}-x)\end{cases}}}$ = ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}{\big (}|x-x_{i}|+{\frac {1}{2}}\,x{\big )}}$	mx((¾(y-x), ¼(x-y)) = |y-x| + ½y	1/4			Ordinal	2	Not really an average.
Upper quartile (Q3)	The value such that three-quarters of the data is less than (or equal to?) it, and one-quarter is greater than (or equal to?) it	${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}\operatorname {mx} {\begin{cases}{\frac {1}{4}}(x-x_{i})\\{\frac {3}{4}}(x_{i}-x)\end{cases}}}$ = ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}{\big (}|x-x_{i}|-{\frac {1}{2}}\,x{\big )}}$	mx(¼(y-x), ¾(x-y)) = |y-x| - ½y	3/4			Ordinal	7	Not really an average.
Midhinge	The arithmetic mean of the lower and upper quartiles: ${\displaystyle {\frac {1}{2}}\left(Q_{1}+Q_{3}\right)}$						Ordinal	(2+7) / 2 = 9 / 2 = 4.5
Median (Q2)	In between the greater half and the lesser half of the data set; not unique if the data set contains an even number of points and the two in the middle are not equal	Minimizing the average absolute deviation, or: ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}|x-x_{i}|}$	|y - x|	1/2			Ordinal	3
Trimean	The sum of the two quartiles and twice the median, divided by four; or equivalently, the arithmetic mean of the midhinge and the median: ${\displaystyle {\frac {1}{4}}\left(Q_{1}+2Q_{2}+Q_{3}\right)}$${\displaystyle ={\frac {1}{2}}\left(\mathrm {MH} +\mathrm {med} \right)}$						Ordinal
Geometric median, spatial median, Euclidean minisum point, Torricelli point, or 1-median	A rotation-invariant generalization of the median for points in ${\displaystyle \mathbb {R} ^{d}}$. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.	Minimizing the sum of the L₂ norms (Euclidean distances): ${\displaystyle {\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {amn} }}\,\sum _{i=1}^{n}||{\vec {x}}-{\vec {x}}_{i}||_{2}}$	${\displaystyle ||{\vec {y}}-{\vec {x}}||_{2}}$					⟨3⟩
Tukey median	Another rotation-invariant generalization of the median for points in ${\displaystyle \mathbb {R} ^{d}}$; a point with the property that every halfspace containing it also contains many sample points	Maximizing the Tukey depth: ${\displaystyle {\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmax} }}\,{\underset {{\vec {u}}\in \mathbb {R} ^{d}}{\operatorname {min} }}\,\sum _{i=1}^{n}\mathbf {1} _{({\vec {x}}_{i}-{\vec {x}})\cdot {\vec {u}}\geq 0}}$						⟨3⟩ (with u⃗ = ⟨u⟩ for any non-zero real number u)
Mode	The most frequent value in the data set	Minimizing the variation ratio, or:${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmax} }}\,\sum _{i=1}^{n}\mathbf {1} _{x_{i}=x}}$	${\displaystyle {\begin{cases}-1,&x=y\\0,&x\neq y\end{cases}}}$				Nominal	2
Lehmer mean	The sum of pth powers divided by the sum of (p-1)th powers:${\displaystyle {\frac {\sum _{i=1}^{n}x_{i}^{p}}{\sum _{i=1}^{n}x_{i}^{p-1}}}}$	${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}x_{i}^{p-1}(x-x_{i})^{2}}$	${\displaystyle x^{p-1}(y-x)^{2}}$	p			(depends on p)	(depends on p)
Contraharmonic mean	The sum of squares divided by the sum: ${\displaystyle {\frac {x_{1}^{2}+\cdots +x_{n}^{2}}{{x_{1}}+\cdots +{x_{n}}}}}$	${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {amn} }}\,\sum _{i=1}^{n}x_{i}(x-x_{i})^{2}}$	x (y-x)²	2			Ratio	(1²+...+9²)/(1+...+9) = 164/28 ≈ 5.857
Arithmetic mean	see above		(y-x)²	1			Interval	(1+...+9)/(1+...+1) = 28/7 = 4
Harmonic mean	see above		(y-x)² / x	0			Ratio	(1+...+1)/(⅟₁+...+⅟₉) ≈ 7/2.837 ≈ 2.467

The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by ${\displaystyle X}$, then the mean is also known as the expected value of ${\displaystyle X}$ (denoted ${\displaystyle E(X)}$). For a discrete probability distribution, the mean is given by ${\displaystyle \textstyle \sum xP(x)}$, where the sum is taken over all possible values of the random variable and ${\displaystyle P(x)}$ is the probability mass function. For a continuous distribution, the mean is ${\displaystyle \textstyle \int _{-\infty }^{\infty }xf(x)\,dx}$, where ${\displaystyle f(x)}$ is the probability density function.^[8] In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite (+∞ or −∞), while for others the mean is undefined.

In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. This can happen when calculating the mean value ${\displaystyle y_{\text{avg}}}$ of a function ${\displaystyle f(x)}$. Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely by integration. The integration formula is written as:

${\displaystyle y_{\text{avg}}(a,b)={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx.}$

In this case, care must be taken to make sure that the integral converges. But the mean may be finite even if the function itself tends to infinity at some points.

[table with columns "Name", "${\displaystyle f}$", "${\displaystyle f^{-1}}$", "Formula", "Optimization problem"]}}

The arithmetic mean, the geometric mean, and the harmonic mean are known collectively as the Pythagorean means.^[9]

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians^[10] because of their importance in geometry and music.

The arithmetic mean (or simply mean or average) of a list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$, usually denoted by ${\displaystyle {\bar {x}}}$, is the sum of the sampled values divided by the number of items in the sample.

${\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}{x_{i}}={\frac {x_{1}+x_{2}+\cdots +x_{n}}{n}}}$

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:

${\displaystyle {\frac {4+36+45+50+75}{5}}={\frac {210}{5}}=42.}$

The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean):^[1]

${\displaystyle {\bar {x}}=\left(\prod _{i=1}^{n}{x_{i}}\right)^{\frac {1}{n}}=\left(x_{1}x_{2}\cdots x_{n}\right)^{\frac {1}{n}}}$

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

${\displaystyle (4\times 36\times 45\times 50\times 75)^{\frac {1}{5}}={\sqrt[{5}]{24\;300\;000}}=30.}$

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, as in the case of speed (i.e., distance per unit of time):

${\displaystyle {\bar {x}}=n\left(\sum _{i=1}^{n}{\frac {1}{x_{i}}}\right)^{-1}}$

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

${\displaystyle {\frac {5}{{\tfrac {1}{4}}+{\tfrac {1}{36}}+{\tfrac {1}{45}}+{\tfrac {1}{50}}+{\tfrac {1}{75}}}}={\frac {5}{\;{\tfrac {1}{3}}\;}}=15.}$

If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean of ${\displaystyle 15}$ tells us that these five different pumps working together will pump at the same rate as five pumps that can each empty the tank in ${\displaystyle 15}$ minutes.

The generalized mean, also known as the power mean or Hölder mean, abstracts several other means. It is defined for positive numbers ${\displaystyle x_{1},\dots ,x_{n}}$ by^[1]

${\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{p}\right)^{1/p}.}$

This, as a function of ${\displaystyle p}$, is well defined on ${\displaystyle \mathbb {R} \setminus \{0\}}$, but can be extended continuously to ${\displaystyle \mathbb {R} \cup \{-\infty ,+\infty \}}$.^[11] By choosing different values for ${\displaystyle p}$, other well known means are retrieved.

Name	Exponent	Value
Minimum	${\displaystyle p=-\infty }$	${\displaystyle \min\{x_{1},\dots ,x_{n}\}}$
Harmonic mean	${\displaystyle p=-1}$	${\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}}$
Geometric mean	${\displaystyle p=0}$	${\displaystyle {\sqrt[{n}]{x_{1}\dots x_{n}}}}$
Arithmetic mean	${\displaystyle p=1}$	${\displaystyle {\frac {x_{1}+\dots +x_{n}}{n}}}$
Root mean square	${\displaystyle p=2}$	${\displaystyle {\sqrt {\frac {x_{1}^{2}+\dots +x_{n}^{2}}{n}}}}$
Cubic mean	${\displaystyle p=3}$	${\displaystyle {\sqrt[{3}]{\frac {x_{1}^{3}+\dots +x_{n}^{3}}{n}}}}$
Maximum	${\displaystyle p=+\infty }$	${\displaystyle \max\{x_{1},\dots ,x_{n}\}}$

A similar approach to the power mean is the ${\displaystyle f}$-mean, also known as the quasi-arithmetic mean. For an injective function ${\displaystyle f\colon I\rightarrow \mathbb {R} }$ on an interval ${\displaystyle I\subset \mathbb {R} }$ and real numbers ${\displaystyle x_{1},\dots ,x_{n}\in I}$ we define their ${\displaystyle f}$-mean as

${\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({{\frac {1}{n}}\sum _{i=1}^{n}{f\left(x_{i}\right)}}\right).}$

By choosing different functions ${\displaystyle f}$, other well known means are retrieved.

Mean	${\displaystyle I}$	Function[note 1]
Arithmetic mean	${\displaystyle \mathbb {R} }$	${\displaystyle x\mapsto x}$
Geometric mean	${\displaystyle ]0,+\infty [}$[note 2]	${\displaystyle x\mapsto \ln(x)}$
Harmonic mean	${\displaystyle \mathbb {R} \setminus \{0\}}$	${\displaystyle x\mapsto x^{-1}}$
Power mean	${\displaystyle \mathbb {R} \setminus \{0\}}$[note 3]	${\displaystyle x\mapsto x^{m}}$

[table with columns "Name", "Formula or description", "Optimization problem"]}}

Instead of a single central point, one can ask for multiple points such that the variation from these points is minimized. This leads to cluster analysis, where each point in the data set is clustered with the nearest "center". Most commonly, using the 2-norm generalizes the mean to k-means clustering, while using the 1-norm generalizes the (geometric) median to k-medians clustering. Using the 0-norm simply generalizes the mode (most common value) to using the k most common values as centers.

Unlike the single-center statistics, this multi-center clustering cannot in general be computed in a closed-form expression, and instead must be computed or approximated by an iterative method; one general approach is expectation–maximization algorithms.

The notion of a "center" as minimizing variation can be generalized in information geometry as a distribution that minimizes divergence (a generalized distance) from a data set. The most common case is maximum likelihood estimation, where the maximum likelihood estimate (MLE) maximizes likelihood (minimizes expected surprisal), which can be interpreted geometrically by using entropy to measure variation: the MLE minimizes cross-entropy (equivalently, relative entropy, Kullback–Leibler divergence).

A simple example of this is for the center of nominal data: instead of using the mode (the only single-valued "center"), one often uses the empirical measure (the frequency distribution divided by the sample size) as a "center". For example, given binary data, say heads or tails, if a data set consists of 2 heads and 1 tails, then the mode is "heads", but the empirical measure is 2/3 heads, 1/3 tails, which minimizes the cross-entropy (total surprisal) from the data set. This perspective is also used in regression analysis, where least squares finds the solution that minimizes the distances from it, and analogously in logistic regression, a maximum likelihood estimate minimizes the surprisal (information distance).

(these are listed separately because they can be applied to any kind of mean)

The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population, and is define by^[1]

${\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}{w_{i}x_{i}}}{\sum _{i=1}^{n}w_{i}}},}$

where ${\displaystyle x_{i}}$ and ${\displaystyle w_{i}}$ are the mean and size of sample ${\displaystyle i}$ respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values.

Weighted mean: ${\displaystyle {\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}}$, ${\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}w_{i}(x-x_{i})^{2}}$

Weighted arithmetic mean: an arithmetic mean that incorporates weighting to certain data elements.

The weights may be assigned based on the original arrangement of the data points, the ranking of the values, or some combination.^[12]

Winsorized mean: Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain

Winsorized mean: an arithmetic mean in which extreme values are replaced by values closer to the median.

Truncated mean: The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded

Sometimes, a set of numbers might contain outliers. 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. A specific example of a truncated mean is the interquartile mean.

Interquartile mean: A special case of the truncated mean, using the interquartile range. A special case of the inter-quantile truncated mean, which operates on quantiles (often deciles or percentiles) that are equidistant but on opposite sides of the median.

Truncated mean or trimmed mean: the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.

Interquartile mean: a truncated mean based on data within the interquartile range.

Given a time series, such as daily stock market prices or yearly temperatures, people often want to create a smoother series.^[13] This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the moving average: one chooses a number n and creates a new series by taking the arithmetic mean of the first n values, then moving forward one place by dropping the oldest value and introducing a new value at the other end of the list, and so on. This is the simplest form of moving average. More complicated forms involve using a weighted average. The weighting can be used to enhance or suppress various periodic behaviors and there is extensive analysis of what weightings to use in the literature on filtering. In digital signal processing the term "moving average" is used even when the sum of the weights is not 1.0 (so the output series is a scaled version of the averages).^[14] The reason for this is that the analyst is usually interested only in the trend or the periodic behavior.

For unimodal distributions the following bounds are known and are sharp:^[15]

${\displaystyle {\frac {|\theta -\mu |}{\sigma }}\leq {\sqrt {3}},}$

${\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {0.6}},}$

${\displaystyle {\frac {|\theta -\nu |}{\sigma }}\leq {\sqrt {3}},}$

where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.

For every distribution,^[16][17]

${\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq 1.}$

AM, GM, and HM of nonnegative real numbers satisfy these inequalities:^[18]

${\displaystyle \mathrm {AM} \geq \mathrm {GM} \geq \mathrm {HM} \,}$

Equality holds if all the elements of the given sample are equal.

The most commonly used definition of the average is the arithmetic mean,^[20] i.e. the sum divided by the count, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5.

However, other meanings are sometimes used depending on the context, which can lead to confusion; for instance, in teaching, "average" sometimes refers to "the three Ms": mean, median, and mode.^{[21][22][23][24][25][26][27]}

The median, defined as the value in the center after sorting the group, is usually used as the average in situations where the data is skewed or has outliers, in order to focus on the main part of the group rather than the long tail. For example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people.^[28]

The harmonic mean, defined as the reciprocal of the mean of the reciprocals, is used in a variety of situations involving rates or ratios, such as computing the average speed from multiple measurements taken over the same distance^[29]. Indeed, unlike an arithmetic mean or median of speeds, a harmonic mean of speeds will give the value of the constant speed that would cause one to travel the same distance in the same amount of time.

The mode represents the most common value found in the group. It can be used when the data is categorical rather than numeric,^[30] when the frequency of each value is relevant (such as where a histogram, bar chart, or probability density function is being referenced),^[31] or to find a value that represents the majority of the group.^[32]

Other statistics that can be used as an average include the mid-range, the quadratic mean or the geometric mean, but they are rarely referred to as "the average".

These different quantities all estimate the central tendency of a group, with each having their advantages and issues. Mathematically, they can be thought as solving different variational problems.

A type of average used in finance is the average percentage return, which is an example of a geometric mean. When the returns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is −10% and the return in the second year is +60%, then the average percentage return or CAGR, R, can be obtained by solving the equation: (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + R) × (1 + R). The value of R that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. The order of the years makes no difference – the average percentage returns of +60% and −10% is the same result as that for −10% and +60%.

This method can be generalized to examples in which the periods are not equal. For example, consider a period of a half of a year for which the return is −23% and a period of two and a half years for which the return is +13%. The average percentage return for the combined period is the single year return, R, that is the solution of the following equation: (1 − 0.23)^0.5 × (1 + 0.13)^2.5 = (1 + R)^0.5+2.5, giving an average return R of 0.0600 or 6.00%.

In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may colloquially be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically 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. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions.

Angles, times of day, and other cyclical quantities require modular arithmetic to add and otherwise combine numbers. These quantities can be averaged using the circular mean. In all these situations, it is possible that no mean exists, for example if all points being averaged are equidistant. Consider a color wheel—there is no mean to the set of all colors. Additionally, there may not be a unique mean for a set of values: for example, when averaging points on a clock, the mean of the locations of 11:00 and 13:00 is 12:00, but this location is equivalent to that of 00:00.

The Fréchet mean 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.

In geometry, there are thousands of different definitions for the center of a triangle that can all be interpreted as the mean of a triangular set of points in the plane.^[34]

This is an approximation to the mean for a moderately skewed distribution.^[35] It is used in hydrocarbon exploration and is defined as:

${\displaystyle m=0.3P_{10}+0.4P_{50}+0.3P_{90}}$

where ${\textstyle P_{10}}$, ${\textstyle P_{50}}$ and ${\textstyle P_{90}}$ are the 10th, 50th and 90th percentiles of the distribution, respectively.

The term central tendency dates from the late 1920s.^[4]

The first recorded time that the arithmetic mean was extended from 2 to n cases for the use of estimation was in the sixteenth century. From the late sixteenth century onwards, it gradually became a common method to use for reducing errors of measurement in various areas.^[36][37] At the time, astronomers wanted to know a real value from noisy measurement, such as the position of a planet or the diameter of the moon. Using the mean of several measured values, scientists assumed that the errors add up to a relatively small number when compared to the total of all measured values. The method of taking the mean for reducing observation errors was mainly developed in astronomy.^[36][38] A possible precursor to the arithmetic mean is the mid-range (the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation.^[37]

However, there are various older vague references to the use of the arithmetic mean (which are not as clear, but might reasonably have to do with our modern definition of the mean). In a text from the 4th century, it was written that (text in square brackets is a possible missing text that might clarify the meaning):^[39]

In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total to see if it is already naturally present among the numbers in the row; and we will find that the property of being [one] ninth [of the sum] only belongs to the [arithmetic] mean itself...

Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves.^[38] This might have been calculated using the average, although there seem to be no direct record of the calculation.

The root is found in Arabic as عوار ʿawār, a defect, or anything defective or damaged, including partially spoiled merchandise; and عواري ʿawārī (also عوارة ʿawāra) = "of or relating to ʿawār, a state of partial damage".^[c] Within the Western languages the word's history begins in medieval sea-commerce on the Mediterranean. 12th and 13th century Genoa Latin avaria meant "damage, loss and non-normal expenses arising in connection with a merchant sea voyage"; and the same meaning for avaria is in Marseille in 1210, Barcelona in 1258 and Florence in the late 13th.^[d] 15th-century French avarie had the same meaning, and it begot English "averay" (1491) and English "average" (1502) with the same meaning. Today, Italian avaria, Catalan avaria and French avarie still have the primary meaning of "damage". The transformation of the meaning in English began in later medieval and early modern Western merchant-marine law contracts under which if the ship met a bad storm and some of the goods had to be thrown overboard to make the ship lighter and safer, then all merchants whose goods were on the ship were to suffer proportionately (and not whoever's goods were thrown overboard); and more generally there was to be proportionate distribution of any avaria^{[citation needed]}. From there the word was adopted by British insurers, creditors, and merchants for talking about their losses as being spread across their whole portfolio of assets and having a mean proportion.^[d][40]

Marine damage is either particular average, which is borne only by the owner of the damaged property, or general average, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".

A second English usage, documented as early as 1674 and sometimes spelled "averish", is as the residue and second growth of field crops, which were considered suited to consumption by draught animals ("avers").^[41]

There is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term for a tenant's day labour obligation to a sheriff, probably anglicised from "avera" found in the English Domesday Book (1085).

The Oxford English Dictionary, however, says that derivations from German hafen haven, and Arabic ʿawâr loss, damage, have been "quite disposed of" and the word has a Romance origin.^[42]

Due to the aforementioned colloquial nature of the term "average", the term can be used to obfuscate the true meaning of data and suggest varying answers to questions based on the averaging method (most frequently arithmetic mean, median, or mode) used. In his article "Framed for Lying: Statistics as In/Artistic Proof", University of Pittsburgh faculty member Daniel Libertz comments that statistical information is frequently dismissed from rhetorical arguments for this reason.^[43] However, due to their persuasive power, averages and other statistical values should not be discarded completely, but instead used and interpreted with caution. Libertz invites us to engage critically not only with statistical information such as averages, but also with the language used to describe the data and its uses, saying: "If statistics rely on interpretation, rhetors should invite their audience to interpret rather than insist on an interpretation."^[43] In many cases, data and specific calculations are provided to help facilitate this audience-based interpretation.

• Average absolute deviation
• Central limit theorem
• Expected value
• Law of averages
• Population mean
• Sample mean
• Central moment
• Location parameter
• Statistical dispersion
• Median
• Mode
• Descriptive statistics
• Kurtosis
• Mean value theorem
• Moment (mathematics)
• Summary statistics
• Taylor's law
• Simplicial depth

• Bakker, Arthur (March 2003). "The Early History of Average Values and Implications for Education". Journal of Statistics Education. 11 (1). doi:10.1080/10691898.2003.11910694.
• Edgeworth, F. Y. (1889) [Read May 25, 1885.]. "VII. Observations and Statistics. An Essay on the Theory of Errors of Observation and the First Principles of Statistics". Transactions of the Cambridge Philosophical Society. XIV (II): 138–169. Google Books TdIsAAAAYAAJ. HathiTrust record 000526741, item mdp.39015008919659; record 100324784, item hvd.32044092879402.
  • Edgeworth, F. Y. (May 25, 1885). "(4) Observations and statistics : Abstract". Proceedings of the Cambridge Philosophical Society. V (IV): 310–312. Google Books KdIsAAAAYAAJ, zpw1AAAAIAAJ. HathiTrust record 100528415, item hvd.hwhqbe. Internet Archive proceedingscambr05camb, proceedingsofcam5188386camb, proceedingscamb09socigoog.
  • "Corrigendum on Mr F. Y. Edgeworth's paper". Proceedings of the Cambridge Philosophical Society. VI (II): 101–102. May 30, 1887. Google Books E501AAAAIAAJ. Internet Archive proceedingscamb12socigoog.
• Kennedy, Christopher; Stanley, Jason (July 2009). "On 'Average'". Mind. 118 (471): 583–646. doi:10.1093/mind/fzp094. eISSN 1460-2113. ISSN 0026-4423.

• Calculations and comparison between arithmetic and geometric mean of two values

Category:Arithmetic functions Category:Means Category:Summary statistics Category:Probability theory Category:Moments (mathematics)