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| {{About|the mathematical concept}}
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| In [[mathematics]], '''mean''' has several different definitions depending on the context.
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| In [[probability]] and [[statistics]], '''mean''' and [[expected value]] are used synonymously to refer to one measure of the [[central tendency]] either of a [[probability distribution]] or of the [[random variable]] characterized by that distribution.<ref>{{cite book|last=Feller|first=William|title=Introduction to Probability Theory and its Applications, Vol I|year=1950|publisher=Wiley|isbn=0471257087|pages=221}}</ref> In the case of a [[discrete probability distribution]] of a random variable ''X'', the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value ''x'' of ''X'' and its probability P(''x''), and then adding all these products together, giving <math>\mu = \sum x P(x)</math>.<ref>Elementary Statistics by Robert R. Johnson and Patricia J. Kuby, [http://books.google.com/books?id=DWCAh7jWO98C&lpg=PP1&pg=PA279#v=onepage&q&f=false p. 279]</ref> An analogous formula applies to the case of a [[continuous probability distribution]]. Not every probability distribution has a defined mean; see the [[Cauchy distribution]] for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value <math>2^n</math> is <math>\tfrac{1}{2^n}</math> for n = 1, 2, 3, ....
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| For a [[data set]], the terms [[arithmetic mean]], [[Expected value|mathematical expectation]], and sometimes [[average]] are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x<sub>n</sub>'' is typically denoted by <math>\bar{x}</math>, pronounced "''x'' bar". If the data set were based on a series of observations obtained by [[sampling (statistics)|sampling]] from a [[statistical population]], the arithmetic mean is termed the '''sample mean''' (denoted <math>\bar{x}</math>) to distinguish it from the '''population mean''' (denoted '''<math>\mu</math>''' or '''<math>\mu_x</math>''').<ref>Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd. ISBN 0-7021-3838-X [http://books.google.com/books?id=f6TlVjrSAsgC&lpg=PP1&pg=PA181#v=onepage&q&f=false p. 181]</ref>
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| For a finite population, the '''population mean''' of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. 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.<ref>Schaum's Outline of Theory and Problems of Probability by Seymour Lipschutz and Marc Lipson, [http://books.google.com/books?id=ZKdqlw2ZnAMC&lpg=PP1&pg=PA141#v=onepage&q&f=false p. 141]</ref>
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| Outside of probability and statistics, a wide range of other notions of "mean" are often used in [[geometry]] and [[Mathematical analysis|analysis]]; examples are given below.
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| ==Types of mean==
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| ===Pythagorean means===
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| {{main|Pythagorean means}}
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| ==== Arithmetic mean (AM) ====
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| The ''arithmetic mean'' (or simply "mean") of a sample <math>x_1,x_2,\ldots,x_n</math> is the sum the sampled values divided by the number of items in the sample:
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| :<math> \bar{x} = \frac{x_1+x_2+\cdots +x_n}{n} </math>
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| For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is
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| :<math>\frac{4 + 36 + 45 + 50 + 75}{5} = \frac{210}{5} = 42.</math>
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| [[Image:Comparison mean median mode.svg|thumb|Comparison of the arithmetic mean, [[median]] and [[mode (statistics)|mode]] of two skewed ([[log-normal distribution|log-normal]]) distributions.]]
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| The '''mean''' may often be confused with the [[median]], [[Mode (statistics)|mode]] or range. The mean is the arithmetic average of a set of values, or distribution; however, for [[skewness|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.
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| Nevertheless, many skewed distributions are best described by their mean – such as the [[Exponential distribution|exponential]] and [[Poisson distribution|Poisson]] distributions.
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| ==== Geometric mean (GM) ====
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| The [[geometric mean]] 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.
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| :<math> \bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^\tfrac1n</math>
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| For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
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| :<math>(4 \times 36 \times 45 \times 50 \times 75)^{^1/_5} = \sqrt[5]{24\;300\;000} = 30.</math>
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| ==== Harmonic mean (HM) ====
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| The [[harmonic mean]] is an average which is useful for sets of numbers which are defined in relation to some [[Unit of measurement|unit]], for example [[speed]] (distance per unit of time).
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|
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| :<math> \bar{x} = n \cdot \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}</math>
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| For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is
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| :<math>\frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75}} = \frac{5}{\;\tfrac{1}{3}\;} = 15.</math>
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| ==== Relationship between AM, GM, and HM ====
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| {{Main|Inequality of arithmetic and geometric means}}
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| AM, GM, and HM satisfy these inequalities:
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| :<math> AM \ge GM \ge HM \, </math>
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| Equality holds only when all the elements of the given sample are equal.
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| ===Generalized means===
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| ====Power mean====
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| The [[generalized mean]], 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''<sub>i</sub> by
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| :<math> \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^\tfrac1m</math>
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| By choosing different values for the parameter ''m'', the following types of means are obtained:
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| :{|
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| |-
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| | <math>m\rightarrow\infty</math> || [[maximum]]
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| |-
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| | <math>m=2</math> || [[quadratic mean]]
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| |-
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| | <math>m=1</math> || [[arithmetic mean]]
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| |-
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| | <math>m\rightarrow0</math> || [[geometric mean]]
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| |-
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| | <math>m=-1</math> || [[harmonic mean]]
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| |-
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| | <math>m\rightarrow-\infty</math> || [[minimum]]
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| |}
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| ==== ''ƒ''-mean ====
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| This can be generalized further as the [[generalized f-mean]]
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| :<math> \bar{x} = f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) </math>
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| and again a suitable choice of an invertible ''ƒ'' will give
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| {|
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| |-
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| | <math>f(x) = x</math> || [[arithmetic mean]],
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| |-
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| | <math>f(x) = \frac{1}{x}</math> || [[harmonic mean]],
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| |-
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| | <math>f(x) = x^m</math> || [[power mean]],
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| |-
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| | <math>f(x) = \ln x</math> || [[geometric mean]].
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| |}
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| ===Weighted arithmetic mean===
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| The [[weighted mean|weighted arithmetic mean]] (or weighted average) is used if one wants to combine average values from samples of the same population with different sample sizes:
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| :<math> \bar{x} = \frac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}}. </math>
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| The weights <math>w_i</math> represent the sizes of the different samples. In other applications they represent a measure for the reliability of the influence upon the mean by the respective values.
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| ===Truncated mean===
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| Sometimes a set of numbers might contain outliers, i.e., data values which are much lower or much higher than the others.
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| Often, outliers are erroneous data caused by [[artifact (observational)|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.
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| ===Interquartile mean===
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| The [[interquartile mean]] is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
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| :<math> \bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i} </math>
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| assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.
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| ===Mean of a function===
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| In [[calculus]], and especially [[multivariable calculus]], the mean of a function is loosely defined as the average value of the function over its [[domain (mathematics)|domain]]. In one variable, the mean of a function ''f''(''x'') over the interval (''a,b'') is defined by
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| : <math>\bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx.</math>
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| Recall that a defining property of the average value <math>\bar{y}</math> of finitely many numbers <math>y_1, y_2, \dots, y_n</math>
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| is that <math>n\bar{y} = y_1 + y_2 + \cdots + y_n</math>. In other words, <math>\bar{y}</math> is the ''constant'' value which when
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| ''added'' to itself <math>n</math> times equals the result of adding the <math>n</math> terms of <math>y_i</math>. By analogy, a
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| defining property of the average value <math>\bar{f}</math> of a function over the interval <math>[a,b]</math> is that
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| : <math>\int_a^b\bar{f}\,dx = \int_a^bf(x)\,dx</math>
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| In other words, <math>\bar{f}</math> is the ''constant'' value which when ''integrated'' over <math>[a,b]</math> equals the result of
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| integrating <math>f(x)</math> over <math>[a,b]</math>. But by the second [[fundamental theorem of calculus]], the integral of a constant
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| <math>\bar{f}</math> is just
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| : <math>\int_a^b\bar{f}\,dx = \bar{f}x\bigr|_a^b = \bar{f}b - \bar{f}a = (b - a)\bar{f}</math>
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| See also the [[Mean_value_theorem#First_mean_value_theorem_for_integration|first mean value theorem for integration]], which guarantees
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| that if <math>f</math> is continuous then there exists a point <math>c \in (a, b)</math> such that
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| : <math>\int_a^bf(x)\,dx = f(c)(b - a)</math>
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| The point <math>f(c)</math> is called the mean value of <math>f(x)</math> on <math>[a,b]</math>. So we write
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| <math>\bar{f} = f(c)</math> and rearrange the preceding equation to get the above definition.
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| In several variables, the mean over a [[relatively compact]] [[neighborhood (mathematics)|domain]] ''U'' in a [[Euclidean space]] is defined by
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| :<math>\bar{f}=\frac{1}{\hbox{Vol}(U)}\int_U f.</math>
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| 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
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| :<math>\exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).</math>
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| 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.
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| There is also a ''harmonic average'' of functions and a ''quadratic average'' (or ''root mean square'') of functions.
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| ===Mean of a probability distribution===
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| See [[expected value]].
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| ===Mean of angles===
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| Sometimes the usual calculations of means fail on cyclical quantities such as [[angle]]s, times of day, and other situations where [[modular arithmetic]] is used. For those quantities it might be appropriate to use a [[mean of circular quantities]] to take account of the modular values, or to adjust the values before calculating the mean.
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| === Fréchet mean ===
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| 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.
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| It is sometimes also known as the '''Karcher mean''' (named after Hermann Karcher).
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| ===Other means===
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| <div style="-moz-column-count:3; column-count:3;">
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| *[[Arithmetic-geometric mean]]
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| *[[Arithmetic-harmonic mean]]
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| *[[Cesàro mean]]
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| *[[Chisini mean]]
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| *[[Contraharmonic mean]]
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| *[[Distance-weighted estimator]]
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| *[[Elementary symmetric mean]]
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| *[[Geometric-harmonic mean]]
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| *[[Heinz mean]]
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| *[[Heronian mean]]
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| *[[Identric mean]]
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| *[[Lehmer mean]]
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| *[[Logarithmic mean]]
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| *[[Median]] <!-- is a mean according to the definition below -->
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| *[[Moving average]]
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| *[[Root mean square]]
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| *[[Rényi's entropy]] (a [[generalized f-mean]])
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| *[[Stolarsky mean]]
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| *[[Weighted geometric mean]]
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| *[[Weighted harmonic mean]]
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| </div>
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| ==Distribution of the population mean==
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| {{main|Standard error of the mean}}
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| ===Using the sample mean===
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| The arithmetic mean of a [[Statistical population|population]], or population mean, is denoted ''μ''. The sample mean (the arithmetic mean of a sample of values drawn from the population) makes a good [[estimator]] of the population mean, as its expected value is equal to the population mean (that is, it is an [[unbiased estimator]]). The sample mean is a [[random variable]], not a constant, since its calculated value will randomly differ depending on which members of the population are sampled, 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 [[normal distribution|normally distributed]] with mean and variance as follows:
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| : <math>\bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.</math>
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| Often, since the population ''variance'' is an unknown parameter, it is estimated by the [[Sum of squares (statistics)|mean sum of squares]]; when this estimated value is used, the distribution of the sample mean is no longer a normal distribution but rather a [[Student's t distribution]] with ''n'' − 1 [[Degrees of freedom (statistics)|degrees of freedom]].
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| ==See also==
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| {{Portal|Statistics}}
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| *[[Algorithms for calculating variance]]
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| *[[Average]]
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| *[[Central tendency]]
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| *[[Descriptive statistics]]
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| *[[Kurtosis]]
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| *[[Law of averages]]
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| *[[Mean value theorem]]
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| *[[Median]]
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| *[[Mode (statistics)]]
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| *[[Spherical mean]]
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| *[[Summary statistics]]
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| *[[Taylor's law]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{MathWorld | urlname= Mean | title= Mean}}
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| * {{MathWorld | urlname= ArithmeticMean | title= Arithmetic Mean}}
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| * [http://www.sengpielaudio.com/calculator-geommean.htm Comparison between arithmetic and geometric mean of two numbers]
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| * [http://www.math.uni-bielefeld.de/~sillke/PUZZLES/means-trapezoid Some relationships involving means]
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| {{Statistics|descriptive}}
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| [[Category:Means| ]]
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| [[Category:Statistical terminology]]
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| [[Category:Theory of probability distributions]]
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| [[es:Media (matemáticas)]]
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| [[it:Media (statistica)]]
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| [[lt:Vidurkis]]
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| [[ar:متوسط]]
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