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It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, median, and mode. For instance, average house prices almost always use the median value for the average. In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...

For a real-valued 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. In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are... A probability distribution describes the values and probabilities that a random event can take place. ... This article is about mathematics. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...

For a data set, the mean is the sum of the observations divided by the number of observations. 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. A data set (or dataset) is a collection of data, usually presented in tabular form. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...

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. The absolute deviation of an element of a data set is the absolute difference between that element and a given point. ...

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.

## Examples of means GA_googleFillSlot("encyclopedia_square");

### Arithmetic mean

Main article: Arithmetic mean

The arithmetic mean is the "standard" average, often simply called the "mean". In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

$bar{x} = frac{1}{n}cdot sum_{i=1}^n{x_i}$

The mean may often be confused with the median or mode. 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. This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...

That said, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions. In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ...

For example, the arithmetic mean of 34, 27, 45, 55, 22, 34 (six values) is (34+27+45+55+22+34)/6 = 217/6 ≈ 36.167.

### Geometric mean

The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth. The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

$bar{x} = left ( prod_{i=1}^n{x_i} right ) ^{1/n}$

For example, the geometric mean of 34, 27, 45, 55, 22, 34 (six values) is (34×27×45×55×22×34)1/6 = 1,699,493,4001/6 = 34.545.

### Harmonic mean

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). In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... Measurement is the determination of the size or magnitude of something. ... This article does not cite any references or sources. ...

$bar{x} = n cdot left ( sum_{i=1}^n frac{1}{x_i} right ) ^{-1}$

For example, the harmonic mean of the numbers 34, 27, 45, 55, 22, and 34 is

$frac{6}{frac{1}{34}+frac{1}{27}+frac{1}{45} + frac{1}{55} + frac{1}{22}+frac{1}{34}}approx 33.0179836.$

### Generalized means

#### Power mean

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 by A generalized mean, also known as power mean or HÃ¶lder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ...

$bar{x}(m) = left ( frac{1}{n}cdotsum_{i=1}^n{x_i^m} right ) ^{1/m}$

By choosing the appropriate value for the parameter m we get

 $mrightarrowinfty$ maximum m = 2 quadratic mean, m = 1 arithmetic mean, $mrightarrow0$ geometric mean, m = − 1 harmonic mean, $mrightarrow-infty$ minimum.

The largest and the smallest element of a set are called extreme values, or extreme records. ... In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity: its specially useful when variates are positive and negative, ie the cases of waves. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... The largest and the smallest element of a set are called extreme values, or extreme records. ...

#### f-mean

This can be generalized further as the generalized f-mean In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ...

$bar{x} = f^{-1}left({frac{1}{n}cdotsum_{i=1}^n{f(x_i)}}right)$

and again a suitable choice of an invertible f will give

 $f(x) = frac{1}{x}$ harmonic mean, f(x) = xm power mean, f(x) = lnx geometric mean.

In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

### Weighted arithmetic mean

The weighted arithmetic mean is used, if one wants to combine average values from samples of the same population with different sample sizes: In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ...

$bar{x} = frac{sum_{i=1}^n{w_i cdot x_i}}{sum_{i=1}^n {w_i}}$

The weights wi 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 (the data) might be contaminated by inaccurate outliers, i.e. values which are much too low or much too high. 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. For other uses, see Data (disambiguation). ... A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. ...

### Interquartile mean

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. The interquartile mean (IQM) is a statistical measure of central tendency, much like the mean (in more popular terms called the average), the median, and the mode. ...

$bar{x} = {2 over n} sum_{i=(n/4)+1}^{3n/4}{x_i}$

assuming the values have been ordered.

### Mean of a function

In calculus, 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 For other uses, see Calculus (disambiguation). ... Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ... In mathematics, the domain of a function is the set of all input values to the function. ...

$bar{f}=frac{1}{b-a}int_a^bf(x),dx.$

(See also mean value theorem.) In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ... In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

$bar{f}=frac{1}{hbox{Vol}(U)}int_U f.$

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

$expleft(frac{1}{hbox{Vol}(U)}int_U log fright).$

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. In mathematics, a measure is a function that assigns a number, e. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In mathematics, Jensens 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. ...

### Mean of angles

Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. For those quantities you need a mean of circular quantities. This article is about angles in geometry. ... Look up daytime in Wiktionary, the free dictionary. ... In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a mean of circular quantities is a mean which suited for quantities like angles, daytimes, fractional parts of real numbers. ...

## Properties

The most general method for defining a mean or average, y, takes any function of a list g(x_1, x_2, ..., x_n), which is symmetric under permutation of the members of the list, and equates it to the same function with the value of the mean replacing each member of the list: g(x_1, x_2, ..., x_n) = g(y, y, ..., y). All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.

### Weighted mean

A weighted mean M is a function which maps tuples of positive numbers to a positive number ($mathbb{R}_{>0}^ntomathbb{R}_{>0}$).

• "Fixed point": $M(1,1,dots,1) = 1$
• Homogenity: $foralllambda forall x M(lambdacdot x_1, dots, lambdacdot x_n) = lambda cdot M(x_1, dots, x_n)$
(using vector notation: $foralllambda forall x M(lambdacdot x) = lambda cdot M x$)
• Monotony: $forall x forall y (forall i x_i le y_i) Rightarrow M x le M y$

It follows In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ... In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ... In 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. ... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ...

• Boundedness: $forall x M x in [min x, max x]$
• Continuity: $lim_{xto y} M x = M y$
Sketch of a proof: Because $forall x forall y left(||x-y||_inftylevarepsiloncdotmin x Rightarrow forall i |x_i-y_i|levarepsiloncdot x_iright)$ and $M((1+varepsilon)cdot x) = (1+varepsilon)cdot M x$ it follows $forall x forall varepsilon>0 forall y ||x-y||_inftylevarepsiloncdotmin x Rightarrow |Mx-My|levarepsilon$.
• There are means, which are not differentiable. 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 median), but is not differentiable.
• All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties.
• If f is bijective, then the generalized f-mean satisfies the fixed point property.
• If f is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
• In general a generalized f-mean will miss homogenity.

If $C, M_1, dots, M_m$ are weighted means, p is a positive real number, then A,B with In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

$forall x A x = C(M_1 x, dots, M_m x)$
$forall x B x = sqrt[p]{C(x_1^p, dots, x_n^p)}$

are also a weighted mean.

### 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 being an unweighted mean if it is a weighted mean and for each permutation π of inputs, the result is the same. Let P be the set of permutations of n-tuples. Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...

Symmetry: $forall x forall piin P M x = M(pi x)$

Analogously to the weighted means, if C is a weighted mean and $M_1, dots, M_m$ are unweighted means, p is a positive real number, then A,B with In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

$forall x A x = C(M_1 x, dots, M_m x)$
$forall x B x = sqrt[p]{M_1(x_1^p, dots, x_n^p)}$

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 $a_1,dots,a_n$. (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean A is obtained by In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...

$A(x_1,dots,x_n) = M(underbrace{x_1,dots,x_1}_{a_1},x_2,dots,x_{n-1},underbrace{x_n,dots,x_n}_{a_n}).$

### Means of tuples of different sizes

If a mean M is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely

• Given an arbitrary tuple x, which is partitioned into $y_1, dots, y_k$, then it holds $M x in mathrm{convexhull}(M y_1, dots, M y_k)$. (See Convex hull)

A partition of U into 6 blocks: an Euler diagram representation. ... Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ...

## Population and sample means

The mean of a normal distribution population 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 Probability density function of Gaussian distribution (bell curve). ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

$bar{x} thicksim Nleft{mu, frac{sigma^2}{n}right}.$

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. Sum of squares is a concept that permeates much of inferential statistics and descriptive statistics. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ...

## Mathematics education

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