Mean - Citizendia

For other uses, see Mean (disambiguation).

In statistics, mean has two related meanings:

the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. 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 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 In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average.
the expected value of a random variable, which is also called the population mean. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way

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. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. For instance, average house prices almost always use the median value for the average.

For a real-valued random variable X, the mean is the expectation of X. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way Note that not every probability distribution has a defined mean (or variance); see the Cauchy distribution for an example. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution.

For a data set, the mean is the sum of the observations divided by the number of observations. A data set (or dataset) is a collection of Data, usually presented in tabular form 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. In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values

An alternative measure of dispersion is the mean deviation, equivalent to the average absolute deviation from the mean. In Statistics, the absolute deviation of an element of a Data set is the absolute difference between that element and a given point It is less sensitive to outliers, but less mathematically tractable.

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

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

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

The mean may often be confused with the median or mode. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. 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). In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued 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 distributions. WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one In Probability theory and Statistics, the Poisson distribution is a Discrete probability distribution that expresses the probability of a number of events

For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is:

$\frac{34+27+45+55+22+34}{6} = \frac{217}{6} \approx 36.167.$

Geometric mean

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). 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 For example rates of growth.

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

For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:

$(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^{1/6} = 1,699,493,400^{1/6} = 34.545.$

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. Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of

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

For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is

$\frac{6}{\frac{1}{34}+\frac{1}{27}+\frac{1}{45} + \frac{1}{55} + \frac{1}{22}+\frac{1}{34}} = \frac{60588}{1835} \approx 33.0179836.$

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. A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric It is defined for a set of n positive numbers x_i by

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

By choosing the appropriate value for the parameter m we get

$m\rightarrow\infty$	maximum
$m = 2$	quadratic mean,
$m = 1$	arithmetic mean,
$m\rightarrow0$	geometric mean,
$m = - 1$	harmonic mean,
$m\rightarrow-\infty$	minimum. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the 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 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 In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average.

f-mean

This can be generalized further as the generalized f-mean

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

and again a suitable choice of an invertible $f$ will give

$f(x) = \frac{1}{x}$	harmonic mean,
$f (x) = x m$	power mean,
$f (x) = ln x$	geometric mean. In Mathematics and Statistics, the quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such 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 Pythagorean means including arithmetic, geometric 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

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:

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

The weights $w i$ represent the bounds of the partial sample. The weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average 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. Debt AIDS Trade in Africa (or DATA) is a Multinational non-government organization founded in January 2002 in London by U2 's Often, outliers are erroneous data caused by artifacts. In Natural science and Signal processing, an artifact is any perceived Distortion or other Data error caused by the instrument of observation In this case one can use a truncated mean. A truncated mean or trimmed mean is a Statistical Measure of central tendency, much like the Mean and Median. 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 mean is a specific example of a truncated mean. The interquartile mean (IQM is a statistical measure of central tendency, much like the Mean (in more popular terms called the Average) the It is simply the arithmetic mean after removing the lowest and the highest quarter of values.

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

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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve In Mathematics, the domain of a given function is the set of " Input " 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

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

(See also mean value theorem. In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative ) In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

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

This generalizes the arithmetic mean. In Mathematics, a relatively compact subspace (or relatively compact subset) Y of a Topological space X is a subset whose closure In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be

$\exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).$

More generally, in measure theory and probability theory either sort of mean plays an important role. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Probability theory is the branch of Mathematics concerned with analysis of random phenomena 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, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a Convex function of an Integral

Mean of angles

Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The DAYTIME service is an Internet protocol defined in RFC 867 In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The In Mathematics, the real numbers may be described informally in several different ways For those quantities you need a 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

Other means

Properties

The most general method for defining a mean or average, y, takes any function of a list g(x_1, x_2, . In Mathematics, the arithmetic-geometric mean (AGM of two positive Real numbers x and y is defined as follows First compute the Arithmetic 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 In Mathematics, the Cesàro means (also called Cesàro averages) of a Sequence ( a n) are the terms of the sequence ( In Mathematics, a function f of n variables x 1. x n leads to a Chisini Contraharmonic mean describes a mean of a set of numbers that is complementary to the Harmonic mean. The elementary symmetric mean is based on Elementary symmetric polynomials If x is a Tuple with x = (x_1\dotsx_n and\alpha_p(x In Mathematics, the geometric-harmonic mean M( x, y) of two positive Real numbers x and y is defined as follows we first The Heinz mean of two non-negative Real numbers A and B was defined by Bhatia as H_x(A B = \frac{A^x B^{1-x} + A^{1-x} B^x}{2} The Heronian mean H of two non-negative Real numbers A and B is given by the formula H = \frac{1}{3}(A + \sqrt{A B} +B\ The Identric mean of two positive Real numbers xy is defined as \begin{matrix}I(xy&=&\frac{1}{e}\cdot\lim_{(\xi\eta\to(xy}\sqrt{\frac{\xi^\xi}{\eta^\eta}}\\&=&\lim_{(\xi\eta\to(xy The Lehmer mean of a Tuple x of positive Real numbers is defined as L_p(x = \frac{\sum_{k=1}^{n} x_k^p}{\sum_{k=1}^{n} x_k^{p-1}} In Mathematics, the logarithmic mean is a function of two Numbers which is equal to their Difference divided by the Logarithm of their In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the In Mathematics, the Stolarsky mean of two positive Real numbers xy is defined as \begin{matrix}S_p(xy&=&\lim_{(\xi\eta\to(xy} In Statistics, given a set of data X = { x 1 x 2. x n } and corresponding In Information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity uncertainty or randomness In Mathematics and Statistics, the quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such . . , 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 ( $\mathbb{R}_{>0}^n\to\mathbb{R}_{>0}$ ).

"Fixed point": $M(1,1,\dots,1) = 1$
Homogenity: $\forall\lambda\ \forall x\ M(\lambda\cdot x_1, \dots, \lambda\cdot x_n) = \lambda \cdot M(x_1, \dots, x_n)$

(using vector notation: $\forall\lambda\ \forall x\ M(\lambda\cdot x) = \lambda \cdot M x$ )

Monotony: $\forall x\ \forall y\ (\forall i\ x_i \le y_i) \Rightarrow M x \le M y$

It follows

Boundedness: $\forall x\ M x \in [\min x, \max x]$
Continuity: $\lim_{x\to y} M x = M y$

Sketch of a proof: Because $\forall x\ \forall y\ \left(||x-y||_\infty\le\varepsilon\cdot\min x \Rightarrow \forall i\ |x_i-y_i|\le\varepsilon\cdot x_i\right)$ and $M((1+\varepsilon)\cdot x) = (1+\varepsilon)\cdot M x$ it follows $\forall x\ \forall \varepsilon>0\ \forall y\ ||x-y||_\infty\le\varepsilon\cdot\min x \Rightarrow |Mx-My|\le\varepsilon$ . 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 a factor then the result is multiplied by some power 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 In Mathematics, especially in Order theory, an upper bound of a Subset S of some Partially ordered set ( P, &le In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

There are means, which are not differentiable. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change 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. A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution
All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties. In Mathematics and Statistics, the quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such
- 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.

The above properties imply techniques to construct more complex means:

If $C, M_1, \dots, M_m$ are weighted means, $p$ is a positive real number, then $A, B$ with

$\forall x\ A x = C(M_1 x, \dots, M_m x)$

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

are also a weighted mean. In Mathematics, the real numbers may be described informally in several different ways

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. In several fields of Mathematics the term permutation is used with different but closely related meanings Let $P$ be the set of permutations of $n$ -tuples.

Symmetry: $\forall x\ \forall \pi\in P \ M x = M(\pi x)$

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

$\forall x\ A x = C(M_1 x, \dots, M_m x)$

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

are also unweighted means. In Mathematics, the term "symmetric function" can mean two different things In Mathematics, the real numbers may be described informally in several different ways

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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the Least common multiple of the Denominators ) Then the corresponding weighted mean $A$ is obtained by

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

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 $M x \in \mathrm{convexhull}(M y_1, \dots, M y_k)$ . In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " (See Convex hull)

Population and sample means

The mean of a normally distributed population has an expected value of μ, known as the population mean. In Mathematics, the convex hull or convex envelope for a set of points X in a Real Vector space V is the minimal Convex The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Statistics, a statistical population is a set of entities concerning which Statistical inferences are to be drawn often based on a Random sample The sample mean makes a good estimator of the population mean, as its expected value is the same as the population mean. In Statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population Parameter (which is called the The sample mean of a population is a random variable, not a constant, and consequently it will have its own distribution. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way For a random sample of n observations from a normally distributed population, the sample mean distribution is

$\bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.$

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, Student's t -distribution (or simply the t -distribution) is a Probability distribution

Mathematics education

In many state and government curriculum standards, students are traditionally expected to learn either the meaning or formula for computing the mean by the fourth grade. However, in many standards-based mathematics curricula, students are encouraged to invent their own methods, and may not be taught the traditional method. Principles and Standards for School Mathematics is a book produced by the National Council of Teachers of Mathematics (NCTM in 2000 to set forth a national vision for precollege Reform based texts such as TERC in fact discourage teaching the traditional "add the numbers and divide by the number of items" method in favor of spending more time on the concept of median, which does not require division. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution However, mean can be computed with a simple four-function calculator, while median requires a computer. The same teacher guide devotes several pages on how to find the median of a set, which is judged to be simpler than finding the mean.

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