Mean

In mathematics, mean has several different definitions depending on the context.

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.[1] 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 $\mu = \sum x P(x)$.[2] 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 $2^n$ is $\tfrac{1}{2^n}$ for n = 1, 2, 3, ....

For a data set, the terms arithmetic mean, 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 x1, x2, ..., xn is typically denoted by $\bar{x}$, pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted $\bar{x}$) to distinguish it from the population mean (denoted $\mu$ or $\mu_x$).[3]

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.[4]

Outside of probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.

Types of mean

Pythagorean means

Main article: Pythagorean means

Arithmetic mean (AM)

The arithmetic mean (or simply "mean") of a sample $x_1,x_2,\ldots,x_n$ is the sum of the sampled values divided by the number of items in the sample:

$\bar{x} = \frac{x_1+x_2+\cdots +x_n}{n}$

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

$\frac{4 + 36 + 45 + 50 + 75}{5} = \frac{210}{5} = 42.$
Comparison of the arithmetic mean, median and mode of two skewed (log-normal) distributions.

The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.

Nevertheless, many skewed distributions are best described by their mean – such as the exponential and Poisson distributions.

Geometric mean (GM)

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.

$\bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^\tfrac1n$

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

$(4 \times 36 \times 45 \times 50 \times 75)^{^1/_5} = \sqrt[5]{24\;300\;000} = 30.$

Harmonic mean (HM)

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).

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

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

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

Relationship between AM, GM, and HM

AM, GM, and HM satisfy these inequalities:

$AM \ge GM \ge HM \,$

Equality holds only when all the elements of the given sample are equal.

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 for a set of n positive numbers xi by

$\bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^\tfrac1m$

By choosing different values for the parameter m, the following types of means are obtained:

 $m\rightarrow\infty$ maximum of $x_i$ $m=2$ quadratic mean $m=1$ arithmetic mean $m\rightarrow0$ geometric mean $m=-1$ harmonic mean $m\rightarrow-\infty$ minimum of $x_i$

ƒ-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 ƒ will give

 $f(x) = x$ arithmetic mean, $f(x) = \frac{1}{x}$ harmonic mean, $f(x) = x^m$ power mean, $f(x) = \ln x$ geometric mean.

Weighted arithmetic mean

The 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:

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

The weights $w_i$ 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.

Truncated mean

Sometimes a set of numbers might contain outliers, i.e., data values which are much lower or much higher than the others. Often, outliers are erroneous data caused by artifacts. In this case, one can use a truncated mean. 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. 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, so is simply a specific example of a weighted mean for a specific set of weights.

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

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

Recall that a defining property of the average value $\bar{y}$ of finitely many numbers $y_1, y_2, \dots, y_n$ is that $n\bar{y} = y_1 + y_2 + \cdots + y_n$. In other words, $\bar{y}$ is the constant value which when added to itself $n$ times equals the result of adding the $n$ terms of $y_i$. By analogy, a defining property of the average value $\bar{f}$ of a function over the interval $[a,b]$ is that

$\int_a^b\bar{f}\,dx = \int_a^bf(x)\,dx$

In other words, $\bar{f}$ is the constant value which when integrated over $[a,b]$ equals the result of integrating $f(x)$ over $[a,b]$. But by the second fundamental theorem of calculus, the integral of a constant $\bar{f}$ is just

$\int_a^b\bar{f}\,dx = \bar{f}x\bigr|_a^b = \bar{f}b - \bar{f}a = (b - a)\bar{f}$

See also the first mean value theorem for integration, which guarantees that if $f$ is continuous then there exists a point $c \in (a, b)$ such that

$\int_a^bf(x)\,dx = f(c)(b - a)$

The point $f(c)$ is called the mean value of $f(x)$ on $[a,b]$. So we write $\bar{f} = f(c)$ and rearrange the preceding equation to get the above definition.

In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

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

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

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

More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

There is also a harmonic average of functions and a quadratic average (or root mean square) of functions.

Mean of a probability distribution

See expected value.

Mean of angles

Sometimes the usual calculations of means fail on cyclical quantities such as angles, 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.

Fréchet mean

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. It is sometimes also known as the Karcher mean (named after Hermann Karcher).

Distribution of the population mean

Using the sample mean

The arithmetic mean of a 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 normally distributed with mean and variance as follows:

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

Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares; 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.