Mean

In statistics, mean has two related meanings:
 * the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. The average is also called sample mean.
 * the expected value of a random variable, which is also called the population mean.

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. See the Other means section below for a list of means.

Sample mean is often used as an estimator of the central tendency such as the population mean. However, other estimators are also used. For example, the median is a more robust estimator of the central tendency than the sample mean.

For a real-valued random variable X, the mean is the expectation of X. If the expectation does not exist, then the random variable has no mean.

For a data set, the mean is just the sum of all the observations divided by the number of observations. Once we have chosen this method of describing the communality of a data set, we usually use the standard deviation to describe how the observations differ. The standard deviation is the square root of the average of squared deviations from the mean.

The mean is the unique value about which the sum of squared deviations is a minimum. If you calculate the sum of squared deviations from any other measure of central tendency, it will be larger than for the mean. This explains why the standard deviation and the mean are usually cited together in statistical reports.

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 tractable when combining data sets.

Note that not every probability distribution has a defined mean or variance &mdash; see the Cauchy distribution for an example.

The following is a summary of some of the multiple methods for calculating the mean of a set of n numbers. See the table of mathematical symbols for explanations of the symbols used.

Arithmetic mean
The arithmetic mean is the "standard" average, often simply called the "mean".


 * $$ \bar{x} = {1 \over n} \sum_{i=1}^n{x_i} $$

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 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.

That said, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions.

An example
An experiment yields the following data: 34,27,45,55,22,34 To get the arithmetic mean
 * 1) How many items? There are 6. Therefore n=6
 * 2) What is the sum of all items? It is 217. Therefore Sigma x (Sigma means sum) is 217
 * 3) To get the arithmetic mean divide sigma by n, here 217/6= 36.1666666667

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.


 * $$ \bar{x} = \sqrt[n]{\prod_{i=1}^n{x_i}} $$

An example
An experiment yields the following data: 34,27,45,55,22,34 To get the geometric mean
 * 1) How many items? There are 6. Therefore n=6
 * 2) What is the product of all items? It is 1699493400.
 * 3) To get the geometric mean take the nth (the 6th) root of that product; it is 34.5451100372

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).


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

An example
An experiment yields the following data: 34,27,45,55,22,34 To get the harmonic mean
 * 1) How many items? There are 6. Therefore n=6
 * 2) What is the sum on the bottom of the fraction? It is 0.181719152307
 * 3) Get the reciprocal of that sum. It is 5.50299727522
 * 4) To get the harmonic mean multiply that by n to get 33.0179836513

Generalized mean
The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. It is defined by


 * $$ \bar{x}(m) = \sqrt[m]{\frac{1}{n}\sum_{i=1}^n{x_i^m}} $$

By choosing the appropriate value for the parameter m we can get the arithmetic mean (m = 1), the geometric mean (m &rarr; 0) or the harmonic mean (m = &minus;1)

This can be generalized further as the generalized f-mean
 * $$ \bar{x} = f^{-1}\left({\frac{1}{n}\sum_{i=1}^n{f(x_i)}}\right) $$

and again a suitable choice of an invertible f(x) will give the arithmetic mean with f(x) = x, the geometric mean with f(x) = log(x), and the harmonic mean with f(x) = 1/x.

Weighted mean
The weighted 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. 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.

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.

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

(See also mean value theorem.) 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.

Other means

 * Arithmetic-geometric mean
 * Arithmetic-harmonic mean
 * Cesàro mean
 * Chisini mean
 * Geometric-harmonic mean
 * Heronian mean
 * Identric mean
 * Lehmer mean
 * Quadratic mean
 * root mean square
 * Stolarsky mean
 * weighted geometric mean
 * weighted harmonic mean
 * Rényi's entropy (a generalized f-mean)