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# Mean

In statistics, mean has two related meanings:

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.

The mean value of a function, f(x), on an interval, a < x < b, can also be calculated (using a limiting process on the data set definition) thus:

$E(f(X))=\frac{\int_a^b f(x)\,dx}{b-a}.$

Note that not every probability distribution has a defined mean or variance - 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.

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## Arithmetic mean

The arithmetic mean is the "standard" average, often simply called the "mean". It is used for many purposes but also often abused by incorrectly using it to describe skewed distributions, with highly misleading results. The classic example is average income - using the arithmetic mean makes it appear to be much higher than is in fact the case. Consider the scores {1, 2, 2, 2, 3, 9}. The arithmetic mean is 3.16, but five out of six scores are below this!

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

## Geometric mean

The geometric mean is an average which is useful for sets of numbers which are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth.

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

## 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).

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

## Generalized mean

The generalized mean is an abstraction of the Arithmetic, Geometric and Harmonic Means.

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

By choosing the appropriate value for the parameter m we can get the arithmetic mean (m = 1), the geometric mean (m -> 0) or the harmonic mean (m = -1)

This could be generalised further as

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

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 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.

## 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.