Mean

In statistics, mean has two related meanings:

• what we call the average in ordinary English. It is sometimes called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. Also called sample mean.
• the expected value of a random variable. Also called population mean.

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 SD (as we abbreviate it) 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.

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:

<math>E(f(X))=\frac{\int_a^b f(x)\,dx}{b-a}.</math>

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

See also: Central tendency, Summary statistics, Descriptive statistics, Variance, Skewness, Kurtosis