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• 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 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.

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

Note that not every probability distributionIn mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra has a defined mean or varianceThis article is about mathematics. Alternate meaning: variance (land use). In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically - see the Cauchy distributionThe Cauchy distribution is a probability distribution with probability density function : where x is the location parameter and s is the scale parameter''. The special case when x 0 and s 1 is called the standard Cauchy distribution with the probability d 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 symbolsIn mathematics, a set of symbols is frequently used in mathematical expressions. As mathematicians are familiar with these symbols, they are not explained each time they are used. So, for mathematical novices, the following table lists many common symbols for explanations of the symbols used.