Mean

In statistics, mean has two related meanings:

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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 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.

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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.

Related Topics:
Estimator - Central tendency - Median - Robust

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For a real-valued random variable X, the mean is the expectation of X.

Related Topics:
Random variable - Expectation

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If the expectation does not exist, then the random variable has no mean.

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For a data set, the mean is just the sum of all the observations divided by the number of observations.

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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.

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The standard deviation is the square root of the average of squared deviations from the mean.

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The mean is the unique value about which the sum of squared deviations is a minimum.

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If you calculate the sum of squared deviations from any other measure of central tendency, it will be larger than for the mean.

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This explains why the standard deviation and the mean are usually cited together in statistical reports.

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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.

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The mean value of a function, f(x), on an interval, a

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:E(f(X))=rac{int_a^b f(x),dx}{b-a}.

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

Related Topics:
Probability distribution - Variance - Cauchy distribution

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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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Mean: Arithmetic mean ►