Arithmetic mean

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In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term "arithmetic mean" is preferred in mathematics and statistics because it helps distinguish it from other averages such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent. For example, per capita GDP gives an approximation of the arithmetic average income of a nation's population.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers. Notably, for skewed distributions, the arithmetic mean may not accord with one's notion of "middle", and robust statistics such as the median may be a better description of central tendency.

Contents

Definition

Suppose we have sample space \{x_1,\ldots,x_n\}. Then the arithmetic mean A is defined via the equation

If the list is a statistical population, then the mean of that population is called a population mean. If the list is a statistical sample, we call the resulting statistic a sample mean.

Motivating properties

The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include:

  • If numbers x_1,\ldots,x_n have mean X, then (x_1-X) + \ldots + (x_n-X) = 0. Since xiX is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals defined this way sum to zero.
  • If it is required to use a single number X as an estimate for the value of numbers x_1,\ldots,x_n, then the arithmetic mean does this best, in the sense of minimizing the sum of squares (xi − X)2 of the residuals. (It follows that the mean is also the best single predictor in the sense of having the lowest root mean squared error.)
  • For a normal distribution, the arithmetic mean is equal to both the median and the mode, other measures of central tendency.

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