Arithmetic mean

In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmiːn/ arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection.^[1] The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic.

"X̄" redirects here. For the character, see macron (diacritic).

In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent. For example, per capita income is 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: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.

If numbers $x_{1},\dotsc ,x_{n}$ have mean ${\bar {x}}$ , then $(x_{1}-{\bar {x}})+\dotsb +(x_{n}-{\bar {x}})=0$ . Since $x_{i}-{\bar {x}}$ is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which the (deviations from the estimate) sum to zero. This can also be interpreted as saying that the mean is translationally invariant in the sense that for any real number $a$ , ${\overline {x+a}}={\bar {x}}+a$ .

residuals

If it is required to use a single number as a "typical" value for a set of known numbers $x_{1},\dotsc ,x_{n}$ , then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of $(x_{i}-{\bar {x}})^{2}$ . The sample mean is also the best single predictor because it has the lowest .^[3] If the arithmetic mean of a population of numbers is desired, then the estimate of it that is unbiased is the arithmetic mean of a sample drawn from the population.

root mean squared error

Firstly, angle measurements are only defined up to an additive constant of 360° ( $2\pi$ or $\tau$ , if measuring in ). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.

radians

Secondly, in this situation, 0° (or 360°) is geometrically a better average value: there is lower about it (the points are both 1° from it and 179° from 180°, the putative average).

dispersion

Symbols and encoding[edit]

The arithmetic mean is often denoted by a bar (vinculum or macron), as in ${\bar {x}}$ .^[3]

Some software (text processors, web browsers) may not display the "x̄" symbol correctly. For example, the HTML symbol "x̄" combines two codes — the base letter "x" plus a code for the line above ( ̄ or ¯).^[7]

In some document formats (such as PDF), the symbol may be replaced by a "¢" (cent) symbol when copied to a text processor such as Microsoft Word.

Fréchet mean

Generalized mean

Inequality of arithmetic and geometric means

Sample mean and covariance

Standard deviation

Standard error of the mean

Summary statistics

Huff, Darrell (1993). . W. W. Norton. ISBN 978-0-393-31072-6.