Arithmetic mean

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In mathematics and statistics, the arithmetic mean ([ærɪθˈmɛtɪk miːn], stress on third syllable), or simply 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. The collection is often a set of results of an experiment or an observational study.

Definition

The arithmetic mean $\bar{x}$ (read $x$ bar) is the mean of the $n$ values $x_1, x_2, \ldots, x_n$.

The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation divided by the total number of observations. Symbolically, if we have a data set consisting of the values $a_1, a_2, \ldots, a_n$, then the arithmetic mean $A$ is defined by the formula:

$A=\frac{1}{n}\sum_{i=1}^n a_i=\frac{a_1+a_2+\cdots+a_n}{n}$

If the data set is a statistical population (i. e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean. If the data set is a statistical sample (a subset of the population), we call the statistic resulting from this calculation a sample mean.

The arithmetic mean is a special case of the weighted arithmetic mean in which all the weights are equal to each other.