# Definition:Arithmetic Mean

## Definition

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers.

The arithmetic mean of $x_1, x_2, \ldots, x_n$ is defined as:

$\ds A_n := \dfrac 1 n \sum_{k \mathop = 1}^n x_k$

That is, to find out the arithmetic mean of a set of numbers, add them all up and divide by how many there are.

## Also known as

The arithmetic mean is the quantity that is usually referred to in natural language as the "average" of a set of numbers.

It is also known as:

the common average
the common mean
the mean value
just the mean.

In the context of probability theory, the arithmetic mean of a random variable is often used as a synonym for its expectation.

## Also see

Note that there are several kinds of average and indeed, several different kinds of mean. This is just one of them.

Other measures of central tendency that are often introduced in schools at the same time:

• Results about arithmetic mean can be found here.

## Linguistic Note

In the context of an arithmetic mean, the word arithmetic is pronounced with the stress on the first and third syllables: a-rith-me-tic, rather than on the second syllable: a-rith-me-tic.

This is because the word is being used in its adjectival form.