# Definition:Arithmetic Mean

*This page is about the arithmetic mean. For other uses, see Definition: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:

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

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

This 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 mean**, or just **mean value** or **mean**.

## Also see

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

- Cauchy's Mean Theorem
- Arithmetic Mean Never Less than Harmonic Mean
- Definition:Average Value of Function

- 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-ticThis is because the word is being used in its adjectival form.

## Sources

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of Mathematical Functions*... (previous) ... (next): $3.1.11$: Arithmetic Mean of $n$ quantities $A$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.12 \ (5)$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 3$: Natural Numbers: $\S 3.10$: Example - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.3$: Functions of discrete random variables: $(18)$