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# arithmetic mean

If $a_{1},\,a_{2},\,\ldots,\,a_{n}$ are real numbers, their *arithmetic mean* is defined as

$A.M.\;=\;\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}.$ |

The arithmetic mean is what is commonly called the *average* of the numbers. The value of $A.M.$ is always between the least and the greatest of the numbers $a_{j}$. If the numbers $a_{j}$ are all positive, then $A.M.\,>\,\frac{a_{j}}{n}$ for all $j$.

A generalization of this concept is that of *weighted mean*, also known as
*weighted average*. Let $w_{1},\ldots,w_{n}$ be numbers whose sum is not zero,
which will be known as *weights*. (Typically, these will be strictly
positive numbers, so their sum will automatically differ from zero.) Then the
weighted mean of $a_{1},a_{2},\ldots,a_{n}$ is defined to be

$W.M.\;=\;\frac{w_{1}a_{1}+w_{2}a_{2}+\ldots+w_{n}a_{n}}{w_{1}\!+\!w_{2}\!+\!% \ldots+\!w_{n}}.$ |

In the special case where all the weights are equal to each other, the weighted mean equals the arithmetic mean.

## Mathematics Subject Classification

26D15*no label found*11-00

*no label found*

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## Additional References

You may also be interested to read about averages, weighted averages, and means in the following book and article.

* Jane Grossman, Michael Grossman, Robert Katz. "Averages: A New Approach", ISBN 0977117049, 1983. (Available for reading at Google Book Search:

http://books.google.com/books?q=%22Non-Newtonian+Calculus%22&btnG=Search...).

* Michael Grossman and Robert Katz. "A new approach to means of two positive numbers", International Journal of Mathematical Education in Science and Technology, Volume 17, Number 2, March 1986, pages 205 - 208.