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 (http://planetmath.org/MinimalAndMaximalNumber) $a_{j}$ . 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.

Title	arithmetic mean
Canonical name	ArithmeticMean
Date of creation	2013-03-22 11:50:42
Last modified on	2013-03-22 11:50:42
Owner	drini (3)
Last modified by	drini (3)
Numerical id	14
Author	drini (3)
Entry type	Definition
Classification	msc 26D15
Classification	msc 11-00
Synonym	average
Synonym	mean
Related topic	GeometricMean
Related topic	HarmonicMean
Related topic	ArithmeticGeometricMeansInequality
Related topic	GeneralMeansInequality
Related topic	WeightedPowerMean
Related topic	PowerMean
Related topic	GeometricDistribution2
Related topic	RootMeanSquare3
Related topic	ProofOfGeneralMeansInequality
Related topic	ProofOfArithmeticGeometricHarmonicMeansInequality
Related topic	DerivationOfHarm
Defines	weighted mean
Defines	weighted average