# 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