arithmetic mean

If  a1,a2,,an  are real numbers, their arithmetic meanMathworldPlanetmath is defined as


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 ( aj.  If the numbers aj are all positive, then  A.M.>ajn  for all j.

A generalizationPlanetmathPlanetmath of this concept is that of weighted mean, also known as weighted average.  Let w1,,wn 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 a1,a2,,an is defined to be


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