Loosely speaking, a mean is a way to describe a collectionMathworldPlanetmath of numbers such that the mean in some sense describe the “averageMathworldPlanetmath” entry of these numbers. The most familiar mean is the arithmetic mean, and unless otherwise noted, by mean, we always mean the arithmetic mean.


The mean of the numbers {1, 2,,n} is n+12.

Mathematically, we define a mean as follows:


A mean is a functionMathworldPlanetmath f whose domain is the collection of all finite multisets of and whose codomain is , such that

  • f is a homogeneous function of degree 1.  That is, if {x1,,xn} is a multiset, then

  • For any set S={x1,,xn} of real numbers,


Pythagoras identified three types of means: the arithmetic mean (, the geometric meanMathworldPlanetmath, and the harmonic meanMathworldPlanetmath. However, in the sense of the above definition, there is a wealth of ther means too. For instance, the minimum function and maximum functions can be seen as “trivial” means. Other well-known means include:

Title mean
Canonical name Mean
Date of creation 2013-03-22 12:43:43
Last modified on 2013-03-22 12:43:43
Owner matte (1858)
Last modified by matte (1858)
Numerical id 16
Author matte (1858)
Entry type Definition
Classification msc 11-00
Classification msc 62-07
Related topic ArithmeticMean
Related topic GeometricMean
Related topic ContraharmonicProportion
Related topic OrderOfSixMeans
Related topic AverageValueOfFunction