mean
Loosely speaking, a mean is a way to describe a collection^{} of numbers such that the mean in some sense describe the “average^{}” entry of these numbers. The most familiar mean is the arithmetic mean, and unless otherwise noted, by mean, we always mean the arithmetic mean.
Example
The mean of the numbers $\{1,\mathrm{\hspace{0.17em}2},\mathrm{\dots},n\}$ is $\frac{n+1}{2}$.
Mathematically, we define a mean as follows:
Definition
A mean is a function^{} $f$ whose domain is the collection of all finite multisets of $\mathbb{R}$ and whose codomain is $\mathbb{R}$, such that

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$f$ is a homogeneous function of degree 1. That is, if $\{{x}_{1},\mathrm{\dots},{x}_{n}\}$ is a multiset, then
$$f(\{\lambda {x}_{1},\mathrm{\dots},\lambda {x}_{n}\})=\lambda f(\{{x}_{1},\mathrm{\dots},{x}_{n}\}),\lambda \ge 0.$$ 
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For any set $S=\{{x}_{1},\mathrm{\dots},{x}_{n}\}$ of real numbers,
$$\mathrm{min}\{{x}_{1},\mathrm{\dots},{x}_{n}\}\le f(S)\le \mathrm{max}\{{x}_{1},\mathrm{\dots},{x}_{n}\}.$$
Pythagoras identified three types of means: the arithmetic mean (http://planetmath.org/ArithmeticMean), the geometric mean^{}, and the harmonic mean^{}. 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 wellknown means include:

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median,

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mode,
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power mean^{}

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Lehmer mean^{}
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rootmeansquare^{} (sometimes called the quadratic mean),
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Heronian mean^{},

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Cesaro mean,

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maximum function, minimum function (http://planetmath.org/MinimalAndMaximalNumber)
Title  mean 
Canonical name  Mean 
Date of creation  20130322 12:43:43 
Last modified on  20130322 12:43:43 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  16 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 1100 
Classification  msc 6207 
Related topic  ArithmeticMean 
Related topic  GeometricMean 
Related topic  ContraharmonicProportion 
Related topic  OrderOfSixMeans 
Related topic  AverageValueOfFunction 