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 $ℝ$ and whose codomain is $ℝ$, such that

• $f$ is a homogeneous function of degree 1.  That is, if $\{x_1,\mathrm{\dots },x_n\}$ is a multiset, then

  $$f(\{\lambda x_1,\mathrm{\dots },\lambda x_n\})=\lambda f(\{x_1,\mathrm{\dots },x_n\}),\lambda \ge 0.$$

• For any set $S=\{x_1,\mathrm{\dots },x_n\}$ of real numbers,

  $$\min \{x_1,\mathrm{\dots },x_n\}\le f(S)\le \max \{x_1,\mathrm{\dots },x_n\}.$$

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 well-known means include:

• median,

• mode,

• generalized mean

• power mean

• Lehmer mean

• arithmetic-geometric mean,

• arithmetic-harmonic mean,

• harmonic-geometric mean,

• root-mean-square (sometimes called the quadratic mean),

• identric mean,

• contraharmonic mean,

• Heronian mean,

• Cesaro mean,

• maximum function, minimum function (http://planetmath.org/MinimalAndMaximalNumber)

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