# 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 $\{1,\,2,\,\ldots,\,n\}$ 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

• $f$ is a homogeneous function of degree 1.  That is, if $\{x_{1},\ldots,x_{n}\}$ is a multiset, then

 $f(\{\lambda x_{1},\ldots,\lambda x_{n}\})=\lambda f(\{x_{1},\ldots,x_{n}\}),% \quad\lambda\geq 0.$
• For any set $S=\{x_{1},\ldots,x_{n}\}$ of real numbers,

 $\min\{x_{1},\ldots,x_{n}\}\leq f(S)\leq\max\{x_{1},\ldots,x_{n}\}.$

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,

• root-mean-square (sometimes called the quadratic 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