# 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,

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