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 $$ f(\{ \lambda x_1, \ldots, \lambda x_n\}) = \lambda f(\{x_1, \ldots, x_n\}),\quad \lambda\ge 0. $$
• 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, 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