mean
Loosely speaking, a mean is a way to describe a collection of numbers suchthat 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 is .
Mathematically, we define a mean as follows:
Definition
A mean is a function whose domain is the collection ofall finite multisets of and whose codomain is ,such that
- •
is a homogeneous function of degree 1. That is, if is a multiset, then
- •
For any set of real numbers,
Pythagoras identified three types of means: the arithmetic mean (http://planetmath.org/ArithmeticMean), the geometricmean, 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 maximumfunctions 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 |