Mean

In Statistics , ''mean'' has two related meanings:

the Arithmetic Mean (and is distinguished from the Geometric Mean or Harmonic Mean ).

the Expected Value of a Random Variable , which is also called the ''population mean''.

It is sometimes stated that the 'mean' means average. This is incorrect as there are different types of averages: the mean, median, and mode. For instance, average house prices almost always use the median value for the average.

For a real-valued Random Variable ''X'', the mean is the Expectation of ''X''.
Note that not every Probability Distribution has a defined mean (or Variance ); see the Cauchy Distribution for an example.

For a Data Set , the mean is the sum of the observations divided by the number of observations. The mean is often quoted along with the Standard Deviation : the mean describes the central location of the data, and the standard deviation describes the spread.

An alternative measure of dispersion is the mean deviation, equivalent to the average Absolute Deviation from the mean. It is less sensitive to outliers, but less mathematically tractable.

As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below.

EXAMPLES OF MEANS

Arithmetic mean

The Arithmetic Mean is the "standard" average, often simply called the "mean".

:

\bar{x} = rac{1}{n}\cdot \sum_{i=1}^n{x_i}

The mean may often be confused with the Median or Mode . The mean is the arithmetic average of a set of values, or distribution; however, for Skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.

That said, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions.

For example, the arithmetic mean of 34, 27, 45, 55, 22, 34 (six values) is (34+27+45+55+22+34)/6 = 217/6 ≈ 36.167.

Geometric mean

The Geometric Mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth.

:

\bar{x} = \left ( \prod_{i=1}^n{x_i} 
ight ) ^{1/n}

For example, the geometric mean of 34, 27, 45, 55, 22, 34 (six values) is (34×27×45×55×22×34)^1/6 = 1,699,493,400^1/6 ≈ 34.545.

Harmonic mean

The Harmonic Mean is an average which is useful for sets of numbers which are defined in relation to some Unit , for example Speed (distance per unit of time).

:

\bar{x} = n \cdot \left ( \sum_{i=1}^n rac{1}{x_i} 
ight ) ^{-1}

For example, the harmonic mean of the numbers 34, 27, 45, 55, 22, and 34 is
:

rac{6}{rac{1}{34}+rac{1}{27}+rac{1}{45} + rac{1}{55} + rac{1}{22}+rac{1}{34}}\approx 33.0179836.

Generalized means

Power mean

The Generalized Mean , also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means. It is defined by

:

\bar{x}(m) = \left ( rac{1}{n}\cdot\sum_{i=1}^n{x_i^m} 
ight ) ^{1/m}

By choosing the appropriate value for the parameter ''m'' we get

$m
ightarrow\infty$ - Maximum ,

$m=2$ - Quadratic Mean ,

$m=1$ - Arithmetic Mean ,

$m
ightarrow0$ - Geometric Mean ,

$m=-1$ - Harmonic Mean ,

$m
ightarrow-\infty$ - Minimum .

f-mean

This can be generalized further as the Generalized F-mean
:

\bar{x} = f^{-1}\left({rac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}
ight)

and again a suitable choice of an invertible

f

will give

$f(x) = x$ - Arithmetic Mean ,

$f(x) = rac{1}{x}$ - Harmonic Mean ,

$f(x) = x^m$ - Power Mean ,

$f(x) = \ln x$ - Geometric Mean .

Weighted arithmetic mean

The Weighted Arithmetic Mean is used, if one wants to combine average values from samples of the same population with different sample sizes:

:

\bar{x} = rac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}}

The weights

w_i

represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.

Truncated mean

Sometimes a set of numbers (the Data ) might be contaminated by inaccurate outliers, i.e. values which are much too low or much too high. In this case one can use a Truncated Mean . It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values.

Interquartile mean

The Interquartile Mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
:

\bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i}

assuming the values have been ordered.

Mean of a function

In Calculus , and especially Multivariable Calculus , the mean of a function is loosely defined as the average value of the function over its Domain . In one variable, the mean of a function ''f''(''x'') over the interval (''a,b'') is defined by

:

\bar{f}=rac{1}{b-a}\int_a^bf(x)\,dx.

(See also Mean Value Theorem .) In several variables, the mean over a Relatively Compact Domain ''U'' in a Euclidean Space is defined by

:

\bar{f}=rac{1}{\hbox{Vol}(U)}\int_U f.

This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the '''geometric''' mean to functions by defining the geometric mean of ''f'' to be

:

\exp\left(rac{1}{\hbox{Vol}(U)}\int_U \log f
ight).

More generally, in Measure Theory and Probability Theory either sort of mean plays an important role. In this context, Jensen's Inequality places Sharp Estimate s on the relationship between these two different notions of the mean of a function.

Mean of angles

Most of the usual means fail on circular quantities, like Angle s, Daytime s, Fractional Part s of Real Number s.
For those quantities you need a Mean Of Circular Quantities .

Other means

Arithmetic-geometric Mean

Arithmetic-harmonic Mean

Cesàro Mean

Chisini Mean

Contraharmonic Mean

Elementary Symmetric Mean

Geometric-harmonic Mean

Heinz Mean

Heronian Mean

Identric Mean

Lehmer Mean

Logarithmic Mean

Median

Root Mean Square

Stolarsky Mean

Temporal Mean

Weighted Geometric Mean

Weighted Harmonic Mean

Rényi's Entropy (a Generalized F-mean )

PROPERTIES

Except some examples there seems to be no consensus, what a mean actually is.
However many means share some properties, that we collect here as a trial of defining the term ''mean''.

Weighted mean

A weighted mean

M

is a function which maps tuples of positive numbers to a positive number
(

\mathbb{R}_{>0}^n	o\mathbb{R}_{>0}

" Fixed Point ": $M(1,1,\dots,1) = 1$

Homogenity : $orall\lambda\ orall x\ M(\lambda\cdot x_1, \dots, \lambda\cdot x_n) = \lambda \cdot M(x_1, \dots, x_n)$

Vector

orall\lambda\ orall x\ M(\lambda\cdot x) = \lambda \cdot M x

Monotony : $orall x\ orall y\ (orall i\ x_i \le y_i) \Rightarrow M x \le M y$

It follows

Boundedness : $orall x\ M x \in x, \max x$

Continuity : $\lim_{x o y} M x = M y$

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT MEAN

	means

	mathematics education

Information About

Mean