Wikia

Psychology Wiki

Changes: Mean

Edit

Back to page

 
 
Line 1: Line 1:
 
{{StatsPsy}}
 
{{StatsPsy}}
   
In [[statistics]], '''''mean''''' has two related meanings:
+
In [[statistics]], '''mean''' has two related meanings:
* the ''[[average]]'' in ordinary English, which is more correctly called the [[arithmetic mean]], to distinguish it from [[geometric mean]] or [[harmonic mean]]. The average is also called [[sample (statistics)|sample]] mean.
+
* 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''.
 
* 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 if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, [[median]], and [[mode (statistics)|mode]]. For instance, average house prices almost always use the median value for the average. These three types of averages are all measures of locations. Other simple statistical analyses use measures of spread, such as [[range (mathematics)|range]], [[interquartile range]], or [[standard deviation]].
  +
For a real-valued [[random variable]] ''X'', the mean is the [[expectation operator|expectation]] of ''X''.
  +
Note that not every [[probability distribution]] has a defined mean (or [[variance]]); see the [[Cauchy distribution]] for an example.
   
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. See the '''[[#Other means|Other means]]''' section below for a list of means.
+
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.
   
Sample mean is often used as an [[estimator]] of the [[central tendency]] such as the population mean. However, other estimators are also used. For example, the [[median]] is a more [[robust]] estimator of the central tendency than the sample mean.
+
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.
   
For a real-valued [[random variable]] ''X'', the mean is the [[expectation operator|expectation]] of ''X''.
+
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.
If the expectation does not exist, then the random variable has no mean.
 
   
For a [[data set]], the mean is just the sum of all the observations divided by the number of observations.
+
==Examples of means==
Once we have chosen this method of describing the communality of a data set, we usually use the [[standard deviation]] to describe how the observations differ.
+
===Arithmetic mean===
The standard deviation is the square root of the average of squared deviations from the mean.
+
{{main|Arithmetic mean}}
  +
The ''arithmetic mean'' is the "standard" average, often simply called the "mean".
   
The mean is the unique value about which the sum of squared deviations is a minimum.
+
:<math> \bar{x} = \frac{1}{n}\cdot \sum_{i=1}^n{x_i} </math>
If you calculate the sum of squared deviations from any other measure of [[central tendency]], it will be larger than for the mean.
 
This explains why the standard deviation and the mean are usually cited together in statistical reports.
 
   
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 tractable when combining data sets.
+
The '''mean''' may often be confused with the [[median]], [[Mode (statistics)|mode]] or range. The mean is the arithmetic average of a set of values, or distribution; however, for [[skewness|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.
   
Note that not every [[probability distribution]] has a defined mean or [[variance]] &mdash; see the [[Cauchy distribution]] for an example.
+
Nevertheless, many skewed distributions are best described by their mean - such as the [[Exponential distribution|Exponential]] and [[Poisson distribution|Poisson]] distributions.
   
The following is a summary of some of the multiple methods for calculating the mean of a set of ''n'' numbers. See the [[table of mathematical symbols]] for explanations of the symbols used.
+
For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is:
  +
:<math>\frac{34+27+45+55+22+34}{6} = \frac{217}{6} \approx 36.167.</math>
   
==Arithmetic mean==
+
===Geometric mean===
The [[arithmetic mean]] is the "standard" average, often simply called the "mean".
+
{{main|Geometric mean}}
  +
The [[geometric mean]] is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.
   
:<math> \bar{x} = {1 \over n} \sum_{i=1}^n{x_i} </math>
+
:<math> \bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^{1/n}</math>
   
The '''mean''' may often be confused with the [[median]] or [[Mode_(statistics)|mode]]. The mean is the arithmetic average of a set of values, or distribution; however, for [[skewness|skewed]] distributions, the mean is not necessarily the same as the middle value (median), or 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.
+
For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:
  +
:<math>(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^{1/6} = 1,699,493,400^{1/6} = 34.545.</math>
   
That said, many skewed distributions are best described by their mean - such as the [[Exponential_distribution|Exponential]] and [[Poisson_distribution|Poisson]] distributions.
+
===Harmonic mean===
  +
{{main|Harmonic mean}}
  +
The [[harmonic mean]] is an average which is useful for sets of numbers which are defined in relation to some [[Unit of measurement|unit]], for example [[speed]] (distance per unit of time).
   
===An example===
+
:<math> \bar{x} = n \cdot \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}</math>
   
An experiment yields the following data: 34,27,45,55,22,34
+
For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is
To get the '''arithmetic mean'''
+
:<math>\frac{6}{\frac{1}{34}+\frac{1}{27}+\frac{1}{45} + \frac{1}{55} + \frac{1}{22}+\frac{1}{34}} = \frac{60588}{1835} \approx 33.0179836.</math>
# How many items? There are 6. Therefore n=6
 
# What is the sum of all items? It is 217. Therefore Sigma x (Sigma means sum) is 217
 
# To get the arithmetic mean divide sigma by n, here 217/6=<u>36.1666666667</u>
 
   
==Geometric mean==
+
===Generalized means===
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.
+
====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 for a set of ''n'' positive numbers ''x''<sub>i</sub> by
   
:<math> \bar{x} = \sqrt[n]{\prod_{i=1}^n{x_i}} </math>
+
:<math> \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^{1/m} </math>
   
===An example===
+
By choosing the appropriate value for the parameter ''m'' we get
+
{|
An experiment yields the following data: 34,27,45,55,22,34
+
|<math>m\rightarrow\infty</math> || [[maximum]]
To get the geometric mean
+
|-
# How many items? There are 6. Therefore n=6
+
| <math>m=2</math> || [[quadratic mean]],
# What is the product of all items? It is 1699493400.
+
|-
# To get the geometric mean take the nth (the 6th) root of that product; it is <U>34.5451100372</U>
+
| <math>m=1</math> || [[arithmetic mean]],
+
|-
==Harmonic mean==
+
| <math>m\rightarrow0</math> || [[geometric 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).
+
|-
+
| <math>m=-1</math> || [[harmonic mean]],
:<math> \bar{x} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} </math>
+
|-
+
| <math>m\rightarrow-\infty</math> || [[minimum]].
===An example===
+
|}
 
An experiment yields the following data: 34,27,45,55,22,34
 
To get the '''harmonic mean'''
 
# How many items? There are 6. Therefore n=6
 
# What is the sum on the bottom of the fraction? It is 0.181719152307
 
# Get the reciprocal of that sum. It is 5.50299727522
 
# To get the harmonic mean multiply that by n to get <U>33.0179836513</U>
 
 
==Generalized mean==
 
The [[generalized mean]], also known as the [[power mean]] or [[Hölder mean]], is an abstraction of the arithmetic, geometric and harmonic means. It is defined by
 
 
:<math> \bar{x}(m) = \sqrt[m]{\frac{1}{n}\sum_{i=1}^n{x_i^m}} </math>
 
 
By choosing the appropriate value for the parameter ''m'' we can get the arithmetic mean (''m'' = 1), the geometric mean (''m'' &rarr; 0) or the harmonic mean (''m'' = &minus;1)
 
   
  +
====f-mean====
 
This can be generalized further as the [[generalized f-mean]]
 
This can be generalized further as the [[generalized f-mean]]
:<math> \bar{x} = f^{-1}\left({\frac{1}{n}\sum_{i=1}^n{f(x_i)}}\right) </math>
+
:<math> \bar{x} = f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) </math>
   
and again a suitable choice of an invertible ''f''(''x'') will give the arithmetic mean with ''f''(''x'') = ''x'', the geometric mean with ''f''(''x'') = log(''x''), and the harmonic mean with ''f''(''x'') = 1/''x''.
+
and again a suitable choice of an invertible <math>f</math> will give
  +
{| <math>f(x) = x</math> || [[arithmetic mean]],
  +
|-
  +
| <math>f(x) = \frac{1}{x}</math> || [[harmonic mean]],
  +
|-
  +
| <math>f(x) = x^m</math> || [[power mean]],
  +
|-
  +
| <math>f(x) = \ln x</math> || [[geometric mean]].
  +
|}
   
==Weighted mean==
+
===Weighted arithmetic mean===
The [[weighted mean]] is used, if one wants to combine average values from samples of the same population with different sample sizes:
+
The [[weighted mean|weighted arithmetic mean]] is used, if one wants to combine average values from samples of the same population with different sample sizes:
   
:<math> \bar{x} = \frac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}} </math>
+
:<math> \bar{x} = \frac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}}. </math>
   
The weights <math>w_i</math> 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.
+
The weights <math>w_i</math> 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==
+
===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.
+
Sometimes a set of numbers might contain outliers, i.e. a [[datum]] which is much lower or much higher than the others.
  +
Often, outliers are erroneous data caused by [[artifact (observational)|artifacts]]. 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==
+
===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.
 
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.
 
:<math> \bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i} </math>
 
:<math> \bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i} </math>
 
assuming the values have been ordered.
 
assuming the values have been ordered.
   
==Mean of a function==
+
===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 (mathematics)|domain]]. In one variable, the mean of a function ''f''(''x'') over the interval (''a,b'') is defined by
 
In [[calculus]], and especially [[multivariable calculus]], the mean of a function is loosely defined as the average value of the function over its [[domain (mathematics)|domain]]. In one variable, the mean of a function ''f''(''x'') over the interval (''a,b'') is defined by
: <math>\bar{f}=\frac{1}{b-a}\int_a^bf(x)dx.</math>
+
  +
: <math>\bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx.</math>
  +
 
(See also [[mean value theorem]].) In several variables, the mean over a [[relatively compact]] [[neighborhood (mathematics)|domain]] ''U'' in a [[Euclidean space]] is defined by
 
(See also [[mean value theorem]].) In several variables, the mean over a [[relatively compact]] [[neighborhood (mathematics)|domain]] ''U'' in a [[Euclidean space]] is defined by
  +
 
:<math>\bar{f}=\frac{1}{\hbox{Vol}(U)}\int_U f.</math>
 
:<math>\bar{f}=\frac{1}{\hbox{Vol}(U)}\int_U f.</math>
   
 
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
 
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
:<math>\exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right)</math>
 
   
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.
+
:<math>\exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).</math>
  +
  +
More generally, in [[measure theory]] and [[probability theory]] either sort of mean plays an important role. In this context, [[Jensen's inequality]] places sharp estimates 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 ==
+
=== Other means ===
  +
<div style="-moz-column-count:3; column-count:3;">
 
*[[Arithmetic-geometric mean]]
 
*[[Arithmetic-geometric mean]]
 
*[[Arithmetic-harmonic mean]]
 
*[[Arithmetic-harmonic mean]]
 
*[[Cesàro mean]]
 
*[[Cesàro mean]]
 
*[[Chisini mean]]
 
*[[Chisini mean]]
  +
*[[Contraharmonic mean]]
  +
*[[Elementary symmetric mean]]
 
*[[Geometric-harmonic mean]]
 
*[[Geometric-harmonic mean]]
  +
*[[Heinz mean]]
 
*[[Heronian mean]]
 
*[[Heronian mean]]
 
*[[Identric mean]]
 
*[[Identric mean]]
  +
*[[Least squares mean]]
 
*[[Lehmer mean]]
 
*[[Lehmer mean]]
*[[Quadratic mean]]
+
*[[Logarithmic mean]]
*[[root mean square]]
+
*[[Median]] <!-- is a mean according to the definition below -->
  +
*[[Moving average]]
  +
*[[Root mean square]]
 
*[[Stolarsky mean]]
 
*[[Stolarsky mean]]
*[[weighted geometric mean]]
+
*[[Weighted geometric mean]]
*[[weighted harmonic mean]]
+
*[[Weighted harmonic mean]]
 
*[[Rényi's entropy]] (a [[generalized f-mean]])
 
*[[Rényi's entropy]] (a [[generalized f-mean]])
  +
</div>
   
  +
== Properties ==
   
  +
All means share some properties and additional properties are shared by the most common means.
  +
Some of these properties are collected here.
  +
  +
=== Weighted mean ===
  +
  +
A '''weighted mean''' <math>M</math> is a function which maps tuples of positive numbers to a positive number
  +
(<math>\mathbb{R}_{>0}^n\to\mathbb{R}_{>0}</math>).
  +
  +
* "[[Fixed point (mathematics)|Fixed point]]": <math> M(1,1,\dots,1) = 1 </math>
  +
* [[Homogeneous function|Homogeneity]]: <math> \forall\lambda\ \forall x\ M(\lambda\cdot x_1, \dots, \lambda\cdot x_n) = \lambda \cdot M(x_1, \dots, x_n) </math>
  +
:: (using [[coordinate vector|vector]] notation: <math> \forall\lambda\ \forall x\ M(\lambda\cdot x) = \lambda \cdot M x </math>)
  +
* [[Monotonic function|Monotony]]: <math> \forall x\ \forall y\ (\forall i\ x_i \le y_i) \Rightarrow M x \le M y </math>
  +
  +
It follows
  +
* [[Upper bound|Boundedness]]: <math> \forall x\ M x \in [\min x, \max x] </math>
  +
* [[Continuous function|Continuity]]: <math> \lim_{x\to y} M x = M y </math>
  +
<!-- for all convergent sequences <math>x</math> it holds <math> \lim (M \circ x) = M (\lim x) </math> -->
  +
:Sketch of a proof: Because <math>\forall x\ \forall y\ \left(||x-y||_\infty\le\lambda\cdot\min y \Rightarrow (\forall i\ |x_i-y_i|\le\lambda\cdot y_i) \Rightarrow (\forall i\ x_i\le y_i+\lambda\cdot y_i) \Rightarrow M x \le(1+\lambda)\cdot M y\right)</math> it follows <math>\forall x\ \forall y\ Mx\ge My\ \Rightarrow\ \forall \varepsilon>0\ ||x-y||_\infty\le\frac{\varepsilon\cdot\min y}{M y} \Rightarrow |Mx-My|\le\varepsilon</math>.
  +
* There are means which are not [[derivative|differentiable]]. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the [[power mean]], or as a special case of a [[median]]), but is not differentiable.
  +
* All means listed above, with the exception of most of the [[Generalized f-mean]]s, satisfy the presented properties.
  +
** If <math>f</math> is bijective, then the generalized f-mean satisfies the fixed point property.
  +
** If <math>f</math> is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
  +
** In general a generalized f-mean will miss homogeneity.
  +
  +
The above properties imply techniques to construct more complex means:
  +
  +
If <math>C, M_1, \dots, M_m</math> are weighted means,
  +
<math>p</math> is a positive [[real number]],
  +
then <math>A, B</math> with
  +
: <math> \forall x\ A x = C(M_1 x, \dots, M_m x) </math>
  +
: <math> \forall x\ B x = \sqrt[p]{C(x_1^p, \dots, x_n^p)} </math>
  +
are also a weighted mean.
  +
  +
=== Unweighted mean ===
  +
  +
Intuitively spoken, an '''unweighted mean''' is a weighted mean with equal weights.
  +
Since our definition of ''weighted mean'' above does not expose particular weights,
  +
equal weights must be asserted by a different way.
  +
A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.
  +
  +
Thus we define <math>M</math> being an unweighted mean if it is a weighted mean
  +
and for each [[permutation]] <math>\pi</math> of inputs, the result is the same.
  +
Let <math>P</math> be the set of permutations of <math>n</math>-tuples.
  +
: [[Symmetric function|Symmetry]]: <math> \forall x\ \forall \pi\in P \ M x = M(\pi x) </math>
  +
  +
Analogously to the weighted means,
  +
if <math>C</math> is a weighted mean and <math>M_1, \dots, M_m</math> are unweighted means,
  +
<math>p</math> is a positive [[real number]],
  +
then <math>A, B</math> with
  +
: <math> \forall x\ A x = C(M_1 x, \dots, M_m x) </math>
  +
: <math> \forall x\ B x = \sqrt[p]{M_1(x_1^p, \dots, x_n^p)} </math>
  +
are also unweighted means.
  +
  +
=== Convert unweighted mean to weighted mean ===
  +
  +
An unweighted mean can be turned into a weighted mean by repeating elements.
  +
This connection can also be used to state that a mean is the weighted version of an unweighted mean.
  +
Say you have the unweighted mean <math>M</math> and
  +
weight the numbers by natural numbers <math>a_1,\dots,a_n</math>.
  +
(If the numbers are [[rational number|rational]], then multiply them with the [[least common denominator]].)
  +
Then the corresponding weighted mean <math>A</math> is obtained by
  +
:<math>A(x_1,\dots,x_n) = M(\underbrace{x_1,\dots,x_1}_{a_1},x_2,\dots,x_{n-1},\underbrace{x_n,\dots,x_n}_{a_n}).</math>
  +
  +
=== Means of tuples of different sizes ===
  +
  +
If a mean <math>M</math> is defined for tuples of several sizes,
  +
then one also expects that the mean of a tuple is bounded by the means of partitions.
  +
More precisely
  +
* Given an arbitrary tuple <math>x</math>, which is [[Partition of a set|partitioned]] into <math>y_1, \dots, y_k</math>, then it holds <math>M x \in \mathrm{convexhull}(M y_1, \dots, M y_k)</math>. (See [[Convex hull]])
  +
  +
==Population and sample means==
  +
The mean of a [[Statistical population|population]] has an expected value of μ, known as the population mean. The sample mean makes a good [[estimator]] of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a [[random variable]], not a constant, and consequently it will have its own distribution. For a random sample of ''n'' observations from a normally distributed population, the sample mean distribution is
  +
  +
: <math>\bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.</math>
  +
  +
Often, since the population variance is an unknown parameter, it is estimated by the [[sum of squares|mean sum of squares]], which changes the distribution of the sample mean from a normal distribution to a [[Student's t distribution]] with ''n''&nbsp;&minus;&nbsp;1 [[degrees of freedom]].
  +
  +
==Mathematics education==
  +
{{Unreferencedsection|date=February 2008}}{{globalize/USA}}
  +
In many state and government curriculum standards, students are traditionally expected to learn either the meaning or formula for computing the mean by the fourth grade. However, in many [[standards-based mathematics]] curricula, students are encouraged to invent their own methods, and may not be taught the traditional method. Reform based texts such as [[Investigations in Numbers, Data, and Space|TERC]] in fact discourage teaching the traditional "add the numbers and divide by the number of items" method in favor of spending more time on the concept of [[median]], which does not require division. However, mean can be computed with a simple four-function calculator, while median requires an abacus. The same teacher guide devotes several pages on how to find the median of a set, which is judged to be simpler than finding the mean.
   
 
==See also==
 
==See also==
*[[Average]]
+
*[[Average]], same as ''central tendency''
*[[Central tendency]]
+
*[[Central tendency measures]]
 
*[[Descriptive statistics]]
 
*[[Descriptive statistics]]
 
*[[Kurtosis]]
 
*[[Kurtosis]]
Line 114: Line 204:
 
*[[Mode (statistics)]]
 
*[[Mode (statistics)]]
 
*[[Summary statistics]]
 
*[[Summary statistics]]
  +
*[[Standard scores]]
  +
For an independent identical distribution from the reals, the mean of a sample is an [[Bias of an estimator|unbiased]] estimator for the mean of the population.
  +
  +
==References==
  +
*{{citation|first1=G.H.|last1=Hardy|authorlink1=G.H. Hardy|first2=J.E.|last2=Littlewood|authorlink2=John Edensor Littlewood|first3=G.|last3=Pólya|authorlink3=George Pólya|title=Inequalities|year=1988|publisher=Cambridge University Press|edition=2nd|isbn=978-0521358804}}
   
 
==External links==
 
==External links==
  +
* [http://www.stats4students.com/Essentials/Measures-Central-Tendency/Overview.php An easy-to-follow guide to understanding & calculating the mean]
 
* [http://www.sengpielaudio.com/calculator-geommean.htm Comparison between arithmetic and geometric mean of two numbers]
 
* [http://www.sengpielaudio.com/calculator-geommean.htm Comparison between arithmetic and geometric mean of two numbers]
   
[[Category:Statistics]][[Category:Means|*]]
+
{{Statistics}}
   
  +
[[Category:Central tendency measures]]
  +
[[Category:Means| ]]
  +
  +
<!--
  +
[[da:Gennemsnit]]
 
[[de:Mittelwert]]
 
[[de:Mittelwert]]
[[fi:artimeettinen keskiarvo]]
+
[[es:Promedio]]
  +
[[eo:Meznombro]]
  +
[[fa:میانگین]]
  +
[[fi:Keskiluku]]
  +
[[fr:Moyenne]]
 
[[he:ממוצע]]
 
[[he:ממוצע]]
[[it:media]]
+
[[it:Media (statistica)]]
[[ja:&#24179;&#22343;]]
+
[[ja:平均]]
[[nl:gemiddelde]]
+
[[ko:평균]]
  +
[[lo:ຄ່າສະເຫຼ່ຍ]]
  +
[[nl:Gemiddelde]]
  +
[[no:Gjennomsnitt]]
  +
[[pl:Średnia]]
 
[[pt:Média]]
 
[[pt:Média]]
+
[[ru:Среднее значение]]
  +
[[sl:Srednja vrednost]]
  +
[[su:Mean]]
  +
[[sv:Medelvärde]]
  +
[[th:มัชฌิม]]
  +
[[tr:Ortalama]]
  +
[[zh:平均数]]
  +
-->
 
{{enWP|Mean}}
 
{{enWP|Mean}}

Latest revision as of 00:37, February 20, 2009

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory


In statistics, mean has two related meanings:

It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" 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. These three types of averages are all measures of locations. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. 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 meansEdit

Arithmetic meanEdit

Main article: Arithmetic mean

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

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

The mean may often be confused with the median, mode or range. 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.

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

For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is:

\frac{34+27+45+55+22+34}{6} = \frac{217}{6} \approx 36.167.

Geometric meanEdit

Main article: Geometric mean

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

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

For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:

(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^{1/6} = 1,699,493,400^{1/6} = 34.545.

Harmonic meanEdit

Main article: 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 \frac{1}{x_i} \right ) ^{-1}

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

\frac{6}{\frac{1}{34}+\frac{1}{27}+\frac{1}{45} + \frac{1}{55} + \frac{1}{22}+\frac{1}{34}} = \frac{60588}{1835} \approx 33.0179836.

Generalized meansEdit

Power meanEdit

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 for a set of n positive numbers xi by

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

By choosing the appropriate value for the parameter m we get

m\rightarrow\infty maximum
m=2 quadratic mean,
m=1 arithmetic mean,
m\rightarrow0 geometric mean,
m=-1 harmonic mean,
m\rightarrow-\infty minimum.

f-meanEdit

This can be generalized further as the generalized f-mean

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

and again a suitable choice of an invertible f will give

f(x) = \frac{1}{x} harmonic mean,
f(x) = x^m power mean,
f(x) = \ln x geometric mean.

Weighted arithmetic meanEdit

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} = \frac{\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 meanEdit

Sometimes a set of numbers might contain outliers, i.e. a datum which is much lower or much higher than the others. Often, outliers are erroneous data caused by artifacts. 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 meanEdit

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 functionEdit

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}=\frac{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}=\frac{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(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).

More generally, in measure theory and probability theory either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

Mean of anglesEdit

Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. For those quantities you need a mean of circular quantities.

Other means Edit

Properties Edit

All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.

Weighted mean Edit

A weighted mean M is a function which maps tuples of positive numbers to a positive number (\mathbb{R}_{>0}^n\to\mathbb{R}_{>0}).

(using vector notation:  \forall\lambda\ \forall x\ M(\lambda\cdot x) = \lambda \cdot M x )

It follows

Sketch of a proof: Because \forall x\ \forall y\ \left(||x-y||_\infty\le\lambda\cdot\min y \Rightarrow (\forall i\ |x_i-y_i|\le\lambda\cdot y_i) \Rightarrow (\forall i\ x_i\le y_i+\lambda\cdot y_i) \Rightarrow M x \le(1+\lambda)\cdot M y\right) it follows \forall x\ \forall y\ Mx\ge My\ \Rightarrow\ \forall \varepsilon>0\ ||x-y||_\infty\le\frac{\varepsilon\cdot\min y}{M y} \Rightarrow |Mx-My|\le\varepsilon.
  • There are means which are not differentiable. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the power mean, or as a special case of a median), but is not differentiable.
  • All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties.
    • If f is bijective, then the generalized f-mean satisfies the fixed point property.
    • If f is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
    • In general a generalized f-mean will miss homogeneity.

The above properties imply techniques to construct more complex means:

If C, M_1, \dots, M_m are weighted means, p is a positive real number, then A, B with

 \forall x\ A x = C(M_1 x, \dots, M_m x)
 \forall x\ B x = \sqrt[p]{C(x_1^p, \dots, x_n^p)}

are also a weighted mean.

Unweighted mean Edit

Intuitively spoken, an unweighted mean is a weighted mean with equal weights. Since our definition of weighted mean above does not expose particular weights, equal weights must be asserted by a different way. A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.

Thus we define M being an unweighted mean if it is a weighted mean and for each permutation \pi of inputs, the result is the same. Let P be the set of permutations of n-tuples.

Symmetry:  \forall x\ \forall \pi\in P \ M x = M(\pi x)

Analogously to the weighted means, if C is a weighted mean and M_1, \dots, M_m are unweighted means, p is a positive real number, then A, B with

 \forall x\ A x = C(M_1 x, \dots, M_m x)
 \forall x\ B x = \sqrt[p]{M_1(x_1^p, \dots, x_n^p)}

are also unweighted means.

Convert unweighted mean to weighted mean Edit

An unweighted mean can be turned into a weighted mean by repeating elements. This connection can also be used to state that a mean is the weighted version of an unweighted mean. Say you have the unweighted mean M and weight the numbers by natural numbers a_1,\dots,a_n. (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean A is obtained by

A(x_1,\dots,x_n) = M(\underbrace{x_1,\dots,x_1}_{a_1},x_2,\dots,x_{n-1},\underbrace{x_n,\dots,x_n}_{a_n}).

Means of tuples of different sizes Edit

If a mean M is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely

Population and sample meansEdit

The mean of a population has an expected value of μ, known as the population mean. The sample mean makes a good estimator of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a random variable, not a constant, and consequently it will have its own distribution. For a random sample of n observations from a normally distributed population, the sample mean distribution is

\bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.

Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares, which changes the distribution of the sample mean from a normal distribution to a Student's t distribution with n − 1 degrees of freedom.

Mathematics educationEdit

In many state and government curriculum standards, students are traditionally expected to learn either the meaning or formula for computing the mean by the fourth grade. However, in many standards-based mathematics curricula, students are encouraged to invent their own methods, and may not be taught the traditional method. Reform based texts such as TERC in fact discourage teaching the traditional "add the numbers and divide by the number of items" method in favor of spending more time on the concept of median, which does not require division. However, mean can be computed with a simple four-function calculator, while median requires an abacus. The same teacher guide devotes several pages on how to find the median of a set, which is judged to be simpler than finding the mean.

See alsoEdit

For an independent identical distribution from the reals, the mean of a sample is an unbiased estimator for the mean of the population.

ReferencesEdit

External linksEdit



This page uses Creative Commons Licensed content from Wikipedia (view authors).

Around Wikia's network

Random Wiki