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{{Short description|Moment of a random variable minus its mean}}
{{Use American English|date = January 2019}}
{{Refimprove|date=September 2014}}
In [[probability theory]] and [[statistics]], a '''central moment''' is a [[moment (mathematics)|moment]] of a [[probability distribution]] of a [[random variable]] about the random variable's [[mean]]; that is, it is the [[expected value]] of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its [[location parameter|location]].

Sets of central moments can be defined for both univariate and multivariate distributions.

==Univariate moments==

The ''n''th [[moment (mathematics)|moment]] about the [[mean]] (or ''n''th '''central moment''') of a real-valued [[random variable]] ''X'' is the quantity ''μ''<sub>''n''</sub> := E[(''X''&nbsp;&minus;&nbsp;E[''X''])<sup>''n''</sup>], where E is the [[expected value|expectation operator]]. For a [[continuous probability distribution|continuous]] [[univariate]] [[probability distribution]] with [[probability density function]] ''f''(''x''), the ''n''th moment about the mean ''μ'' is
:<math> \mu_n = \operatorname{E} \left[ ( X - \operatorname{E}[X] )^n \right]  = \int_{-\infty}^{+\infty} (x - \mu)^n f(x)\,\mathrm{d} x. </math><ref name="ProbRand">{{cite book | title=Probability and Random Processes | publisher=Oxford University Press |author1=Grimmett, Geoffrey  |author2=Stirzaker, David | year=2009 | location=Oxford, England | isbn=978-0-19-857222-0}}</ref>

For random variables that have no mean, such as the [[Cauchy distribution]], central moments are not defined.

The first few central moments have intuitive interpretations:
* The "zeroth" central moment ''μ''<sub>0</sub> is 1.
* The first central moment ''μ''<sub>1</sub> is 0 (not to be confused with the first [[raw moment]] or the [[expected value]] ''μ'').
* The second central moment ''μ''<sub>2</sub> is called the [[variance]], and is usually denoted ''σ''<sup>2</sup>, where σ represents the [[standard deviation]].
* The third and fourth central moments are used to define the [[standardized moment]]s which are used to define [[skewness]] and [[kurtosis]], respectively.

===Properties===
For all ''n'', the ''n''th central moment is [[Homogeneous function|homogeneous]] of degree ''n'':

:<math>\mu_n(cX)=c^n\mu_n(X).\,</math>

''Only'' for ''n'' such that n equals 1, 2, or 3 do we have an additivity property for random variables ''X'' and ''Y'' that are [[statistical independence|independent]]:

:<math>\mu_n(X+Y)=\mu_n(X)+\mu_n(Y)\,</math> provided ''n'' ∈ {{math|{1, 2, 3}<nowiki/>}}.

A related functional that shares the translation-invariance and homogeneity properties with the ''n''th central moment, but continues to have this additivity property even when ''n''&nbsp;≥&nbsp;4 is the ''n''th [[cumulant]] κ<sub>''n''</sub>(''X'').  For ''n''&nbsp;=&nbsp;1, the ''n''th cumulant is just the [[expected value]]; for ''n''&nbsp;= either 2 or 3, the ''n''th cumulant is just the ''n''th central moment; for ''n''&nbsp;≥&nbsp;4, the ''n''th cumulant is an ''n''th-degree monic polynomial in the first ''n'' moments (about zero), and is also a (simpler) ''n''th-degree polynomial in the first ''n'' central moments.

===Relation to moments about the origin===
Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the ''n''th-order moment about the origin to the moment about the mean is

:<math>
\mu_n = \operatorname{E}\left[\left(X - \operatorname{E}[X]\right)^n\right] = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j \mu^{n-j},
</math>

where ''μ'' is the mean of the distribution, and the moment about the origin is given by

:<math>
\mu'_m = \int_{-\infty}^{+\infty} x^m f(x)\,dx = \operatorname{E}[X^m] = \sum_{j=0}^m {m \choose j}  \mu_j \mu^{m-j}.
</math>

For the cases ''n'' = 2, 3, 4 — which are of most interest because of the relations to [[variance]], [[skewness]], and [[kurtosis]], respectively — this formula becomes (noting that <math>\mu = \mu'_1</math> and <math>\mu'_0=1</math>):

:<math>\mu_2 = \mu'_2 - \mu^2\,</math> which is commonly referred to as <math> \operatorname{Var}(X) = \operatorname{E}[X^2] - \left(\operatorname{E}[X]\right)^2</math>

:<math>\mu_3 = \mu'_3 - 3 \mu \mu'_2 +2 \mu^3\,</math>

:<math>\mu_4 = \mu'_4 - 4 \mu \mu'_3 + 6 \mu^2 \mu'_2 - 3 \mu^4.\,</math>

... and so on,<ref>{{Cite web|url=http://mathworld.wolfram.com/CentralMoment.html|title = Central Moment}}</ref> following [[Pascal's triangle]], i.e.

:<math>\mu_5 = \mu'_5 - 5 \mu \mu'_4 + 10 \mu^2 \mu'_3 - 10 \mu^3 \mu'_2 + 4 \mu^5.\,</math>

because <math> 5\mu^4\mu'_1 - \mu^5 \mu'_0 = 5\mu^4\mu - \mu^5 = 5 \mu^5 - \mu^5 = 4 \mu^5</math>

The following sum is a stochastic variable having a '''''compound distribution'''''

:<math>W = \sum_{i=1}^M Y_i, </math>

where the <math>Y_i</math> are mutually independent random variables sharing the same common distribution and <math>M</math> a random integer variable independent of the <math>Y_k</math> with its own distribution. The moments of <math>W</math> are obtained as 

:<math>\operatorname{E}[W^n]= \sum_{i=0}^n\operatorname{E}\left[{M \choose i}\right]\sum_{j=0}^i {i \choose j}(-1)^{i-j}\operatorname{E} \left[ \left(\sum_{k=1}^j Y_k\right)^n \right],         </math>

where <math>\operatorname{E} \left[ \left(\sum_{k=1}^j Y_k\right)^n\right] </math> is defined as zero for <math>j=0</math>.

===Symmetric distributions===

In distributions that are [[symmetric distribution|symmetric about their means]] (unaffected by being [[reflection (mathematics)|reflected]] about the mean), all odd central moments equal zero whenever they exist, because in the formula for the ''n''th moment, each term involving a value of ''X'' less than the mean by a certain amount exactly cancels out the term involving a value of ''X'' greater than the mean by the same amount.

==Multivariate moments==

For a [[continuous probability distribution|continuous]] [[Joint probability distribution|bivariate]] [[probability distribution]] with [[probability density function]] ''f''(''x'',''y'') the (''j'',''k'') moment about the mean ''μ''&nbsp;=&nbsp;(''μ''<sub>''X''</sub>,&nbsp;''μ''<sub>''Y''</sub>) is
:<math> \mu_{j,k} = \operatorname{E} \left[ ( X - \operatorname{E}[X] )^j ( Y - \operatorname{E}[Y] )^k \right]  = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} (x - \mu_X)^j (y - \mu_Y)^k f(x,y )\,dx \,dy. </math>

==Central moment of complex random variables==
The ''n''th central moment for a complex random variable ''X'' is defined as <ref>{{cite book
| last1 = Eriksson| first1 = Jan 
| last2 = Ollila| first2 = Esa 
| last3 = Koivunen| first3 = Visa
| title = 2009 IEEE International Conference on Acoustics, Speech and Signal Processing 
| chapter = Statistics for complex random variables revisited 
| doi = 10.1109/ICASSP.2009.4960396
| pages = 3565–3568
| year = 2009| isbn = 978-1-4244-2353-8 
| s2cid = 17433817 
}}</ref>
{{Equation box 1
|indent =
|title=
|equation = <math>\alpha_n = \operatorname{E} \left[ ( X - \operatorname{E}[X] )^n \right],</math>
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}
The absolute ''n''th central moment of ''X'' is defined as
{{Equation box 1
|indent =
|title=
|equation = <math>\beta_n = \operatorname{E} \left[ |( X - \operatorname{E}[X] )|^n \right].</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

The 2nd-order central moment ''β''<sub>2</sub> is called the [[Complex random variable#Variance and pseudo-variance|''variance'']] of ''X'' whereas the 2nd-order central moment ''α''<sub>2</sub> is the [[Complex random variable#Variance and pseudo-variance|''pseudo-variance'']] of ''X''.

==See also==
*[[Standardized moment]]
*[[Image moment]]
*{{Slink|Normal distribution|Moments}}
*[[Complex random variable]]

==References==
{{Reflist}}

{{Theory of probability distributions}}

{{DEFAULTSORT:Central Moment}}
[[Category:Statistical deviation and dispersion]]
[[Category:Moment (mathematics)]]

[[fr:Moment (mathématiques)#Moment centré]]