Central moment

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In probability theory and statistics, a central moment is a 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 characterised. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from the zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.

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

Univariate moments[edit]

The nth moment about the mean (or nth central moment) of a real-valued random variable X is the quantity μn := E[(X − E[X])n], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x), the nth moment about the mean μ is

 \mu_n = \operatorname{E} \left[ ( X - \operatorname{E}[X] )^n \right]  = \int_{-\infty}^{+\infty} (x - \mu)^n f(x)\,dx.

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:






Only for n such that 1 ≤ n ≤ 3 do we have an additivity property for random variables X and Y that are independent:

\mu_n(X+Y)=\mu_n(X)+\mu_n(Y)\text{ provided }1 \leq n\leq 3.\,


Relation to moments about the origin[edit]


\mu_n = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j \mu'^{n-j},

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


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 \mu = \mu'_1):

\mu_2 = \mu'_2 - \mu^2\,
\mu_3 = \mu'_3 - 3 \mu \mu'_2 + 2 \mu^3\,
\mu_4 = \mu'_4 - 4 \mu \mu'_3 + 6 \mu^2 \mu'_2 - 3 \mu^4.\,

Symmetric distributions[edit]

In a symmetric distribution (one that is unaffected by being reflected about its mean), all odd moments equal zero, because in the formula for the nth 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[edit]

For a continuous bivariate probability distribution with probability density function f(x,y) the (j,k) moment about the mean μ = (μX, μY) is

 \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.

See also[edit]