.

In probability theory and statistics, central 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 because then the values' higher-order quantities relate only to the spread and shape of the distribution, rather than to its location.

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

Univariate moments

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

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

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:

The "zeroth" central moment μ₀ is one.
The first central moment μ₁ is zero (not to be confused with the first moment itself, the expected value or mean).
The second central moment μ₂ is called the variance, and is usually denoted σ², where σ represents the standard deviation.
The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.

Properties

The nth central moment is translation-invariant, i.e. for any random variable X and any constant c, we have

\( \mu_n(X+c)=\mu_n(X).\, \)

For all n, the nth central moment is homogeneous of degree n:

\( \mu_n(cX)=c^n\mu_n(X).\, \)

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

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

A related functional that shares the translation-invariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κn(X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central moment; for n ≥ 4, the nth cumulant is an nth-degree monic polynomial in the first n moments (about zero), and is also a (simpler) nth-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 nth-order moment about the origin to the moment about the mean is

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

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

\( \mu'_j = \int_{-\infty}^{+\infty} x^j f(x)\,dx. \)

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 \( \mu = \mu'_1 \)):

\( \mu_2 = \mu'_2 - \mu^2\, \)

\( \mu_3 = \mu'_3 - 3 \mu \mu'_2 + 2 \mu^3\, \)

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

Multivariate moments

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

\( \mu_{j,k} = \operatorname{E} \left[ ( X - \operatorname{E}[X] )^j ( Y - \operatorname{E}[Y] )^k \right] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} (x - \mu_X)^j (y - \mu_Y)^k f(x,y )\,dx \,dy. \)