Definitions

# Central moment

In probability theory and statistics, 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 = leftlangle (X - langle X rangle )^k rightrangle = int_{-infty}^{+infty} (x - mu)^k f(x),dx.

Note that $langle X rangle$ is equivalent to E(X) (i.e the expectation of X); it is the notation preferred by physicists.

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 first central moment is zero.
• The second moment about the mean is called the variance, and is usually denoted σ2, where σ represents the standard deviation.
• The third and fourth moments about the mean 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

$mu_n\left(X+c\right)=mu_n\left(X\right).,$

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

$mu_n\left(cX\right)=c^nmu_n\left(X\right).,$

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

$mu_n\left(X+Y\right)=mu_n\left(X\right)+mu_n\left(Y\right) mathrm\left\{provided\right\} nleq 3.,$

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 m^{n-j},

where m 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 $n=2,3,text\left\{ and \right\} 4$—which are of most interest because of the relations to variance, skewness, and kurtosis, respectively—this formula becomes:

$mu_2 = mu\text{'}_2 - m^2$

$mu_3 = mu\text{'}_3 - 3 m mu\text{'}_2 + 2 m^3$

$mu_4 = mu\text{'}_4 - 4 m mu\text{'}_3 + 6 m^2 mu\text{'}_2 - 3 m^4.$