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.


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


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


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) mathrm{provided} 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{ and } 4—which are of most interest because of the relations to variance, skewness, and kurtosis, respectively—this formula becomes:

mu_2 = mu'_2 - m^2

mu_3 = mu'_3 - 3 m mu'_2 + 2 m^3

mu_4 = mu'_4 - 4 m mu'_3 + 6 m^2 mu'_2 - 3 m^4.

See also

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