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 zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.
Contents
- Univariate moments
- Properties
- Relation to moments about the origin
- Symmetric distributions
- Multivariate moments
- References
Sets of central moments can be defined for both univariate and multivariate distributions.
Univariate moments
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
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:
Properties
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 such that n equals 1, 2, or 3 do we have an additivity property for random variables X and Y that are independent:
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
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
... and so on, following Pascal's triangle, i.e.
because
The following sum is a stochastic variable having a compound distribution
where the
where
Symmetric distributions
In a symmetric distribution (one that is unaffected by being reflected about its mean), all odd central 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
For a continuous bivariate probability distribution with probability density function f(x,y) the (j,k) moment about the mean μ = (μX, μY) is