There are several kinds of moments in statistics. This page will define these moments and give equations relating them to each other.
Let X be a random variable. Then rth moment of X is the expected value of Xr. The rth moment is also called the rth raw moment to distinguish it from other kinds of moments.
For a given constant a, the rth moment of X about a is the rth (raw) moment of X – a. The rth raw moment is the rth moment about 0.
Let μ be the mean of X, the first moment of X. The rth central moment of X is the rth moment of X about μ, which is the rth (raw) moment of X − μ.
Let σ² be the variance of X, the second central moment of X. The rth standardized moment of X is the rth (raw) moment of (X − μ)/σ.
Denote the rth moment of X about a by μ′r(a).
When r = 1 the subscript is implicit, i.e.
When a = 0 we can also leave it implicit, and so we can denote the rth raw moment by
We remove the prime from μ′r when referring to central moments:
The rth standardized moment is denoted by adding a tilde on top of μ.
Relating raw and central moments
Let a and b be two constants and c = b − a. Then
This is essentially just the binomial theorem, but the application can be a little confusing and error-prone.
If we let a = μ and b = 0, we get
and if we let a = 0 and b = μ we get
Because the raw and central moments up to order 4 come up most frequently in application, the equations relating these moments are given below for convenience.
Central moments in terms of raw moments:
Raw moments in terms of central moments: