There are several kinds of moments in statistics. This page will define these moments and give equations relating them to each other.

## Definitions

Let *X* be a random variable. Then ** rth moment** of

*X*is the expected value of

*X*

^{r}. The

*r*th moment is also called the

*r*th

**raw**moment to distinguish it from other kinds of moments.

For a given constant *a*, the ** rth moment of X about a** is the

*r*th (raw) moment of

*X*−

*a*. The

*r*th raw moment is the

*r*th moment about 0.

Let μ be the mean of *X*, the first moment of *X*. The ** rth central moment** of

*X*is the

*r*th moment of

*X*about μ, which is the

*r*th (raw) moment of

*X*− μ.

Let σ² be the variance of *X*, the second central moment of *X*. The *r***th standardized moment** of *X* is the *r*th (raw) moment of (*X* − μ)/σ.

## Notation

Denote the *r*th 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 *r*th raw moment by

We remove the prime from μ′_{r} when referring to central moments:

The *r*th 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: