# Moments: raw, centralized, and standardized

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 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 Xa. 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.

## Notation

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 = ba. 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: 