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 X^r. 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.

Notation

Denote the rth moment of X about a by μ′_r(a).

$\mu'_r(a) = \text{E}\,(X - a)^r$

When r = 1 the subscript is implicit, i.e.

$\mu'(a) = \mu'_1(a) = \text{E}\,(X - a)$

When a = 0 we can also leave it implicit, and so we can denote the rth raw moment by

$\mu'_r = \mu'_r(0) = \text{E}\,X^r$

We remove the prime from μ′_r when referring to central moments:

$\mu_r = \mu'_r(\mu) = \text{E}\,(X - \mu)^r$
The rth standardized moment is denoted by adding a tilde on top of μ.

$\tilde{\mu}_r = \text{E}\,\left( \frac{X-\mu}{\sigma} \right)^r = \frac{\mu_r}{\sigma^r}$

Relating raw and central moments

Let a and b be two constants and c = b − a. Then

$\mu'_r(a) = \sum_{j=0}^r \binom{r}{j} \mu'_{r-j}(b)c^j$

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

$\mu_r = \sum_{j=0}^r \binom{r}{j} (-1)^j \mu'_{r-j} \,\mu^j$
and if we let a = 0 and b = μ we get

$\mu'_r = \sum_{j=0}^r \binom{r}{j} \mu_{r-j}\,\mu^j$

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:

$\begin{align*} \mu_0 &= 1 \\ \mu_1 &= 0 \\ \mu_2 &= \mu'_2 - \mu^2 \\ \mu_3 &= \mu'_3 - 3\mu'_2\,\mu + 2\mu^3 \\ \mu_4 &= \mu'_4 + 4\mu'_3\,\mu + 6\mu'_2\,\mu^2 -3 \mu^4 \end{align*}$

Raw moments in terms of central moments:

$\begin{align*} \mu'_0 &= 1 \\ \mu'_1 &= \mu \\ \mu'_2 &= \mu_2 + \mu^2 \\ \mu'_3 &= \mu_3 + 3\mu_2\,\mu + \mu^3 \\ \mu'_4 &= \mu_4 + 4\mu_3\,\mu + 6\mu_2\,\mu^2 + \mu^4 \end{align*}$