Definition:Central Moment

Definition

Let $X$ be a random variable on some probability space with mean $\mu$.

Then the $n$th central moment of $X$, usually denoted $\mu_n$, is defined as:

$\mu_n = \expect {\paren {X - \mu}^n}$

where $\expect X$ denotes the expectation of $X$.

That is, the $n$th central moment of $X$ is its $n$th moment about $\mu$.

The second central moment of $X$ is the variance of $X$, and is instead usually denoted $\sigma^2$.

Also known as

Some sources refer to the $n$th central moment of a random variable as its $n$th moment about the mean.

Also see

• Definition:Moment (Probability Theory)
• Definition:Raw Moment
• Definition:Standardized Moment

• Results about central moments can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): moment: 2. (moment about the mean or central moment)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): moment
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): moment

• Weisstein, Eric W. "Central Moment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralMoment.html