 Central moment

For a random variable with finite mean $\mu$, its $n$th central moment is $\mathbb{E}[(X-\mu {)}^n]$ when defined. The first is zero; the second is the variance; the standardized third gives skewness. Translating $X$ by a constant leaves central moments unchanged, because the mean translates by the same amount. For equally likely values one and three, the second central moment is one, whereas the second raw moment is five. Odd central moments can cancel between positive and negative deviations, so an absolute moment provides a different integrability check.

In statistics, the **central moment** of a random variable is a measure of the variability of that variable about its mean. Specifically, the \( n \)-th central moment is defined as the expected value of the \( n \)-th power of the deviation of the random variable from its mean.