Central moment ($\mathbb{E}[(X-\mathbb{E}X{)}^n]$)

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.

For a real random variable with finite third absolute moment and positive variance ${\sigma }^2$, its skewness is ${\gamma }_1=\mathbb{E}[(X-\mathbb{E}X{)}^3]/{\sigma }^3$. It is dimensionless and unchanged by positive rescaling or translation; a reflection reverses its sign. A symmetric distribution has zero skewness, but the converse fails: values $-2,1,3$ with probabilities $0.4,0.5,0.1$ have mean zero and third moment zero without symmetry. Constant variables have no defined standardized skewness because $\sigma =0$. The cosmological reduced skewness $S_3=⟨{\delta }^3⟩/{\sigma }^4$ uses a different normalization.