Central moment

Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$. Let $\mu = E[X]$. The $k$-th central moment of $X$ is $$\mu_k = E\left[(X - \mu)^k\right].$$

Remarks

Always $\mu_1 = 0$, and $\mu_2 = \operatorname{Var}(X)$ is the variance. Central moments do not change when one adds a constant to $X$. For $a \in \mathbb{R}$, the central moments of $aX$ are $a^k \mu_k$. The binomial theorem gives $\mu_k = \sum_{j=0}^{k} \binom{k}{j} (-\mu)^{k - j} m_j$ in terms of the moments $m_j = E[X^j]$, with $m_0 = 1$. For a distribution that is symmetric about its mean, all odd central moments are $0$.

Explanations

L1 human

Central moments measure the shape of a distribution around its center, and not its position. The third gives asymmetry and the fourth gives the weight of the tails.