---
tag: 0ZT24EC90K
kind: definition
name: Central moment
msc: ["60E05"]
defines: 09753QQDCK
uses: [0PH0VHTQS0, 0DCT0QZVQ1, 007QED3E0Y]
sources:
  - url: https://en.wikipedia.org/wiki/Central_moment
    locator: Univariate moments
---

## 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|007QED3E0Y]].
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|0PH0VHTQS0]] $m_j = E[X^j]$, 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.