In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.

Sets of central moments can be defined for both univariate and multivariate distributions.

The n-th moment about the mean (or n-th central moment) of a real-valued random variable X is the quantity μ_n := E[(X − E[X])ⁿ], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x), the n-th moment about the mean μ is $${\displaystyle \mu _{n}=\operatorname {E} \left[{\left(X-\operatorname {E} [X]\right)}^{n}\right]=\int _{-\infty }^{+\infty }(x-\mu )^{n}f(x)\,\mathrm {d} x.}$$

For random variables that have no mean, such as the Cauchy distribution, central moments are not defined.

The first few central moments have intuitive interpretations:

For all n, the n-th central moment is homogeneous of degree n:

$${\displaystyle \mu _{n}(cX)=c^{n}\mu _{n}(X).\,}$$

Only for n such that n equals 1, 2, or 3 do we have an additivity property for random variables X and Y that are independent: