The k^th moment about the mean (or k^th central moment) of a real-valued random variable X is the quantity E[(X-E[X])^k], where E is the expectation operator. Some random variables have no mean, in which case the moment about the mean is not defined. The k^th moment about the mean is often denoted μ_k. For a continuous univariate distribution P(x) the moment about the mean μ is

<math> \mu_k

\left\langle ( x - \langle x \rangle )^k \right\rangle

\int_{-\infty}^{+\infty} (x - \mu)^k P(x)\,dx </math>

Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the n^th order moment about the origin to the moment about the mean is

<math> \mu_n = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j m^{n-j}, </math>

where m is the mean of the distribution, and the moment about the origin is given by

<math> \mu'_j = \int_{-\infty}^{+\infty} x^n P(x)\,dx. </math>

The first moment about the mean is zero. The second moment about the mean is called the variance, and is usually denoted σ², where σ represents the standard deviation. The third and fourth moments about the mean are used to define the standardized moments which are in turn used to define skewness and kurtosis, respectively.