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Moment about the mean

The kth moment about the mean (or kth 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 kth moment about the mean is often denoted μk. 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 nth 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.

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