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**Statistics:**
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In probability theory and statistics, 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 probability distribution with probability density function *f*(*x*) the moment about the mean μ is

- $ \mu_k = \left\langle ( x - \langle x \rangle )^k \right\rangle = \int_{-\infty}^{+\infty} (x - \mu)^k f(x)\,dx. $

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

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

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

- $ \mu'_j = \int_{-\infty}^{+\infty} x^j f(x)\,dx. $

The first moment about the mean is zero. The second moment about the mean is called the variance, and is usually denoted σ^{2}, 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.

For *n* ≥ 2, the *n*th central moment is translation-invariant, i.e. for any random variable *X* and any constant *c*, we have

- $ \mu_n(X+c)=\mu_n(X).\, $

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

- $ \mu_n(cX)=c^n\mu_n(X).\, $

*Only* for *n* ≤ 3 do we have an additivity property for random variables *X* and *Y* that are independent:

- $ \mu_n(X+Y)=\mu_n(X)+\mu_n(Y)\ \mathrm{provided}\ n\leq 3.\, $

A related functional that shares the translation-invariance and homogeneity properties with the *n*th central moment, but continues to have this additivity property even when *n* ≥ 4 is the *n*th cumulant κ_{n}(*X*). For *n* = 1, the *n*th cumulant is just the expected value; for *n* = either 2 or 3, the *n*th cumulant is just the *n*th central moment; for *n* ≥ 4, the *n*th cumulant is an *n*th-degree monic polynomial in the first *n* moments (about zero), and is also a (simpler) *n*th-degree polynomial in the first *n* central moments.

## See alsoEdit

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