In probability theory and statistics, the k^{th} moment about the mean (or k^{th} central moment) of a realvalued random variable X is the quantity μ_{k} := E[(X − E[X])^{k}], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x) the moment about the mean μ is
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:
 The "zeroth" central moment μ_{0} is one.
 The first central moment μ_{1} is zero.
 The second central moment μ_{2} is called the variance, and is usually denoted σ^{2}, where σ represents the standard deviation.
 The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.
Properties
The nth central moment is translationinvariant, i.e. for any random variable X and any constant c, we have
For all n, the nth central moment is homogeneous of degree n:
Only for n ≤ 3 do we have an additivity property for random variables X and Y that are independent:
A related functional that shares the translationinvariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κ_{n}(X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central moment; for n ≥ 4, the nth cumulant is an nthdegree monic polynomial in the first n moments (about zero), and is also a (simpler) nthdegree polynomial in the first n central moments.
Relation to moments about the origin
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
where μ is the mean of the distribution, and the moment about the origin is given by
For the cases n = 2, 3, 4 — which are of most interest because of the relations to variance, skewness, and kurtosis, respectively — this formula becomes:
See also
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