In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity μk := 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 μ0 is one.
- The first central moment μ1 is zero.
- The second central moment μ2 is called the variance, and is usually denoted σ2, 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.
The nth central moment is translation-invariant, 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 translation-invariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κn(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 nth-degree monic polynomial in the first n moments (about zero), and is also a (simpler) nth-degree 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 nth-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:
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