Central moment

In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location. Sets of central moments can be defined for both univariate and multivariate distributions. The nth moment about the mean (or nth central moment) of a real-valued random variable X is the quantity μn := E[(X − E[X])n], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x), the nth 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 1. The first central moment μ1 is 0 (not to be confused with the first raw moment or the expected value μ). 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 such that n equals 1, 2, or 3 do we have an additivity property for random variables X and Y that are independent: provided n ∈ {1, 2, 3}. 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).