In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments.

Standard normalization

Let X be a random variable with a probability distribution P and mean value (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of X. Then the standardized moment of degree k is that is, the ratio of the kth moment about the mean to the kth power of the standard deviation, The power of k is because moments scale as x^k, meaning that they are homogeneous functions of degree k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension; in the above ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers. The first four standardized moments can be written as: For skewness and kurtosis, alternative definitions exist, which are based on the third and fourth cumulant respectively.

Other normalizations

Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation,. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because \mu is the first moment about zero (the mean), not the first moment about the mean (which is zero). See Normalization (statistics) for further normalizing ratios.