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.
Let be a random variable with a probability distribution and mean value (i.e. the first raw moment or moment about zero), the operator denoting the expected value of . Then the standardized moment of degree is that is, the ratio of the -th moment about the mean
to the -th power of the standard deviation,
The power of is because moments scale as meaning that they are homogeneous functions of degree , 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.
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 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.