Standardized moment

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 $\mu = \mathrm{E}[X]$ (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 $$\frac{\mu_k}{\sigma^k},$$ that is, the ratio of the kth moment about the mean



\mu_k = \operatorname{E} \left[ ( X - \mu )^k \right] = \int_{-\infty}^{\infty} (x - \mu)^k P(x)\,dx, $$

to the kth power of the standard deviation,


 * $$\sigma^k = \mu_2^{k/2} = \left(\sqrt{\mathrm{E}\left[(X - \mu)^2\right]}\right)^k.$$

The power of k is because moments scale as $$x^k,$$ meaning that $$\mu_k(\lambda X) = \lambda^k \mu_k(X):$$ 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, $$\frac{\sigma}{\mu}$$. 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.