Standardized moment

In probability theory and statistics, the standardized moment of a probability distribution is a moment (normally a higher degree central moment) that is normalized. The normalization is typically a division by an expression of the standard deviation which renders the moment scale invariant. This has the advantage that such normalized moments differ only in other properties than variability, facilitating e.g. comparison of shape of different probability distributions.

Let X be a random variable with a probability distribution P and mean value ${\textstyle \mu =\mathrm {E} [X]}$ (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 ${\displaystyle {\frac {\mu _{k}}{\sigma ^{k}}}\!}$, that is, a ratio of the kth moment about the mean

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