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.
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Standard normalization
Let X be a random variable with a probability distribution P and mean value
and the standard deviation to the power of k
The power of k is because moments scale as
The first four standardized moments can be written as:
Note that for skewness and kurtosis alternative definitions exist, which are based on the third and fourth cumulant respectively.
The kth standardized moment may be generalized as:
Other normalizations
Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation,
See Normalization (statistics) for further normalizing ratios.