In probability theory and statistics, the kth standardized moment of a probability distribution is where is the kth moment about the mean and σ is the standard deviation.
It is the normalization of the kth moment with respect to standard deviation. The power of k is because moments scale as , meaning that : they are homogeneous polynomials of degree k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension, but in the ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers.
Note that 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.
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