The shape of different probability distributions can be compared using standardized moments.
[1] Let X be a random variable with a probability distribution P and mean value
[2] that is, the ratio of the kth moment about the mean to the kth power of the standard deviation, The power of k is because moments scale as
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.
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