Standardized moment


In probability theory and statistics, a standardized moment of a probability distribution is a 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.

Standard normalization

Let X be a random variable with a probability distribution P and mean value , the operator E denoting the expected value of X. Then the standardized moment of degree k is 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 meaning that 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:
Degree kComment
1The first standardized moment is zero, because the first moment about the mean is always zero.
2The second standardized moment is one, because the second moment about the mean is equal to the variance σ2.
3The third standardized moment is a measure of skewness.
4The fourth standardized moment refers to the kurtosis.

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,. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because is the first moment about zero, not the first moment about the mean.
See Normalization for further normalizing ratios.