Standardized moment

In probability theory and statistics, a standardized moment of a probability distribution is a moment that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of a [probability distribution|shape] of different probability distributions can be compared using standardized moments.

Standard normalization

Let be a random variable with a probability distribution and mean value, the operator denoting the expected value of. Then the standardized moment of degree is that is, the ratio of the -th moment about the mean
to the -th power of the standard deviation,
The power of is because moments scale as meaning that they are homogeneous functions of degree, 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 k		Comment
1		The first standardized moment is zero, because the first moment about the mean is always zero.
2		The second standardized moment is one, because the second moment about the mean is equal to the variance.
3		The third standardized moment is a measure of skewness.
4		The fourth standardized moment refers to the kurtosis.

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, not the first moment about the mean.
See Normalization (statistics) for further normalizing ratios.