Standardized moment

In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments.^[1]

Standard normalization

Let X be a random variable with a probability distribution P and mean value ${\textstyle \mu =\operatorname {E} [X]}$ (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of X. Then the standardized moment of degree k is ${\displaystyle \mu _{k}/\sigma ^{k}}$,^[2] that is, the ratio of the k-th moment about the mean

$${\displaystyle \mu _{k}=\operatorname {E} \left[(X-\mu )^{k}\right]=\int _{-\infty }^{\infty }{\left(x-\mu \right)}^{k}f(x)\,dx,}$$

to the k-th power of the standard deviation,

$${\displaystyle \sigma ^{k}=\mu _{2}^{k/2}=\operatorname {E} \!{\left[{\left(X-\mu \right)}^{2}\right]}^{k/2}.}$$

The power of k is because moments scale as ${\displaystyle x^{k}}$, meaning that ${\displaystyle \mu _{k}(\lambda X)=\lambda ^{k}\mu _{k}(X):}$ 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 k		Comment
1	${\displaystyle {\tilde {\mu }}_{1}={\frac {\mu _{1}}{\sigma ^{1}}}={\frac {\operatorname {E} \left[(X-\mu )^{1}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{1/2}}}={\frac {\mu -\mu }{\sqrt {\operatorname {E} \left[(X-\mu )^{2}\right]}}}=0}$	The first standardized moment is zero, because the first moment about the mean is always zero.
2	${\displaystyle {\tilde {\mu }}_{2}={\frac {\mu _{2}}{\sigma ^{2}}}={\frac {\operatorname {E} \left[(X-\mu )^{2}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{2/2}}}=1}$	The second standardized moment is one, because the second moment about the mean is equal to the variance σ2.
3	${\displaystyle {\tilde {\mu }}_{3}={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\operatorname {E} \left[(X-\mu )^{3}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{3/2}}}}$	The third standardized moment is a measure of skewness.
4	${\displaystyle {\tilde {\mu }}_{4}={\frac {\mu _{4}}{\sigma ^{4}}}={\frac {\operatorname {E} \left[(X-\mu )^{4}\right]}{\left(\operatorname {E} \left[(X-\mu )^{2}\right]\right)^{4/2}}}}$	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.

Other normalizations

Further information: Normalization (statistics)

Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, ${\displaystyle \sigma /\mu }$. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because ${\displaystyle \mu }$ 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.

See also

• Coefficient of variation
• Moment (mathematics)
• Central moment
• Standard score § Other normalizations