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.

Standard normalization

Let Template:Mvar be a random variable with a probability distribution Template:Mvar and mean value $\mu =\mathrm{E}[X]$ (i.e. the first raw moment or moment about zero), the operator Template:Math denoting the expected value of Template:Mvar. Then the standardized moment of degree Template:Mvar is Template:Nowrap that is, the ratio of the Template:Mvar-th moment about the mean

$${\mu }_k=\mathrm{E}[(X-\mu {)}^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(x-\mu )}^{k}f(x)dx,$$

to the Template:Mvar-th power of the standard deviation,

$${\sigma }^k={\mu }_2^{k/2}=\mathrm{E}{\left[{(X-\mu )}^2\right]}^{k/2}.$$

The power of Template:Mvar is because moments scale as Template:Nowrap meaning that ${\mu }_k(\lambda X)={\lambda }^k{\mu }_k(X):$ they are homogeneous functions of degree Template:Mvar, 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	${\alpha }_1=\frac{{\mu }_1}{{\sigma }^1}=\frac{\mathrm{E}[(X-\mu {)}^1]}{{(\mathrm{E}[(X-\mu {)}^2])}^{1/2}}=\frac{\mu -\mu }{\sqrt{\mathrm{E}[(X-\mu {)}^2]}}=0$	The first standardized moment is zero, because the first moment about the mean is always zero.
2	${\alpha }_2=\frac{{\mu }_2}{{\sigma }^2}=\frac{\mathrm{E}[(X-\mu {)}^2]}{{(\mathrm{E}[(X-\mu {)}^2])}^{2/2}}=1$	The second standardized moment is one, because the second moment about the mean is equal to the variance Template:Math.
3	${\alpha }_3=\frac{{\mu }_3}{{\sigma }^3}=\frac{\mathrm{E}[(X-\mu {)}^3]}{{(\mathrm{E}[(X-\mu {)}^2])}^{3/2}}$	The third standardized moment is a measure of skewness.
4	${\alpha }_4=\frac{{\mu }_4}{{\sigma }^4}=\frac{\mathrm{E}[(X-\mu {)}^4]}{{(\mathrm{E}[(X-\mu {)}^2])}^{4/2}}$	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

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