Standardized moment

Short description: Normalized central moments

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 X be a random variable with a probability distribution P and mean value [math]\displaystyle{ \mu = \mathrm{E}[X] }[/math] (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 [math]\displaystyle{ \frac{\mu_k}{\sigma^k}, }[/math] that is, the ratio of the kth moment about the mean

[math]\displaystyle{ \mu_k = \operatorname{E} \left[ ( X - \mu )^k \right] = \int_{-\infty}^{\infty} (x - \mu)^k P(x)\,dx, }[/math]

to the kth power of the standard deviation,

[math]\displaystyle{ \sigma^k = \mu_2^{k/2} = \left(\sqrt{\mathrm{E}\left[(X - \mu)^2\right]}\right)^k. }[/math]

The power of k is because moments scale as [math]\displaystyle{ x^k, }[/math] meaning that [math]\displaystyle{ \mu_k(\lambda X) = \lambda^k \mu_k(X): }[/math] 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	[math]\displaystyle{ \tilde{\mu}_1 = \frac{\mu_1}{\sigma^1} = \frac{\operatorname{E} \left[ ( X - \mu )^1 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{1/2}} = \frac{\mu - \mu}{\sqrt{ \operatorname{E} \left[ ( X - \mu )^2 \right]}} = 0 }[/math]	The first standardized moment is zero, because the first moment about the mean is always zero.
2	[math]\displaystyle{ \tilde{\mu}_2 = \frac{\mu_2}{\sigma^2} = \frac{\operatorname{E} \left[ ( X - \mu )^2 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{2/2}} = 1 }[/math]	The second standardized moment is one, because the second moment about the mean is equal to the variance σ2.
3	[math]\displaystyle{ \tilde{\mu}_3 = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E} \left[ ( X - \mu )^3 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{3/2}} }[/math]	The third standardized moment is a measure of skewness.
4	[math]\displaystyle{ \tilde{\mu}_4 = \frac{\mu_4}{\sigma^4} = \frac{\operatorname{E} \left[ ( X - \mu )^4 \right]}{( \operatorname{E} \left[ ( X - \mu )^2 \right])^{4/2}} }[/math]	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, [math]\displaystyle{ \frac{\sigma}{\mu} }[/math]. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because [math]\displaystyle{ \mu }[/math] 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