Standardized moment

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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.<ref>Template:Cite book</ref>

Standard normalizationEdit

Let Template:Mvar be a random variable with a probability distribution Template:Mvar and mean value <math display="inline">\mu = \operatorname{E}[X]</math> (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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> that is, the ratio of the Template:Mvar-th moment about the mean

<math display="block"> \mu_k = \operatorname{E} \left[ ( X - \mu )^k \right] = \int_{-\infty}^{\infty} {\left(x - \mu\right)}^k f(x)\,dx, </math>

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

<math display="block">\sigma^k = \mu_2^{k/2} = \operatorname{E}\!{\left[ {\left(X - \mu\right)}^2 \right]}^{k/2}.</math>

The power of Template:Mvar is because moments scale as Template:Nowrap meaning that <math>\mu_k(\lambda X) = \lambda^k \mu_k(X):</math> 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 <math>

\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 </math>

The first standardized moment is zero, because the first moment about the mean is always zero.
2 <math>

\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 </math>

The second standardized moment is one, because the second moment about the mean is equal to the variance Template:Math.
3 <math>

\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}} </math>

The third standardized moment is a measure of skewness.
4 <math>

\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}} </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 normalizationsEdit

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

ReferencesEdit

<references /> Template:Statistics