Editing Standardized moment (section)

== Standard normalization ==
Let {{mvar|X}} be a [[random variable]] with a probability distribution {{mvar|P}} and mean value <math display="inline">\mu = \operatorname{E}[X]</math> (i.e. the first [[Raw moments|raw moment or moment about zero]]), the operator {{math|E}} denoting the [[expected value]] of {{mvar|X}}. Then the '''standardized moment''' of degree {{mvar|k}} is {{nowrap|<math>\mu_k/\sigma^k</math>,}}<ref>{{Cite web|url=http://mathworld.wolfram.com/StandardizedMoment.html|title=Standardized Moment | last=Weisstein | first = Eric W.|website=mathworld.wolfram.com|language=en|access-date=2016-03-30}}</ref> that is, the ratio of the {{mvar|k}}-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 {{mvar|k}}-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 {{mvar|k}} is because moments scale as {{nowrap|<math>x^k</math>,}} meaning that <math>\mu_k(\lambda X) = \lambda^k \mu_k(X):</math> they are [[homogeneous function]]s of degree {{mvar|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 number]]s.

The first four standardized moments can be written as:
{| class="wikitable"
!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]] {{math|''σ''<sup>2</sup>}}.
|-
|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.