Editing Standardized moment

{{Short description|Normalized central moments}}
{{Use American English|date = January 2019}}

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 a probability distribution|shape]] of different probability distributions can be compared using standardized moments.<ref>{{Cite book|url=https://books.google.com/books?id=q1clOAAACAAJ|title=The Elements of Statistics: With Applications to Economics and the Social Sciences|last=Ramsey|first=James Bernard|last2=Newton|first2=H. Joseph|last3=Harvill|first3=Jane L. | date = 2002-01-01|publisher=Duxbury/Thomson Learning|isbn=9780534371111|pages=96|language=en|chapter=CHAPTER 4 MOMENTS AND THE SHAPE OF HISTOGRAMS|chapter-url=http://www.econ.nyu.edu/user/ramseyj/textbook/viewtext.htm}}</ref>

== 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.

== Other normalizations ==
{{Details|Normalization (statistics)}}
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 also ==
*[[Coefficient of variation]]
*[[Moment (mathematics)]]
*[[Central moment]]
*{{section link|Standard score|Other normalizations}}

== References ==
<references />
{{Statistics}}

{{DEFAULTSORT:Standardized Moment}}
[[Category:Statistical deviation and dispersion]]
[[Category:Statistical ratios]]
[[Category:Moments (mathematics)]]