View source for Standardized moment

{{Short description|Normalized central moments}}

In [[Probability theory|probability theory]] and [[Statistics|statistics]], a '''standardized moment''' of a [[Probability distribution|probability distribution]] is a moment (often a higher degree [[Central moment|central moment]]) that is normalized, typically by a power of the [[Standard deviation|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 ''X'' be a [[Random variable|random variable]] with a probability distribution ''P'' and mean value <math display="inline">\mu = \mathrm{E}[X]</math> (i.e. the first raw moment or moment about zero), the operator E denoting the [[Expected value|expected value]] of ''X''. Then the '''standardized moment''' of degree ''k'' is <math>\frac{\mu_k}{\sigma^k},</math><ref>{{Cite web|url=http://mathworld.wolfram.com/StandardizedMoment.html|title=Standardized Moment|last=W.|first=Weisstein, Eric|website=mathworld.wolfram.com|language=en|access-date=2016-03-30}}</ref> that is, the ratio of the ''k''th moment about the mean

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

to the ''k''th power of the [[Standard deviation|standard deviation]],

:<math>\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>x^k,</math> meaning that <math>\mu_k(\lambda X) = \lambda^k \mu_k(X):</math> they are [[Homogeneous function|homogeneous function]]s 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:
{| class="wikitable"
!Degree ''k''
!
!Comment
|-
|1
|<math>
\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>
\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|variance]] σ<sup>2</sup>.
|-
|3
|<math>
\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|skewness]].
|-
|4
|<math>
\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|kurtosis]].
|}
For skewness and kurtosis, alternative definitions exist, which are based on the third and fourth [[Cumulant|cumulant]] respectively.

== Other normalizations ==
{{Details|Normalization (statistics)}}
Another scale invariant, dimensionless measure for characteristics of a distribution is the [[Coefficient of variation|coefficient of variation]], <math>\frac{\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:Moment (mathematics)]]

{{Sourceattribution|Standardized moment}}