Standardized moment

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.{{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}}

Standard normalization

Let {{mvar|X}} be a random variable with a probability distribution {{mvar|P}} and mean value $\mu = \operatorname{E}[X]$ (i.e. the first 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| $\mu_k/\sigma^k$ ,}}{{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}} that is, the ratio of the {{mvar|k}}-th moment about the mean

$\mu_k = \operatorname{E} \left[ ( X - \mu )^k \right] = \int_{-\infty}^{\infty} {\left(x - \mu\right)}^k f(x)\,dx,$

to the {{mvar|k}}-th power of the standard deviation,

$\sigma^k = \mu_2^{k/2} = \operatorname{E}\!{\left[ {\left(X - \mu\right)}^2 \right]}^{k/2}.$

The power of {{mvar|k}} is because moments scale as {{nowrap| $x^k$ ,}} meaning that $\mu_k(\lambda X) = \lambda^k \mu_k(X):$ they are homogeneous functions 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 numbers.

The first four standardized moments can be written as:

class="wikitable"

!Degree k

!Comment

| $\tilde{\mu}_1 = \frac{\mu_1}{\sigma^1}= \frac{\operatorname{E} \left[ ( X - \mu )^1 \right]}{\left( \operatorname{E} \left[ ( X - \mu )^2 \right]\right)^{1/2}}
= \frac{\mu - \mu}{\sqrt{ \operatorname{E} \left[ ( X - \mu )^2 \right]}}
= 0$

|The first standardized moment is zero, because the first moment about the mean is always zero.

| $\tilde{\mu}_2 = \frac{\mu_2}{\sigma^2}
= \frac{\operatorname{E} \left[ ( X - \mu )^2 \right]}{\left( \operatorname{E} \left[ ( X - \mu )^2 \right]\right)^{2/2}}
= 1$

|The second standardized moment is one, because the second moment about the mean is equal to the variance {{math|σ²}}.

| $\tilde{\mu}_3 = \frac{\mu_3}{\sigma^3}
= \frac{\operatorname{E} \left[ ( X - \mu )^3 \right]}{\left( \operatorname{E} \left[ ( X - \mu )^2 \right]\right)^{3/2}}$

|The third standardized moment is a measure of skewness.

| $\tilde{\mu}_4 = \frac{\mu_4}{\sigma^4}
= \frac{\operatorname{E} \left[ ( X - \mu )^4 \right]}{\left( \operatorname{E} \left[ (X - \mu)^2 \right]\right)^{4/2}}$

|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

Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, $\sigma / \mu$ . However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because $\mu$ 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.

References

Category:Statistical deviation and dispersion

Category:Statistical ratios

Category:Moments (mathematics)

Standardized moment

Standard normalization

Other normalizations

See also

References