Standardized moment

Short description: Normalized central moments

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.[1]

## Contents

* 1 Standard normalization
* 2 Other normalizations
* 3 See also
* 4 References

## Standard normalization

Let X be a random variable with a probability distribution P and mean value μ=E⁡[X] (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of X. Then the standardized moment of degree k is μk/σk,[2] that is, the ratio of the k-th moment about the mean

μk=E⁡[(X−μ)k]=∫−∞∞(x−μ)kf(x)dx,

to the k-th power of the standard deviation,

σk=μ2k/2=E⁡[(X−μ)2]k/2.

The power of k is because moments scale as xk, meaning that μk(λX)=λkμk(X): they are homogeneous functions 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:

Degree k |  | Comment

|
|

1  | μ~1=μ1σ1=E⁡[(X−μ)1](E⁡[(X−μ)2])1/2=μ−μE⁡[(X−μ)2]=0 | The first standardized moment is zero, because the first moment about the mean is always zero.
2  | μ~2=μ2σ2=E⁡[(X−μ)2](E⁡[(X−μ)2])2/2=1 | The second standardized moment is one, because the second moment about the mean is equal to the variance σ2.
3  | μ~3=μ3σ3=E⁡[(X−μ)3](E⁡[(X−μ)2])3/2 | The third standardized moment is a measure of skewness.
4  | μ~4=μ4σ4=E⁡[(X−μ)4](E⁡[(X−μ)2])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

Further information: Normalization (statistics)

Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, σ/μ. However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because μ 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
* Standard score § Other normalizations

## References

1. ↑ Ramsey, James Bernard; Newton, H. Joseph; Harvill, Jane L. (2002-01-01). "CHAPTER 4 MOMENTS AND THE SHAPE OF HISTOGRAMS" (in en). The Elements of Statistics: With Applications to Economics and the Social Sciences. Duxbury/Thomson Learning. pp. 96. ISBN 9780534371111. http://www.econ.nyu.edu/user/ramseyj/textbook/viewtext.htm.
2. ↑ Weisstein, Eric W.. "Standardized Moment" (in en). http://mathworld.wolfram.com/StandardizedMoment.html.

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Statistics

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Descriptive statistics

| Continuous data|

| Center|

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Dispersion|

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Frequentist inference|

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Rank statistics|

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Bayesian inference|

* Bayesian probability
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* Correlation
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| Correlation|

* Pearson product-moment
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|

Regression analysis|

* Errors and residuals
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Linear regression|

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Non-standard predictors|

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Generalized linear model|

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Partition of variance|

* Analysis of variance (ANOVA, anova)
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Categorical / Multivariate / Time-series / Survival analysis

| Categorical|

* Cohen's kappa
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Multivariate|

* Regression
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Time-series|

| General|

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Specific tests|

* Dickey–Fuller
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Time domain|

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Frequency domain|

* Spectral density estimation
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Survival|

| Survival function|

* Kaplan–Meier estimator (product limit)
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Hazard function|

* Nelson–Aalen estimator

Test|

* Log-rank test

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