Difference between revisions of "Chi-squared distribution"

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[[category:Unimodal distributions]]
 
[[category:Unimodal distributions]]
 
[[category:Univariate distributions]]
 
[[category:Univariate distributions]]
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{{ReleaseBar}}
  
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== ChiSquared(d) ==
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The '''''<math>\Chi^2</math>-squared distribution''''' is a [[category:Continuous distributions|continuous]], [[:category:Semi-bounded distributions|positive only]], [[:category:Unimodal distributions|unimodal]] probability distribution that describes the sum independent [[Normal distribution|normally-distributed]] random variables.
  
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The [[ChiSquared]] distribution with «d» degrees of freedom describes the distribution of a Chi-Squared metric defined as
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<center>[[image:ChiSquared(3).png]]</center>
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The [[ChiSquared distribution]] with «dof» degrees of freedom describes the distribution of a Chi-Squared metric defined as
  
 
:<math>\Chi^2 \sum_{i=1}^n {y_i}^2</math>
 
:<math>\Chi^2 \sum_{i=1}^n {y_i}^2</math>
  
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where each ''y<sub>i</sub>'' is independently sampled from a standard normal distribution and ''d = n - 1'' . The distribution is defined over nonnegative values.
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where each ''y<sub>i</sub>'' is independently sampled from a standard normal distribution and ''«dof» = n - 1'' . The distribution is defined over nonnegative values.
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The Chi-squared distribution is commonly used for analyses of second moments, such as analyses of variance and contingency table analyses. The ratio of two chi-squared-distributed variables follows an [[F-distribution]].
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== Functions ==
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=== <div id="ChiSquared">ChiSquared(dof)</div> ===
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The distribution function. Use this to define a chance variable or other uncertain quantity with an F-distribution with «dof» degrees of freedom.
  
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The Chi-squared distribution is commonly used for analyses of second moments, such as analyses of variance and contingency table analyses. It can also be used to generate the [[F distribution]].
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=== <div id="DensChiSquared">Dens{{Release||5.1|_}}ChiSquared(x, dof)</div> ===
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The analytic probability density at «x». Equal to
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:<math>p(x) = {1\over{2^{d/2} \Gamma(d/2)}} x^{d/2-1} e^{-x/2}</math>
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where <math>d</math> is «dof» and <math>\Gamma(x)</math> is the <code>[[GammaFn]](x)</code>.
  
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=== <div id="CumChiSquared">CumChiSquared(x, dof)</div> ===
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The analytic cumulative density up to «x». This is the probability that a random sample will be less than or equal to «x».
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=== <div id="CumChiSquaredInv">CumChiSquaredInv(p, dof)</div> ===
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The inverse cumulative density (quantile function), which computes the p<sup>th</sup> fractile/quantile/percentile value x, which has a «p» probability of being greater than or equal to a random variate draw from a chi-squared distribution with «dof» degrees of freedom.
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== Statistics ==
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Theoretical statistics (i.e., in the absence of sampling error) are:
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* [[Mean]] = dof
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* Mode = k-2 when k>2, 0 otherwise
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* [[Median]] = ...
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* [[Variance]] = 2 * dof
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* [[Skewness]] = <math>\sqrt( 8 / dof)</math>
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* [[Kurtosis] = 12 / dof
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== Examples ==
 
Suppose
 
Suppose
 
:<code>Variable V := ChiSquared(k)</code>
 
:<code>Variable V := ChiSquared(k)</code>
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:<code>Variable S := (V/k)*(W/m)</code>
 
:<code>Variable S := (V/k)*(W/m)</code>
  
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<code>S</code> is distributed as an F distribution with <code>k</code> and <code>m</code> degrees of freedom.
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<code>S</code> is distributed as an [[F-distribution]] with <code>k</code> and <code>m</code> degrees of freedom.
  
 
The F distribution is useful for the analysis of ratios of variance, such as a one-factor between-subjects analysis of variance.
 
The F distribution is useful for the analysis of ratios of variance, such as a one-factor between-subjects analysis of variance.
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== Library ==
 
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Distributions
 
  
 
== See Also ==
 
== See Also ==
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* [[CumChiSquared]]
 
* [[CumChiSquared]]
 
* [[Normal]]
 
* [[Normal]]
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* [[Gamma]] -- very closely related distribution.
 
* [[Rayleigh]]
 
* [[Rayleigh]]
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* [[F distribution]]
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* [[F-distribution]]
 
* [[Parametric continuous distributions]]
 
* [[Parametric continuous distributions]]
 
* [[Distribution Densities Library]]
 
* [[Distribution Densities Library]]

Revision as of 20:28, 10 October 2018



Release:

4.6  •  5.0  •  5.1  •  5.2  •  5.3  •  5.4  •  6.0  •  6.1  •  6.2  •  6.3  •  6.4  •  6.5


The [math]\displaystyle{ \Chi^2 }[/math]-squared distribution is a, positive only, unimodal probability distribution that describes the sum independent normally-distributed random variables.

ChiSquared(3).png

The ChiSquared distribution with «dof» degrees of freedom describes the distribution of a Chi-Squared metric defined as

[math]\displaystyle{ \Chi^2 \sum_{i=1}^n {y_i}^2 }[/math]

where each yi is independently sampled from a standard normal distribution and «dof» = n - 1 . The distribution is defined over nonnegative values.

The Chi-squared distribution is commonly used for analyses of second moments, such as analyses of variance and contingency table analyses. The ratio of two chi-squared-distributed variables follows an F-distribution.

Functions

ChiSquared(dof)

The distribution function. Use this to define a chance variable or other uncertain quantity with an F-distribution with «dof» degrees of freedom.

DensChiSquared(x, dof)

The analytic probability density at «x». Equal to

[math]\displaystyle{ p(x) = {1\over{2^{d/2} \Gamma(d/2)}} x^{d/2-1} e^{-x/2} }[/math]

where [math]\displaystyle{ d }[/math] is «dof» and [math]\displaystyle{ \Gamma(x) }[/math] is the GammaFn(x).

CumChiSquared(x, dof)

The analytic cumulative density up to «x». This is the probability that a random sample will be less than or equal to «x».

CumChiSquaredInv(p, dof)

The inverse cumulative density (quantile function), which computes the pth fractile/quantile/percentile value x, which has a «p» probability of being greater than or equal to a random variate draw from a chi-squared distribution with «dof» degrees of freedom.

Statistics

Theoretical statistics (i.e., in the absence of sampling error) are:

  • Mean = dof
  • Mode = k-2 when k>2, 0 otherwise
  • Median = ...
  • Variance = 2 * dof
  • Skewness = [math]\displaystyle{ \sqrt( 8 / dof) }[/math]
  • [[Kurtosis] = 12 / dof

Examples

Suppose

Variable V := ChiSquared(k)
Variable W := ChiSquared(m)
Variable S := (V/k)*(W/m)

S is distributed as an F-distribution with k and m degrees of freedom.

The F distribution is useful for the analysis of ratios of variance, such as a one-factor between-subjects analysis of variance.

See Also

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