Difference between revisions of "Chi-squared distribution"
m
 
(13 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
[[category:Distribution Functions]]
+
[[category:Continuous distributions]]
−
[[Category:Doc Status D]] <!-- For Lumina use, do not change -->
+
[[category:Semi-bounded distributions]]
 +
[[category:Unimodal distributions]]
 +
[[category:Univariate distributions]]
 +
{{ReleaseBar}}
  
−
== ChiSquared(d) ==
+
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 of independent [[Normal distribution|normally-distributed]] random variables.
  
−
The ChiSquared distribution with «d» degrees of freedom describes the distribution of a Chi-Squared metric defined as
+
<center>[[image:ChiSquared(3).png]]</center>
  
−
<math>\Chi^2 \sum_{i=1}^n {y_i}^2</math>
+
The [[Chi-squared distribution]] with «dof» degrees of freedom describes the distribution of a Chi-Squared metric defined as
  
−
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.
+
:<math>\Chi^2 \sum_{i=1}^n {y_i}^2</math>
  
−
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.
+
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.
  
 +
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 ==
 +
=== <div id="ChiSquared">ChiSquared(dof)</div> ===
 +
The distribution function. Use this to define a chance variable or other uncertain quantity with an F-distribution with «dof» degrees of freedom.
 +
 +
=== <div id="DensChiSquared">Dens{{Release||5.1|_}}ChiSquared(x, dof)</div> ===
 +
The analytic probability density at «x». Equal to
 +
:<math>p(x) = {1\over{2^{d/2} \Gamma(d/2)}} x^{d/2-1} e^{-x/2}</math>
 +
where <math>d</math> is «dof» and <math>\Gamma(x)</math> is the <code>[[GammaFn]](x)</code>.
 +
 +
=== <div id="CumChiSquared">CumChiSquared(x, dof)</div> ===
 +
The analytic cumulative density up to «x». This is the probability that a random sample will be less than or equal to «x».
 +
 +
=== <div id="CumChiSquaredInv">CumChiSquaredInv(p, dof)</div> ===
 +
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.
 +
 +
== 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>\sqrt{ 8 / dof}</math>
 +
* [[Kurtosis]] = 12 / dof
 +
 +
== Examples ==
 
Suppose
 
Suppose
 
:<code>Variable V := ChiSquared(k)</code>
 
:<code>Variable V := ChiSquared(k)</code>
Line 17: Line 47:
 
:<code>Variable S := (V/k)*(W/m)</code>
 
:<code>Variable S := (V/k)*(W/m)</code>
  
−
<code>S</code> is distributed as an F distribution with <code>k</code> and <code>m</code> degrees of freedom.
+
<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.
−
 
−
== Library ==
 
−
Distributions
 
  
 
== See Also ==
 
== See Also ==
 
* [[Dens_ChiSquared]]
 
* [[Dens_ChiSquared]]
 
* [[CumChiSquared]]
 
* [[CumChiSquared]]
 +
* [[Normal]]
 +
* [[Gamma]] -- very closely related distribution.
 +
* [[Rayleigh]]
 +
* [[F-distribution]]
 
* [[Parametric continuous distributions]]
 
* [[Parametric continuous distributions]]
 
* [[Distribution Densities Library]]
 
* [[Distribution Densities Library]]

Latest revision as of 20:37, 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 continuous, positive only, unimodal probability distribution that describes the sum of independent normally-distributed random variables.

ChiSquared(3).png

The Chi-squared 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:

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

Comments


You are not allowed to post comments.