Chi-square distribution

For other uses, see Chi.

In probability theory and statistics, the chi-square distribution (also chi-squared or <math>\chi^2</math>  distribution) is one of the most widely used theoretical probability distributions. Chi-square distribution with <math>\nu</math> degrees of freedom is written as <math>\chi^2(\nu)</math>. It is a special case of gamma distribution.

Chi-square distribution is primarily used in statistical significance tests and confidence intervals. It is useful, because it is relatively easy to show that certain probability distributions come close to it, under certain conditions. One of these conditions is that the null hypothesis must be true. Another one is that the different random variables (or observations) must be independent of each other.

Related pages

• Chi-squared test

References

chi-square

Probability density function File:Chi-square distributionPDF.png
Cumulative distribution function File:Chi-square distributionCDF.png
Parameters	<math>k > 0\,</math> degrees of freedom
Support	<math>x \in [0; +\infty)\,</math>
Probability density function (pdf)	<math>\frac{(1/2)^{k/2
Cumulative distribution function (cdf)	{{{cdf}}}
Mean	{{{mean}}}
Median	{{{median}}}
Mode	{{{mode}}}
Variance	{{{variance}}}
Skewness	{{{skewness}}}
Excess kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
Moment-generating function (mgf)	{{{mgf}}}
Characteristic function	{{{char}}}

{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,</math>|

 cdf        =<math>\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,</math>|
 mean       =<math>k\,</math>|
 median     =approximately <math>k-2/3\,</math>|
 mode       =<math>k-2\,</math> if <math>k\geq 2\,</math>|
 variance   =<math>2\,k\,</math>|
 skewness   =<math>\sqrt{8/k}\,</math>|
 kurtosis   =<math>12/k\,</math>|
 entropy    =<math>\frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)</math>|
 mgf        =<math>(1-2\,t)^{-k/2}</math> for <math>2\,t<1\,</math>|
 char       =<math>(1-2\,i\,t)^{-k/2}\,</math>

}}

Other websites

• Chi-Square Tutorial by Khan Academy