Chi-square distribution

For other uses, see Chi.

In probability theory and statistics, the chi-square distribution (also chi-squared or [math]\displaystyle{ \chi^2 }[/math]  distribution) is one of the most widely used theoretical probability distributions. Chi-square distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom is written as [math]\displaystyle{ \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.

Chi-square Distribution Media

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  Plot of the chi-square distribution for values of k = {1, 2, 3, 4, 6, 9}. Accurate plotcurves. Labels are embedded in Computer-Modern font.

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  Plot of the cumulative chi-square distribution for values of k = {1, 2, 3, 4, 6, 9}. Accurate plotcurves. Labels are embedded in Computer-Modern font.

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  Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (k = 10)

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  Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (blue) and relative difference (red) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.

Related pages

• Chi-squared test

References

chi-square

Probability density function
Cumulative distribution function
Parameters	[math]\displaystyle{ k \gt 0\, }[/math] degrees of freedom
Support	[math]\displaystyle{ x \in [0; +\infty)\, }[/math]
Probability density function (pdf)	[math]\displaystyle{ \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\, }[/math]
Cumulative distribution function (cdf)	[math]\displaystyle{ \frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\, }[/math]
Mean	[math]\displaystyle{ k\, }[/math]
Median	approximately [math]\displaystyle{ k-2/3\, }[/math]
Mode	[math]\displaystyle{ k-2\, }[/math] if [math]\displaystyle{ k\geq 2\, }[/math]
Variance	[math]\displaystyle{ 2\,k\, }[/math]
Skewness	[math]\displaystyle{ \sqrt{8/k}\, }[/math]
Excess kurtosis	[math]\displaystyle{ 12/k\, }[/math]
Entropy	[math]\displaystyle{ \frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2) }[/math]
Moment-generating function (mgf)	[math]\displaystyle{ (1-2\,t)^{-k/2} }[/math] for [math]\displaystyle{ 2\,t\lt 1\, }[/math]
Characteristic function	[math]\displaystyle{ (1-2\,i\,t)^{-k/2}\, }[/math]

Other websites

• Chi-Square Tutorial by Khan Academy