Chi-Square

The chi-square cumulative distribution function is defined for nonnegative x and μ by the integral

$\displaystyle \func$ChiSquareDist(x;μ) = $\displaystyle {\frac{{1}}{{\Gamma (\frac{\mu }{2})2^{\frac{\mu }{2}}}}}$$\displaystyle \int_{{0}}^{{x}}$u$\scriptstyle {\frac{{\mu }}{{2}}}$-1e-$\scriptstyle {\frac{{u}}{{2}}}$du

of the chi-square probability density function

$\displaystyle \limfunc$ChiSquareDen(u;μ) = $\displaystyle {\frac{{1}}{{\Gamma (\frac{\mu }{2})2^{\frac{\mu }{2}}}}}$u$\scriptstyle {\frac{{\mu }}{{2}}}$-1e-$\scriptstyle {\frac{{u}}{{2}}}$

The indexing parameter μ > 0 is the mean of the distribution; it is referred to as the degrees of freedom.

The following plots show distribution functions $\func$ChiSquareDist(x;μ) and density functions $\limfunc$ChiSquareDen(x;μ) for μ = 1, 5, 10, 15 and 0≤x≤25.


itbpFU2.1577in1.7573in0inChi-square density functions itbpFU2.1577in1.7573in0inChi-square distribution functions

The function, $\limfunc$ChiSquareInv$\left(\vphantom{ t;\nu }\right.$t;ν$\left.\vphantom{ t;\nu }\right)$ gives the value of x for which $\func$ChiSquareDist(x;ν) = t. This relationship is demonstrated in the following examples.

$\displaystyle \func$ChiSquareDist(1.6103;5) =  9.9999×10-2 $\displaystyle \approx$ .1  
$\displaystyle \limfunc$ChiSquareInv(.1;5) =  1.6103  
$\displaystyle \func$ChiSquareDist$\displaystyle \left(\vphantom{ 2.366;3}\right.$2.366;3$\displaystyle \left.\vphantom{ 2.366;3}\right)$ =  .5  
$\displaystyle \limfunc$ChiSquareInv(.5;3) =  2.366  

A chi-square distribution table shows values of ν down the left column and values u of $\func$ChiSquareDist across the top row. The entry in row ν and column u is $\limfunc$ChiSquareInv(u;ν).