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Chi square distribution

Chi-Square Distribution

Probability Density FunctionThe chi-square distribution results when nu independent variables with standard normal distributions are squared and summed. The formula for the probability density function of the chi-square distribution isf(x) = EXP(-x/2)*x**(nu/2 - 1)/(2**(nu/2)*GAMMA(nu/2)) x >= 0
where nu is the shape parameter and GAMMA is the gamma function. The formula for the gamma function is
GAMMA(a) = INTEGRAL[t**(a-1)*EXP(-t)dt] where the integration is from 0 to infinity
In a testing context, the chi-square distribution is treated as a "standardized distribution" (i.e., no location or scale parameters). However, in a distributional modeling context (as with other probability distributions), the chi-square distribution itself can be transformed with a location parameter, mu, and a scale parameter, sigma.
The following is the plot of the chi-square probability density function for 4 different values of the shape parameter.
plot of the chi-square probability density function for 4 different values of the shape parameter
Cumulative Distribution FunctionThe formula for the cumulative distribution function of the chi-square distribution isF(x) = gamma(nu/2,x/2)/GAMMA(nu/2) for x >= 0
where GAMMA is the gamma function defined above and gamma is the incomplete gamma function. The formula for the incomplete gamma function is
GAMMA(a,x) = INTEGRAL[t**(a-1)*EXP(-t)dt] where the integration is from 0 to x
The following is the plot of the chi-square cumulative distribution function with the same values of  as the pdf plots above.
plot of the chi-square cumulative distribution function with the same values of nu as the pdf plots above
Percent Point FunctionThe formula for the percent point function of the chi-square distribution does not exist in a simple closed form. It is computed numerically.The following is the plot of the chi-square percent point function with the same values of nu as the pdf plots above.
plot of the chi-square percent point function with the same values of nu as the pdf plots above
Other Probability FunctionsSince the chi-square distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit the formulas and plots for the hazard, cumulative hazard, survival, and inverse survival probability functions.
Common Statistics
Meannu
Medianapproximately nu - 2/3 for large nu
Modenu -2 for nu > 2
Range0 to positive infinity
Standard DeviationSQRT(2*nu)
Coefficient of VariationSQRT(2/nu)
Skewness2**1.5/SQRT(nu)
Kurtosis3 + 12/nu
Parameter EstimationSince the chi-square distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
CommentsThe chi-square distribution is used in many cases for the critical regions for hypothesis tests and in determining confidence intervals. Two common examples are the chi-square test for independence in an RxC contingency table and the chi-square test to determine if the standard deviation of a population is equal to a pre-specified value.
SoftwareMost general purpose statistical software programs support at least some of the probability functions for the chi-square distribution.

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