Chi Square Distribution

The chi square (χ²) distribution, also known as the chi-squared distribution, is a continuous probability distribution that models the distribution of the sum of the squares of $n$ independent normal variables. It is widely used in hypothesis testing and confidence interval estimation for variance in statistics.

The chi square distribution is characterized by a single parameter, the degrees of freedom (n). While typically a positive integer, n can be any positive real number. The distribution is defined for all non-negative real values.

The probability density function (PDF) is given by:

The cumulative distribution function (CDF) is:

where $\gamma (s,x)$ is the lower incomplete gamma function and $\mathrm{\Gamma }(s)$ is the gamma function.

The chi-square distribution has numerous important applications in statistical analysis and quality control:

• The distribution is fundamental in goodness-of-fit testing, where it helps determine whether sample data conforms to a hypothesized probability distribution.

• In contingency table analysis, the chi-square test evaluates whether there is a significant relationship between categorical variables.

• Statistical process control relies on chi-square distributions to establish confidence intervals for population variance, enabling effective quality monitoring in manufacturing.

• When evaluating regression models, the chi-square test helps assess the goodness-of-fit of the model and validates underlying assumptions.

• The distribution plays a crucial role in hypothesis testing for variance components in analysis of variance (ANOVA) procedures.

The chi-square distribution has several notable properties:

• The sum of independent chi-square variables is also chi-square distributed with degrees of freedom equal to the sum of the individual degrees of freedom.

• For large n, the distribution approaches a normal distribution with mean n and variance 2n.

• If Z is standard normal, then $Z^2$ follows a chi-square distribution with 1 degree of freedom.

• The chi-square distribution is a special case of the gamma distribution with shape $\alpha =n/2$ and scale $\theta =2$.

• The ratio of two chi-square distributions divided by their respective degrees of freedom follows an F-distribution.

The chi square distribution is implemented by the ChiSquareDistribution class. It has one constructor which takes the degrees of freedom as its only argument. The following constructs a chi square distribution with 10 degrees of freedom:

var chiSquare = new ChiSquareDistribution(10);

Dim chi = New ChiSquareDistribution(10)

No code example is currently available or this language may not be supported.

let chiSquare = ChiSquareDistribution(10.0)

The ChiSquareDistribution class has one specific property, DegreesOfFreedom, that returns the degrees of freedom of the distribution.

ChiSquareDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new Pcg32();
double sample = ChiSquareDistribution.Sample(random, 10);

Dim random = New Pcg32()
Dim sample = ChiSquareDistribution.Sample(random, 10)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = ChiSquareDistribution.Sample(random, 10.0)

The above example uses the Pcg32 to generate uniform random numbers.

For more information on the chi-square distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 18. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 11. Wiley Series in Probability and Statistics.

Other Resources