Chi-Square Distribution

Overview

The chi-square (χ²) distribution is a one-parameter family of curves. The chi-square distribution is commonly used in hypothesis testing, particularly the chi-square test for goodness of fit.

Statistics and Machine Learning Toolbox™ offers multiple ways to work with the chi-square distribution.

• Use distribution-specific functions (chi2cdf, chi2inv, chi2pdf, chi2rnd, chi2stat) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple chi-square distributions.

• Use generic distribution functions (cdf, icdf, pdf, random) with a specified distribution name ('Chisquare') and parameters.

Parameters

The chi-square distribution uses the following parameter.

Parameter	Description	Support
nu (ν)	Degrees of freedom	ν = 1, 2, 3,...

The degrees of freedom parameter is typically an integer, but chi-square functions accept any positive value.

The sum of two chi-square random variables with degrees of freedom ν₁ and ν₂ is a chi-square random variable with degrees of freedom ν = ν₁ + ν₂.

Probability Density Function

The probability density function (pdf) of the chi-square distribution is

$$y=f\left(x|\nu \right)=\frac{x^{\left(\nu -2\right)/2}{e}^{-x/2}}{2^{\frac{\nu }{2}}\Gamma \left(\nu /2\right)},$$

where ν is the degrees of freedom and Γ( · ) is the Gamma function.

For an example, see Compute Chi-Square Distribution pdf.

Cumulative Distribution Function

The cumulative distribution function (cdf) of the chi-square distribution is

$$p=F(x|\nu )={\displaystyle {\int }_0^x\frac{t^{(\nu -2)/2}{e}^{-t/2}}{2^{\nu /2}\Gamma (\nu /2)}dt},$$

where ν is the degrees of freedom and Γ( · ) is the Gamma function. The result p is the probability that a single observation from the chi-square distribution with ν degrees of freedom falls in the interval [0, x].

For an example, see Compute Chi-Square Distribution cdf.

Inverse Cumulative Distribution Function

The inverse cumulative distribution function (icdf) of the chi-square distribution is

$$x=F^{-1}(p|\nu )=\{x:F(x|\nu )=p\},$$

where

$$p=F(x|\nu )={\displaystyle {\int }_0^x\frac{t^{(\nu -2)/2}{e}^{-t/2}}{2^{\nu /2}\Gamma (\nu /2)}}dt,$$

ν is the degrees of freedom, and Γ( · ) is the Gamma function. The result p is the probability that a single observation from the chi-square distribution with ν degrees of freedom falls in the interval [0, x].

Descriptive Statistics

The mean of the chi-square distribution is ν.

The variance of the chi-square distribution is 2ν.

Examples

Compute Chi-Square Distribution pdf

Compute the pdf of a chi-square distribution with 4 degrees of freedom.

x = 0:0.2:15;
y = chi2pdf(x,4);

Plot the pdf.

figure;
plot(x,y)
xlabel('Observation')
ylabel('Probability Density')

The chi-square distribution is skewed to the right, especially for few degrees of freedom.

Compute Chi-Square Distribution cdf

Compute the cdf of a chi-square distribution with 4 degrees of freedom.

x = 0:0.2:15;
y = chi2cdf(x,4);

Plot the cdf.

figure;
plot(x,y)
xlabel('Observation')
ylabel('Cumulative Probability')

Related Distributions

• F Distribution — The F distribution is a two-parameter distribution that has parameters ν₁ (numerator degrees of freedom) and ν₂ (denominator degrees of freedom). The F distribution can be defined as the ratio $$F=\frac{\raisebox{1ex}{${\chi }_1^2$}\!\left/ \!\raisebox{-1ex}{${\nu }_1$}\right.}{\raisebox{1ex}{${\chi }_2^2$}\!\left/ \!\raisebox{-1ex}{${\nu }_2$}\right.}$$, where χ²₁ and χ²₂ are both chi-square distributed with ν₁ and ν₂ degrees of freedom, respectively.

• Gamma Distribution — The gamma distribution is a two-parameter continuous distribution that has parameters a (shape) and b (scale). The chi-square distribution is equal to the gamma distribution with 2a = ν and b = 2.

• Noncentral Chi-Square Distribution — The noncentral chi-square distribution is a two-parameter continuous distribution that has parameters ν (degrees of freedom) and δ (noncentrality). The noncentral chi-square distribution is equal to the chi-square distribution when δ = 0.

• Normal Distribution — The normal distribution is a two-parameter continuous distribution that has parameters μ (mean) and σ (standard deviation). The standard normal distribution occurs when μ = 0 and σ = 1.

  If Z₁, Z₂, …, Z_n are standard normal random variables, then $${\displaystyle \sum _{i=1}^n{Z}_i{}^2}$$ has a chi-square distribution with degrees of freedom ν = n – 1.

  If a set of n observations is normally distributed with variance σ² and sample variance s², then $$\frac{\left(n-1\right)s^2}{{\sigma }^2}$$ has a chi-square distribution with degrees of freedom ν = n – 1. This relationship is used to calculate confidence intervals for the estimate of the normal parameter σ² in the function normfit.

• Student's t Distribution — The Student's t distribution is a one-parameter continuous distribution that has parameter ν (degrees of freedom). If Z has a standard normal distribution and χ² has a chi-square distribution with degrees of freedom ν, then $$\text{t = }\frac{Z}{\sqrt{{\chi }^2/\nu }}$$ has a Student's t distribution with degrees of freedom ν.

• Wishart Distribution — The Wishart distribution is a higher dimensional analog of the chi-square distribution.

References

[1] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 9. Dover print.; [Nachdr. der Ausg. von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Devroye, Luc. Non-Uniform Random Variate Generation. New York, NY: Springer New York, 1986. https://doi.org/10.1007/978-1-4613-8643-8

[3] Evans, M., N. Hastings, and B. Peacock. Statistical Distributions. 2nd ed., Hoboken, NJ: John Wiley & Sons, Inc., 1993.

[4] Kreyszig, Erwin. Introductory Mathematical Statistics: Principles and Methods. New York: Wiley, 1970.