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

$y = f (x | ν) = \frac{x^{(ν - 2) / 2} e^{- x / 2}}{2^{\frac{ν}{2}} Γ (ν / 2)},$

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

$p = F (x | ν) = \int_{0}^{x} \frac{t^{(ν - 2) / 2} e^{- t / 2}}{2^{ν / 2} Γ (ν / 2)} d t,$

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

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

where

$p = F (x | ν) = \int_{0}^{x} \frac{t^{(ν - 2) / 2} e^{- t / 2}}{2^{ν / 2} Γ (ν / 2)} d t,$

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

Open Live Script

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')

Figure contains an axes object. The axes object with xlabel Observation, ylabel Probability Density contains an object of type line.

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

Compute Chi-Square Distribution cdf

Open Live Script

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')

Figure contains an axes object. The axes object with xlabel Observation, ylabel Cumulative Probability contains an object of type line.

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 $F = \frac{\frac{χ_{1}^{2}}{ν_{1}}}{\frac{χ_{2}^{2}}{ν_{2}}}$ , where χ²₁and 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 $\sum_{i = 1}^{n} Z_{i}^{2}$ 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 $\frac{(n - 1) s^{2}}{σ^{2}}$ 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 $t = \frac{Z}{\sqrt{χ^{2} / ν}}$ 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.