## Chi-Square Distribution

### Overview

The chi-square (*χ*^{2}) 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
*ν*_{1} and
*ν*_{2} is a chi-square random variable
with degrees of freedom *ν* = *ν*_{1}
+ *ν*_{2}.

### 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 2*ν*.

### 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*ν*_{1}(numerator degrees of freedom) and*ν*_{2}(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*χ*^{2}_{1 }and*χ*^{2}_{2}are both chi-square distributed with*ν*_{1}and*ν*_{2}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

*Z*_{1},*Z*_{2}, …,*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*σ*^{2}and sample variance*s*^{2}, 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*σ*^{2 }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*χ*^{2}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.

## See Also

`chi2cdf`

| `chi2pdf`

| `chi2inv`

| `chi2stat`

| `chi2gof`

| `chi2rnd`