Chi-square distribution

chi-square

Probability density function
Cumulative distribution function
Parameters	Failed to parse (Missing texvc executable; please see math/README to configure.): k > 0\, degrees of freedom
Support	Failed to parse (Missing texvc executable; please see math/README to configure.): x \in [0; +\infty)\,
Probability density function (pdf)	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,
Cumulative distribution function (cdf)	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,
Mean	Failed to parse (Missing texvc executable; please see math/README to configure.): k\,
Median	approximately Failed to parse (Missing texvc executable; please see math/README to configure.): k-2/3\,
Mode	Failed to parse (Missing texvc executable; please see math/README to configure.): k-2\, if Failed to parse (Missing texvc executable; please see math/README to configure.): k\geq 2\,
Variance	Failed to parse (Missing texvc executable; please see math/README to configure.): 2\,k\,
Skewness	Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{8/k}\,
Excess kurtosis	Failed to parse (Missing texvc executable; please see math/README to configure.): 12/k\,
Entropy	Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{k}{2}\!+\!\ln(2\Gamma(k/2))\!+\!(1\!-\!k/2)\psi(k/2)
Moment-generating function (mgf)	Failed to parse (Missing texvc executable; please see math/README to configure.): (1-2\,t)^{-k/2} for Failed to parse (Missing texvc executable; please see math/README to configure.): 2\,t<1\,
Characteristic function	Failed to parse (Missing texvc executable; please see math/README to configure.): (1-2\,i\,t)^{-k/2}\,

In probability theory and statistics, the chi-square distribution (also chi-squared or Failed to parse (Missing texvc executable; please see math/README to configure.): \chi^2   distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests. It is useful because, under reasonable assumptions, easily calculated quantities can be proven to have distributions that approximate to the chi-square distribution if the null hypothesis is true.

If Failed to parse (Missing texvc executable; please see math/README to configure.): X_i

are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable

Failed to parse (Missing texvc executable; please see math/README to configure.): Q = \sum_{i=1}^k X_i^2

is distributed according to the chi-square distribution. This is usually written

Failed to parse (Missing texvc executable; please see math/README to configure.): Q\sim\chi^2_k.\,

The chi-square distribution has one parameter: Failed to parse (Missing texvc executable; please see math/README to configure.): k

- a positive integer that specifies the number of degrees of freedom (i.e. the number of Failed to parse (Missing texvc executable; please see math/README to configure.): X_i

)

The chi-square distribution is a special case of the gamma distribution.

The best-known situations in which the chi-square distribution are used are the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, and of the independence of two criteria of classification of qualitative data. However, many other statistical tests lead to a use of this distribution. One example is Friedman's analysis of variance by ranks.

Characteristics

Probability density function

A probability density function of the chi-square distribution is

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x;k)= \begin{cases}\displaystyle \frac{1}{2^{k/2}\Gamma(k/2)}\,x^{(k/2) - 1} e^{-x/2}&\text{for }x>0,\\ 0&\text{for }x\le0, \end{cases}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma

denotes the Gamma function, which takes particular values at the half-integers.

Cumulative distribution function

Its cumulative distribution function is:

Failed to parse (Missing texvc executable; please see math/README to configure.): F(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)} = P(k/2, x/2)

where Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma(k,z)

is the lower incomplete Gamma function and Failed to parse (Missing texvc executable; please see math/README to configure.): P(k, z)
is the regularized Gamma function.

Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages.

Characteristic function

The characteristic function of the Chi-square distribution is

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi(t;k)=(1-2it)^{-k/2}.\,

Properties

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables divided by their respective degrees of freedom.

Normal approximation

If Failed to parse (Missing texvc executable; please see math/README to configure.): X\sim\chi^2_k , then as Failed to parse (Missing texvc executable; please see math/README to configure.): k

tends to infinity, the distribution of Failed to parse (Missing texvc executable; please see math/README to configure.): X
tends to normality.

However, the tendency is slow (the skewness is Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{8/k}

and the kurtosis excess is Failed to parse (Missing texvc executable; please see math/README to configure.): 12/k

) and two transformations are commonly considered, each of which approaches normality faster than Failed to parse (Missing texvc executable; please see math/README to configure.): X

itself:

Fisher empirically showed that Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2X}

is approximately normally distributed with mean Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2k-1}
and unit variance. It is possible to arrive at the same normal approximation result by using moment matching. To see this, consider the mean and the variance of a Chi-distributed random variable Failed to parse (Missing texvc executable; please see math/README to configure.): z=\sqrt{X}

, which are given by Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_z= \sqrt{2} \frac{\Gamma\left(k/2+1/2\right)}{\Gamma\left(k/2 \right)}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_z^2= k-\mu_z^2 , where Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma(\cdot)

is the Gamma

function. The particular ratio of the Gamma functions in Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_z

has the following series expansion [1]:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\Gamma\left(N+1/2\right)}{\Gamma\left(N \right)}=\sqrt{N}\left(1-\frac{1}{8N}+ \frac{1}{128N^2}+\frac{5}{1024N^3}-\frac{21}{32768N^4}+\ldots\right).

When Failed to parse (Missing texvc executable; please see math/README to configure.): N\gg 1 , this ratio can be approximated as follows: Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\Gamma\left(N+1/2\right)}{\Gamma\left(N \right)}\approx\sqrt{N}\left(1-\frac{1}{8N}\right)\approx\sqrt{N}\left(1-\frac{1}{4N}\right)^{0.5}=\sqrt{N-1/4}.

Then, simple moment matching results in the following approximation of Failed to parse (Missing texvc executable; please see math/README to configure.): z

Failed to parse (Missing texvc executable; please see math/README to configure.): z\sim{\mathcal N}\left(\sqrt{k-1/2}, \frac{1}{2}\right) , from which it follows that Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2X}\sim{\mathcal N}\left(\sqrt{2k-1}, 1\right) .

Wilson and Hilferty showed in 1931 that Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt[3]{X/k}

is approximately normally distributed with mean Failed to parse (Missing texvc executable; please see math/README to configure.): 1-2/(9k)
and variance Failed to parse (Missing texvc executable; please see math/README to configure.): 2/(9k)

.

The expected value of a random variable having chi-square distribution with Failed to parse (Missing texvc executable; please see math/README to configure.): k

degrees of freedom is Failed to parse (Missing texvc executable; please see math/README to configure.): k
and the variance is Failed to parse (Missing texvc executable; please see math/README to configure.): 2k

. The median is given approximately by

Failed to parse (Missing texvc executable; please see math/README to configure.): k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}.

Note that 2 degrees of freedom lead to an exponential distribution.

Information entropy

The information entropy is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): H = \int_{-\infty}^\infty f(x;k)\ln(f(x;k)) dx = \frac{k}{2} + \ln \left( 2 \Gamma \left( \frac{k}{2} \right) \right) + \left(1 - \frac{k}{2}\right) \psi(k/2).

where Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x)

is the Digamma function.

Related distributions

• Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim \mathrm{Exponential}(\lambda = \frac{1}{2})

is an exponential distribution if Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim \chi_2^2
(with 2 degrees of freedom).

• Failed to parse (Missing texvc executable; please see math/README to configure.): Y \sim \chi_k^2

is a chi-square distribution if Failed to parse (Missing texvc executable; please see math/README to configure.): Y = \sum_{m=1}^k X_m^2
for Failed to parse (Missing texvc executable; please see math/README to configure.): X_i \sim N(0,1)
independent that are normally distributed.  

• If the Failed to parse (Missing texvc executable; please see math/README to configure.): X_i\sim N(\mu_i,1)

have nonzero means, then Failed to parse (Missing texvc executable; please see math/README to configure.): Y = \sum_{m=1}^k X_m^2
is drawn from a noncentral chi-square distribution.

• The chi-square distribution Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim \chi^2_\nu

is a special case of the gamma distribution, in that Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim \textrm{Gamma}(\tfrac{\nu}{2}, 2)

.

• Failed to parse (Missing texvc executable; please see math/README to configure.): Y \sim \mathrm{F}(\nu_1, \nu_2)

is an F-distribution if Failed to parse (Missing texvc executable; please see math/README to configure.): Y = \frac{X_1 / \nu_1}{X_2 / \nu_2}
where Failed to parse (Missing texvc executable; please see math/README to configure.): X_1 \sim \chi_{\nu_1}^2
and Failed to parse (Missing texvc executable; please see math/README to configure.): X_2 \sim \chi_{\nu_2}^2
are independent with their respective degrees of freedom.

• Failed to parse (Missing texvc executable; please see math/README to configure.): Y \sim \chi^2(\bar{\nu})

is a chi-square distribution if Failed to parse (Missing texvc executable; please see math/README to configure.): Y = \sum_{m=1}^N X_m
where Failed to parse (Missing texvc executable; please see math/README to configure.): X_m \sim \chi^2(\nu_m)
are independent and Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{\nu} = \sum_{m=1}^N \nu_m

.

• if Failed to parse (Missing texvc executable; please see math/README to configure.): X

is chi-square distributed, then Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{X}
is chi distributed.

• in particular, if Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim \chi_2^2

(chi-square with 2 degrees of freedom), then Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{X}
is Rayleigh distributed.

• if Failed to parse (Missing texvc executable; please see math/README to configure.): X_1, \dots, X_n

are i.i.d. Failed to parse (Missing texvc executable; please see math/README to configure.): N(\mu,\sigma^2)
random variables, then Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}
where Failed to parse (Missing texvc executable; please see math/README to configure.): \bar X = \frac{1}{n} \sum_{i=1}^n X_i

.

• if Failed to parse (Missing texvc executable; please see math/README to configure.): X \sim \mathrm{SkewLogistic}(\frac{1}{2})\,

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{log}(1 + e^{-X}) \sim \chi_2^2\,

See also

• Cochran's theorem
• Inverse-chi-square distribution
• Degrees of freedom (statistics)
• Fisher's method for combining independent tests of significance
• Noncentral chi-square distribution

External links

• Course notes on Chi-Square Goodness of Fit Testing from Yale University Stats 101 class. Example includes hypothesis testing and parameter estimation.
• On-line calculator for the significance of chi-square, in Richard Lowry's statistical website at Vassar College.
• Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
• Chi-Square Calculator for critical values of Chi-Square in R. Webster West's applet website at University of South Carolina
• Chi-Square Calculator from GraphPad
• Table of Chi-squared distribution

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