Chi-squared distribution

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This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see chi-squared test. For the music group, see Chi2 (band).
Probability density function
Chi-square pdf.svg
Cumulative distribution function
Chi-square cdf.svg
Notation \chi^2(k)\! or \chi^2_k\!
Parameters k \in \mathbb{N}^\star~~ (known as "degrees of freedom")
Support x ∈ [0, +∞)
pdf \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}\; x^{\frac{k}{2}-1} e^{-\frac{x}{2}}\,
CDF \frac{1}{\Gamma\left(\frac{k}{2}\right)}\;\gamma\left(\frac{k}{2},\,\frac{x}{2}\right)
Mean k
Median \approx k\bigg(1-\frac{2}{9k}\bigg)^3
Mode max{ k − 2, 0 }
Variance 2k
Skewness \scriptstyle\sqrt{8/k}\,
Ex. kurtosis 12 / k
Entropy \begin{align}\frac{k}{2}&+\ln(2\Gamma(k/2)) \\ &\!+(1-k/2)\psi(k/2)\end{align}
MGF (1 − 2 t)k/2   for  t  < ½
CF (1 − 2 it)k/2      [1]

In probability theory and statistics, the chi-squared distribution (also chi-square or χ²-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. A special case of the gamma distribution, it is one of the most widely used probability distributions in inferential statistics, e.g., in hypothesis testing or in construction of confidence intervals.[2][3][4][5] When there is a need to contrast it with the noncentral chi-squared distribution, this distribution is sometimes called the central chi-squared distribution.

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, like Friedman's analysis of variance by ranks.

History and name[edit]

This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875/1876,[6][7] where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmertsche ("Helmertian") or "Helmert distribution".

The distribution was independently rediscovered by the English mathematician Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chi-squared test, published in (Pearson 1900), with computed table of values published in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII). The name "chi-squared" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing -½χ² for what would appear in modern notation as -½xTΣ−1x (Σ being the covariance matrix).[8] The idea of a family of "chi-squared distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.[6]

Definition[edit]

If Z1, ..., Zk are independent, standard normal random variables, then the sum of their squares,


    Q\ = \sum_{i=1}^k Z_i^2 ,

is distributed according to the chi-squared distribution with k degrees of freedom. This is usually denoted as


    Q\ \sim\ \chi^2(k)\ \ \text{or}\ \ Q\ \sim\ \chi^2_k .

The chi-squared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i.e. the number of Zi’s)

Characteristics[edit]

Further properties of the chi-squared distribution can be found in the box at the upper right corner of this article.

Probability density function[edit]

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


f(x;\,k) =
\begin{cases}
  \frac{x^{(k/2-1)} e^{-x/2}}{2^{k/2} \Gamma\left(\frac{k}{2}\right)},  & x \geq 0; \\ 0, & \text{otherwise}.
\end{cases}

where Γ(k/2) denotes the Gamma function, which has closed-form values for integer k.

For derivations of the pdf in the cases of one, two and k degrees of freedom, see Proofs related to chi-squared distribution.

Differential equation


\left\{2 x f'(x)+f(x) (-k+x+2)=0,f(1)=\frac{2^{-k/2}}{\sqrt{e}
   \Gamma \left(\frac{k}{2}\right)}\right\}

Cumulative distribution function[edit]

Chernoff bound for the CDF and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (k = 10)

Its cumulative distribution function is:


    F(x;\,k) = \frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})} = P\left(\frac{k}{2},\,\frac{x}{2}\right),

where γ(s,t) is the lower incomplete Gamma function and P(s,t) is the regularized Gamma function.

In a special case of k = 2 this function has a simple form:


    F(x;\,2) = 1 - e^{-\frac{x}{2}}

and the form is not much more complicated for other small even k.

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.

Letting z \equiv x/k, Chernoff bounds on the lower and upper tails of the CDF may be obtained.[9] For the cases when 0 < z < 1 (which include all of the cases when this CDF is less than half):


    F(z k;\,k) \leq (z e^{1-z})^{k/2}.

The tail bound for the cases when z > 1, similarly, is


    1-F(z k;\,k) \leq (z e^{1-z})^{k/2}.

For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chi-squared distribution.

Additivity[edit]

It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if {Xi}i=1n are independent chi-squared variables with {ki}i=1n degrees of freedom, respectively, then Y = X1 + ⋯ + Xn is chi-squared distributed with k1 + ⋯ + kn degrees of freedom.

Sample mean[edit]

The sample mean of n i.i.d. chi-squared variables of degree k is distributed according to a gamma distribution with shape α and scale θ parameters:  \bar X = \frac{1}{n} \sum_{i=1}^{n} X_i \sim \mathrm{Gamma}\left(\alpha=n\cdot k /2, \theta= 2/n \right)  \qquad \mathrm{where } \quad X_i \sim \chi^2(k)

Asymptotically, given that for a scale parameter α going to infinity, a Gamma distribution converges towards a Normal distribution with expectation μ = kθ and variance σ2 = kθ2, the sample mean converges towards:  \bar X  \xrightarrow{n \to \infty} N(k, 2\cdot k /n )

Note that we would have obtained the same result invoking instead the central limit theorem, noting that the expectation of the χ² is k, and its variance 2k (and hence the variance of the sample mean being 2k/n).

Entropy[edit]

The differential entropy is given by


    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\!\left[\frac{k}{2}\right],

where ψ(x) is the Digamma function.

The chi-squared distribution is the maximum entropy probability distribution for a random variate X for which E(X)=k and E(\ln(X))=\psi\left(k/2\right)+log(2) are fixed. Since the chi-squared is in the family of gamma distributions, this can be derived by substituting appropriate values in the Expectation of the Log moment of Gamma. For derivation from more basic principles, see the derivation in moment generating function of the sufficient statistic.

Noncentral moments[edit]

The moments about zero of a chi-squared distribution with k degrees of freedom are given by[10][11]


    \operatorname{E}(X^m) = k (k+2) (k+4) \cdots (k+2m-2) = 2^m \frac{\Gamma(m+\frac{k}{2})}{\Gamma(\frac{k}{2})}.

Cumulants[edit]

The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:


    \kappa_n = 2^{n-1}(n-1)!\,k

Asymptotic properties[edit]

By the central limit theorem, because the chi-squared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k. For many practical purposes, for k > 50 the distribution is sufficiently close to a normal distribution for the difference to be ignored.[12] Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of (X-k)/\sqrt{2k} tends to a standard normal distribution. However, convergence is slow as the skewness is \sqrt{8/k} and the excess kurtosis is 12/k.

  • The sampling distribution of ln(χ2) converges to normality much faster than the sampling distribution of χ2,[13] as the logarithm removes much of the asymmetry.[14] Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:
  • If X ~ χ²(k) then \scriptstyle\sqrt{2X} is approximately normally distributed with mean \scriptstyle\sqrt{2k-1} and unit variance (result credited to R. A. Fisher).
  • If X ~ χ²(k) then \scriptstyle\sqrt[3]{X/k} is approximately normally distributed with mean \scriptstyle 1-2/(9k) and variance \scriptstyle 2/(9k) .[15] This is known as the Wilson–Hilferty transformation.

Relation to other distributions[edit]

Approximate formula for median compared with numerical quantile (top) as presented in SAS Software. Difference between numerical quantile and approximate formula (bottom).
  • If X \sim \mathrm{F}(\nu_1, \nu_2) then Y = \lim_{\nu_2 \to \infty} \nu_1 X has the chi-squared distribution \chi^2_{\nu_{1}}
  • As a special case, if X \sim \mathrm{F}(1, \nu_2)\, then Y = \lim_{\nu_2 \to \infty} X\, has the chi-squared distribution \chi^2_{1}
  •  \|\boldsymbol{N}_{i=1,...,k}{(0,1)}\|^2 \sim \chi^2_k (The squared norm of k standard normally distributed variables is a chi-squared distribution with k degrees of freedom)
  • If X_i \sim \mathrm{Laplace}(\mu,\beta)\, then \sum_{i=1}^n{\frac{2 |X_i-\mu|}{\beta}} \sim \chi^2(2n)\,

A chi-squared variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.

If Y is a k-dimensional Gaussian random vector with mean vector μ and rank k covariance matrix C, then X = (Yμ)TC−1(Yμ) is chi-squared distributed with k degrees of freedom.

The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-squared distribution called the noncentral chi-squared distribution.

If Y is a vector of k i.i.d. standard normal random variables and A is a k×k idempotent matrix with rank k−n then the quadratic form YTAY is chi-squared distributed with k−n degrees of freedom.

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

  • Y is F-distributed, Y ~ F(k1,k2) if \scriptstyle Y = \frac{X_1 / k_1}{X_2 / k_2} where X1 ~ χ²(k1) and X2  ~ χ²(k2) are statistically independent.
  • If X1  ~  χ2k1 and X2  ~  χ2k2 are statistically independent, then X1 + X2  ~ χ2k1+k2. If X1 and X2 are not independent, then X1 + X2 is not chi-squared distributed.

Generalizations[edit]

The chi-squared distribution is obtained as the sum of the squares of k independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

Linear combination[edit]

If X_1,...,X_n are iid chi square random variables and a_1,...,a_n\in\mathbb{R}_{>0}, then a closed expression for the distribution of X=\sum_{i=1}^n a_iX_i is not known. However, there exists a computationally efficient algorithm to calculate the pdf f_X to arbitrary precision.[16]

Chi-squared distributions[edit]

Noncentral chi-squared distribution[edit]

The noncentral chi-squared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

Generalized chi-squared distribution[edit]

The generalized chi-squared distribution is obtained from the quadratic form z′Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.

Gamma, exponential, and related distributions[edit]

The chi-squared distribution X ~ χ²(k) is a special case of the gamma distribution, in that X ~ Γ(k/2, 1/2) using the rate parameterization of the gamma distribution (or X ~ Γ(k/2, 2) using the scale parameterization of the gamma distribution) where k is an integer.

Because the exponential distribution is also a special case of the Gamma distribution, we also have that if X ~ χ²(2), then X ~ Exp(1/2) is an exponential distribution.

The Erlang distribution is also a special case of the Gamma distribution and thus we also have that if X ~ χ²(k) with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.

Applications[edit]

The chi-squared distribution has numerous applications in inferential statistics, for instance in chi-squared 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, each divided by their respective degrees of freedom.

Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.

  • The box below shows some statistics based on Xi ∼ Normal(μi, σ2i), i = 1, ⋯, k, independent random variables that have probability distributions related to the chi-squared distribution:
Name Statistic
chi-squared distribution \sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2
noncentral chi-squared distribution \sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2
chi distribution \sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}
noncentral chi distribution \sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}

The chi-squared distribution is also often encountered in Magnetic Resonance Imaging .[17]

Table of χ2 value vs p-value[edit]

The p-value is the probability of observing a test statistic at least as extreme in a chi-squared distribution. Accordingly, since the cumulative distribution function (CDF) for the appropriate degrees of freedom (df) gives the probability of having obtained a value less extreme than this point, subtracting the CDF value from 1 gives the p-value. The table below gives a number of p-values matching to χ2 for the first 10 degrees of freedom.

A low p-value indicates greater statistical significance, i.e. greater confidence that the observed deviation from the null hypothesis is significant. A p-value of 0.05 is often used as a bright-line cutoff between significant and not-significant results.

Degrees of freedom (df) χ2 value[18]
1
0.004 0.02 0.06 0.15 0.46 1.07 1.64 2.71 3.84 6.64 10.83
2
0.10 0.21 0.45 0.71 1.39 2.41 3.22 4.60 5.99 9.21 13.82
3
0.35 0.58 1.01 1.42 2.37 3.66 4.64 6.25 7.82 11.34 16.27
4
0.71 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 13.28 18.47
5
1.14 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 15.09 20.52
6
1.63 2.20 3.07 3.83 5.35 7.23 8.56 10.64 12.59 16.81 22.46
7
2.17 2.83 3.82 4.67 6.35 8.38 9.80 12.02 14.07 18.48 24.32
8
2.73 3.49 4.59 5.53 7.34 9.52 11.03 13.36 15.51 20.09 26.12
9
3.32 4.17 5.38 6.39 8.34 10.66 12.24 14.68 16.92 21.67 27.88
10
3.94 4.87 6.18 7.27 9.34 11.78 13.44 15.99 18.31 23.21 29.59
P value (Probability)
0.95 0.90 0.80 0.70 0.50 0.30 0.20 0.10 0.05 0.01 0.001

See also[edit]

References[edit]

  1. ^ M.A. Sanders. "Characteristic function of the central chi-squared distribution". Retrieved 2009-03-06. 
  2. ^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 940, ISBN 978-0486612720, MR 0167642 .
  3. ^ NIST (2006). Engineering Statistics Handbook - Chi-Squared Distribution
  4. ^ Jonhson, N.L.; S. Kotz, , N. Balakrishnan (1994). Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18). John Willey and Sons. ISBN 0-471-58495-9. 
  5. ^ Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 241-246). McGraw-Hill. ISBN 0-07-042864-6. 
  6. ^ a b Hald 1998, pp. 633–692, 27. Sampling Distributions under Normality.
  7. ^ F. R. Helmert, "Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen", Zeitschrift für Mathematik und Physik 21, 1876, S. 102–219
  8. ^ R. L. Plackett, Karl Pearson and the Chi-Squared Test, International Statistical Review, 1983, 61f. See also Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics.
  9. ^ Dasgupta, Sanjoy D. A.; Gupta, Anupam K. (2002). "An Elementary Proof of a Theorem of Johnson and Lindenstrauss". Random Structures and Algorithms 22: 60–65. doi:10.1002/rsa.10073. Retrieved 2012-05-01. 
  10. ^ Chi-squared distribution, from MathWorld, retrieved Feb. 11, 2009
  11. ^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1
  12. ^ Box, Hunter and Hunter (1978). Statistics for experimenters. Wiley. p. 118. ISBN 0471093157. 
  13. ^ Bartlett, M. S.; Kendall, D. G. (1946). "The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation". Supplement to the Journal of the Royal Statistical Society 8 (1): 128–138. JSTOR 2983618. 
  14. ^ Shoemaker, Lewis H. (2003). "Fixing the F Test for Equal Variances". The American Statistician 57 (2): 105–114. doi:10.1198/0003130031441. JSTOR 30037243. 
  15. ^ Wilson, E. B.; Hilferty, M. M. (1931). "The distribution of chi-squared". Proc. Natl. Acad. Sci. USA 17 (12): 684–688. 
  16. ^ Bausch, J. (2013). "On the efficient calculation of a linear combination of chi-square random variables with an application in counting string vacua". Journal of Physics A: Mathematical and Theoretical 46 (50): 505202. doi:10.1088/1751-8113/46/50/505202. 
  17. ^ den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", Physica Medica, [1]
  18. ^ Chi-Squared Test Table B.2. Dr. Jacqueline S. McLaughlin at The Pennsylvania State University. In turn citing: R.A. Fisher and F. Yates, Statistical Tables for Biological Agricultural and Medical Research, 6th ed., Table IV

Further reading[edit]

External links[edit]