 Chisquared distribution

This article is about the mathematics of the chisquared distribution. For its uses in statistics, see chisquared test. For the music group, see Chi2 (band).
Probability density function
Cumulative distribution function
notation: or parameters: k ∈ N_{1} — degrees of freedom support: x ∈ [0, +∞) pdf: cdf: mean: k median: mode: max{ k − 2, 0 } variance: 2k skewness: ex.kurtosis: 12 / k entropy: mgf: (1 − 2 t)^{−k/2} for t < ½ cf: (1 − 2 i t)^{−k/2} ^{[1]} In probability theory and statistics, the chisquared distribution (also chisquare or χ²distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. 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 chisquared distribution, this distribution is sometimes called the central chisquared distribution.
The chisquared distribution is used in the common chisquared 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.
The chisquared distribution is a special case of the gamma distribution.
Contents
Definition
If Z_{1}, ..., Z_{k} are independent, standard normal random variables, then the sum of their squares,
is distributed according to the chisquared distribution with k degrees of freedom. This is usually denoted as
The chisquared distribution has one parameter: k — a positive integer that specifies the number of degrees of freedom (i.e. the number of Z_{i}’s)
Characteristics
Further properties of the chisquared distribution can be found in the box at the upper right corner of this article.
Probability density function
The probability density function (pdf) of the chisquared distribution is
where Γ(k/2) denotes the Gamma function, which has closedform values at the halfintegers.
For derivations of the pdf in the cases of one and two degrees of freedom, see Proofs related to chisquared distribution.
Cumulative distribution function
Its cumulative distribution function is:
where γ(k,z) is the lower incomplete Gamma function and P(k,z) is the regularized Gamma function.
In a special case of k = 2 this function has a simple form:
Tables of this distribution — usually in its cumulative form — are widely available and the function is included in many spreadsheets and all statistical packages. For a closed form approximation for the CDF, see under Noncentral chisquared distribution.
Additivity
It follows from the definition of the chisquared distribution that the sum of independent chisquared variables is also chisquared distributed. Specifically, if {X_{i}}_{i=1}^{n} are independent chisquared variables with {k_{i}}_{i=1}^{n} degrees of freedom, respectively, then Y = X_{1} + ⋯ + X_{n} is chisquared distributed with k_{1} + ⋯ + k_{n} degrees of freedom.
Information entropy
The information entropy is given by
where ψ(x) is the Digamma function.
The Chisquared distribution is the maximum entropy probability distribution for a random variate X for which E(X) = ν is fixed, and is fixed. ^{[6]}
Noncentral moments
The moments about zero of a chisquared distribution with k degrees of freedom are given by^{[7]}^{[8]}
Cumulants
The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:
Asymptotic properties
By the central limit theorem, because the chisquared 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.^{[9]} Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of tends to a standard normal distribution. However, convergence is slow as the skewness is and the excess kurtosis is 12/k. Other functions of the chisquared distribution converge more rapidly to a normal distribution. Some examples are:
 If X ~ χ²(k) then is approximately normally distributed with mean and unit variance (result credited to R. A. Fisher).
 If X ~ χ²(k) then is approximately normally distributed with mean and variance ^{[10]} This is known as the WilsonHilferty transformation.
Related distributions
 (normal distribution)
 (Noncentral chisquared distribution with noncentrality parameter λ = 0)
 If X∼F(ν_{1},ν_{2}) then has the chisquared distribution
 As a special case, if then has the chisquared distribution
 (The squared norm of n standard normally distributed variables is a chisquared distribution with k degrees of freedom)
 If and , then . (gamma distribution)
 If then (chi distribution)
 If (Rayleigh distribution) then
 If (Maxwell distribution) then
 If X∼χ^{2}(ν) then (Inversechisquared distribution)
 The chisquared distribution is a special case of type 3 Pearson distribution
 If and then (beta distribution)
 If (Uniform distribution (continuous)) then
 is a transformation of Laplace distribution
 If then
 chisquared distribution is a transformation of Pareto distribution
 Student's tdistribution is a transformation of chisquared distribution
 Student's tdistribution can be obtained from chisquared distribution and normal distribution
 Noncentral beta distribution can be obtained as a transformation of chisquared distribution and Noncentral chisquared distribution
 Noncentral tdistribution can be obtained from normal distribution and chisquared distribution
A chisquared 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 kdimensional Gaussian random vector with mean vector μ and rank k covariance matrix C, then X = (Y−μ)^{T}C^{−1}(Y−μ) is chisquared distributed with k degrees of freedom.
The sum of squares of statistically independent unitvariance Gaussian variables which do not have mean zero yields a generalization of the chisquared distribution called the noncentral chisquared 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 Y^{T}AY is chisquared distributed with k−n degrees of freedom.
The chisquared distribution is also naturally related to other distributions arising from the Gaussian. In particular,
 Y is Fdistributed, Y ~ F(k_{1},k_{2}) if where X_{1} ~ χ²(k_{1}) and X_{2} ~ χ²(k_{2}) are statistically independent.
 If X is chisquared distributed, then is chi distributed.
 If X_{1} ~ χ^{2}_{k1} and X_{2} ~ χ^{2}_{k2} are statistically independent, then X_{1} + X_{2} ~ χ^{2}_{k1+k2}. If X_{1} and X_{2} are not independent, then X_{1} + X_{2} is not chisquared distributed.
Generalizations
The chisquared distribution is obtained as the sum of the squares of k independent, zeromean, unitvariance 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.
Chisquared distributions
Noncentral chisquared distribution
Main article: Noncentral chisquared distributionThe noncentral chisquared distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.
Generalized chisquared distribution
Main article: Generalized chisquared distributionThe generalized chisquared distribution is obtained from the quadratic form z′Az where z is a zeromean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.
The chisquared distribution X ~ χ²(k) is a special case of the gamma distribution, in that X ~ Γ(k/2, 2) (using the shape parameterization of the gamma distribution) where k/2 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
The chisquared distribution has numerous applications in inferential statistics, for instance in chisquared 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 tdistribution. It enters all analysis of variance problems via its role in the Fdistribution, which is the distribution of the ratio of two independent chisquared random variables, each divided by their respective degrees of freedom.
Following are some of the most common situations in which the chisquared distribution arises from a Gaussiandistributed sample.
 if X_{1}, ..., X_{n} are i.i.d. N(μ, σ^{2}) random variables, then where .
 The box below shows probability distributions with name starting with chi for some statistics based on X_{i} ∼ Normal(μ_{i}, σ^{2}_{i}), i = 1, ⋯, k, independent random variables:
Name Statistic chisquared distribution noncentral chisquared distribution chi distribution noncentral chi distribution Table of χ^{2} value vs pvalue
The pvalue is the probability of observing a test statistic at least as extreme in a chisquared 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 pvalue. The table below gives a number of pvalues matching to χ^{2} for the first 10 degrees of freedom.
A pvalue of 0.05 or less is usually regarded as statistically significant, i.e. the observed deviation from the null hypothesis is significant.
Degrees of freedom (df) χ^{2} value ^{[11]} 10.004 0.02 0.06 0.15 0.46 1.07 1.64 2.71 3.84 6.64 10.83 20.10 0.21 0.45 0.71 1.39 2.41 3.22 4.60 5.99 9.21 13.82 30.35 0.58 1.01 1.42 2.37 3.66 4.64 6.25 7.82 11.34 16.27 40.71 1.06 1.65 2.20 3.36 4.88 5.99 7.78 9.49 13.28 18.47 51.14 1.61 2.34 3.00 4.35 6.06 7.29 9.24 11.07 15.09 20.52 61.63 2.20 3.07 3.83 5.35 7.23 8.56 10.64 12.59 16.81 22.46 72.17 2.83 3.82 4.67 6.35 8.38 9.80 12.02 14.07 18.48 24.32 82.73 3.49 4.59 5.53 7.34 9.52 11.03 13.36 15.51 20.09 26.12 93.32 4.17 5.38 6.39 8.34 10.66 12.24 14.68 16.92 21.67 27.88 103.94 4.86 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 Nonsignificant Significant See also
 Cochran's theorem
 Fisher's method for combining independent tests of significance
 Generalized chisquared distribution
 Pearson's chisquared test
 Wishart distribution
References
 ^ M.A. Sanders. "Characteristic function of the central chisquared distribution". http://www.planetmathematics.com/CentralChiDistr.pdf. Retrieved 20090306.
 ^ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 940, ISBN 9780486612720, MR0167642, http://www.math.sfu.ca/~cbm/aands/page_940.htm.
 ^ NIST (2006). Engineering Statistics Handbook  ChiSquared Distribution
 ^ Jonhson, N.L.; S. Kotz, , N. Balakrishnan (1994). Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18). John Willey and Sons. ISBN 0471584959.
 ^ Mood, Alexander; Franklin A. Graybill, Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 241246). McGrawHill. ISBN 0070428646.
 ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics (Elsevier): 219–230. http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpapermasterdownload%5C2009519932327055475115776.pdf. Retrieved 20110602.
 ^ Chisquared distribution, from MathWorld, retrieved Feb. 11, 2009
 ^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 9780387346571
 ^ Box, Hunter and Hunter. Statistics for experimenters. Wiley. p. 46.
 ^ Wilson, E.B.; Hilferty, M.M. (1931) "The distribution of chisquared". Proceedings of the National Academy of Sciences, Washington, 17, 684–688.
 ^ ChiSquared 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
External links
 Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history
 Course notes on ChiSquared Goodness of Fit Testing from Yale University Stats 101 class.
 Mathematica demonstration showing the chisquared sampling distribution of various statistics, e.g. Σx², for a normal population
 Simple algorithm for approximating cdf and inverse cdf for the chisquared distribution with a pocket calculator
Some common univariate probability distributions Continuous beta • Cauchy • chisquared • exponential • F • gamma • Laplace • lognormal • normal • Pareto • Student's t • uniform • WeibullDiscrete Categories: Continuous distributions
 Normal distribution
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