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{{About|the mathematics of the chi-squared distribution|its uses in statistics|chi-squared test|the music group|Chi2 (band)}}

{{Probability distribution
| type = density
| pdf_image = [[File:Chi-square pdf.svg|321px]]
| cdf_image = [[File:Chi-square cdf.svg|321px]]
| notation = <math>\chi^2(k)\!</math> or <math>\chi^2_k\!</math>
| parameters = <math>k \in \mathbb{N}~~</math> (known as "degrees of freedom")
| support = ''x'' ∈ [0, +∞)
| pdf = <math>\frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}\; x^{\frac{k}{2}-1} e^{-\frac{x}{2}}\,</math>
| cdf = <math>\frac{1}{\Gamma\left(\frac{k}{2}\right)}\;\gamma\left(\frac{k}{2},\,\frac{x}{2}\right)</math>
| mean = ''k''
| median = <math>\approx k\bigg(1-\frac{2}{9k}\bigg)^3</math>
| mode = max{ ''k'' − 2, 0 }
| variance = 2''k''
| skewness = <math>\scriptstyle\sqrt{8/k}\,</math>
| kurtosis = 12 / ''k''
| entropy = <math>\begin{align}\frac{k}{2}&+\ln(2\Gamma(k/2)) \\ &\!+(1-k/2)\psi(k/2)\end{align}</math>
| mgf = {{nowrap|(1 − 2 ''t'')−''k''/2}}   for  t  < ½
| char = {{nowrap|(1 − 2 ''i'' ''t'')−''k''/2}}      <ref>{{cite web | url=http://www.planetmathematics.com/CentralChiDistr.pdf | title=Characteristic function of the central chi-squared distribution | author=M.A. Sanders | accessdate=2009-03-06}}</ref>
}}

In [[probability theory]] and [[statistics]], the '''chi-squared distribution''' (also '''chi-square''' or {{nowrap|1='''''χ''²-distribution'''}}) with ''k'' [[Degrees of freedom (statistics)|degrees of freedom]] is the distribution of a sum of the squares of ''k'' [[Independence (probability theory)|independent]] [[standard normal]] random variables. It is one of the most widely used [[probability distribution]]s in [[inferential statistics]], e.g., in [[hypothesis testing]] or in construction of [[confidence interval]]s.<ref name=abramowitz>{{Abramowitz_Stegun_ref|26|940}}</ref><ref>NIST (2006). [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Engineering Statistics Handbook - Chi-Squared Distribution]</ref><ref>{{cite book
| last = Jonhson
| first = N.L.
| coauthors = S. Kotz, , N. Balakrishnan
| title = Continuous Univariate Distributions (Second Ed., Vol. 1, Chapter 18)
| publisher = John Willey and Sons
| year = 1994
| isbn = 0-471-58495-9
}}</ref><ref>{{cite book
| last = Mood
| first = Alexander
| coauthors = Franklin A. Graybill, Duane C. Boes
| title = Introduction to the Theory of Statistics (Third Edition, p. 241-246)
| publisher = McGraw-Hill
| year = 1974
| isbn = 0-07-042864-6
}}</ref> 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 test]]s for [[goodness of fit]] of an observed distribution to a theoretical one, the [[statistical independence|independence]] of two criteria of classification of [[data analysis|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 test|Friedman's analysis of variance by ranks]].

The chi-squared distribution is a special case of the [[gamma distribution]].

==History and name==
This distribution was first described by the German statistician [[Helmert|Friedrich Robert Helmert]] in papers of 1875/1876,{{sfn|Hald|1998|pp=633–692|loc=27. Sampling Distributions under Normality}}<ref>F. R. [[Helmert]], "[http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN599415665_0021&DMDID=DMDLOG_0018 Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen]", ''Zeitschrift für Mathematik und Physik'' [http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN599415665_0021 21], 1876, S. 102–219</ref> 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 {{Harv|Pearson|1900}}, with computed table of values published in {{Harv|Elderton|1902}}, collected in {{Harv|Pearson|1914|pp=xxxi–xxxiii, 26–28|loc=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 (letter)|Chi]], writing
-½χ² for what would appear in modern notation as -½'''x'''TΣ-1'''x''' (Σ being the [[covariance matrix]]).<ref>
R. L. Plackett, ''Karl Pearson and the Chi-Squared Test'', International Statistical Review, 1983, [http://www.jstor.org/stable/1402731?seq=3 61f.]
See also Jeff Miller, [http://jeff560.tripod.com/c.html Earliest Known Uses of Some of the Words of Mathematics].
</ref> 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.{{sfn|Hald|1998|pp=633–692|loc=27. Sampling Distributions under Normality}}

==Definition==
If ''Z''1, ..., ''Z''''k'' are [[independence (probability theory)|independent]], [[standard normal]] random variables, then the sum of their squares,
: <math>
Q\ = \sum_{i=1}^k Z_i^2 ,
</math>
is distributed according to the '''chi-squared distribution''' with ''k'' degrees of freedom. This is usually denoted as
: <math>
Q\ \sim\ \chi^2(k)\ \ \text{or}\ \ Q\ \sim\ \chi^2_k .
</math>

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

==Characteristics==
Further properties of the chi-squared 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 chi-squared distribution is
:<math>
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}
</math>
where Γ(''k''/2) denotes the [[Gamma function]], which has [[particular values of the Gamma function|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]].

===Cumulative distribution function===

[[File:Chernoff XS CDF.png|thumb|right|400px|Chernoff bound for the [[Cumulative distribution function|CDF]] and tail (1-CDF) of a chi-squared random variable with ten degrees of freedom (''k'' = 10) ]]

Its [[cumulative distribution function]] is:
: <math>
F(x;\,k) = \frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})} = P\left(\frac{k}{2},\,\frac{x}{2}\right),
</math>
where γ(''s'',''t'') is the [[incomplete Gamma function|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:
: <math>
F(x;\,2) = 1 - e^{-\frac{x}{2}}.
</math>

Tables of the chi-squared cumulative distribution function are widely available and the function is included in many [[spreadsheet]]s and all [[List of statistical packages|statistical packages]].

Letting <math>z \equiv x/k</math>, [[Chernoff_bound#The_first_step_in_the_proof_of_Chernoff_bounds|Chernoff bounds]] on the lower and upper tails of the CDF may be obtained.<ref>{{cite journal |last1=Dasgupta |first1=Sanjoy D. A. |last2=Gupta |first2=Anupam K. |year=2002 |title=An Elementary Proof of a Theorem of Johnson and Lindenstrauss |journal=Random Structures and Algorithms |volume=22 |issue= |pages=60–65 |publisher= |doi= |url=http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf |accessdate=2012-05-01 }}</ref> For the cases when <math>0 < z < 1</math> (which include all of the cases when this CDF is less than half):
: <math>
F(z k;\,k) \leq (z e^{1-z})^{k/2}.
</math>

The tail bound for the cases when <math>z > 1</math>, similarly, is
: <math>
1-F(z k;\,k) \leq (z e^{1-z})^{k/2}.
</math>

For another [[approximation]] for the CDF modeled after the cube of a Gaussian, see [[Noncentral_chi-squared_distribution#Approximation|under Noncentral chi-squared distribution]].

===Additivity===
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''=1''n'' are independent chi-squared variables with {''ki''}''i''=1''n'' degrees of freedom, respectively, then {{nowrap|''Y {{=}} X''1 + ⋯ + ''Xn''}} is chi-squared distributed with {{nowrap|''k''1 + ⋯ + ''kn''}} degrees of freedom.

===Sample mean===
The sample mean of n [[Independent and identically distributed random variables|i.i.d.]] chi-squared variables of degree k is distributed according to a gamma distribution with shape α and scale θ parameters:
<math> \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)</math>

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:
<math> \bar X \xrightarrow{n \to \infty} N(k, 2\cdot k /n ) </math>

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===
The [[differential entropy]] is given by
: <math>
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],
</math>
where ''ψ''(''x'') is the [[Digamma function]].

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

===Noncentral moments===
The moments about zero of a chi-squared distribution with ''k'' degrees of freedom are given by<ref>[http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-squared distribution], from [[MathWorld]], retrieved Feb. 11, 2009</ref><ref>M. K. Simon, ''Probability Distributions Involving Gaussian Random Variables'', New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1</ref>
: <math>
\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})}.
</math>

===Cumulants===
The [[cumulant]]s are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:
: <math>
\kappa_n = 2^{n-1}(n-1)!\,k
</math>

===Asymptotic properties===
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.<ref>{{cite book|title=Statistics for experimenters|author=Box, Hunter and Hunter|publisher=Wiley|year=1978|isbn=0471093157|page=118}}</ref> Specifically, if ''X'' ~ ''χ''²(''k''), then as ''k'' tends to infinity, the distribution of <math>(X-k)/\sqrt{2k}</math> [[convergence of random variables#Convergence in distribution|tends]] to a standard normal distribution. However, convergence is slow as the [[skewness]] is <math>\sqrt{8/k}</math> and the [[excess kurtosis]] is 12/''k''.
* The sampling distribution of ln(''χ''2) converges to normality much faster than the sampling distribution of ''χ''2,<ref>{{cite journal |first=M. S. |last=Bartlett |first2=D. G. |last2=Kendall |title=The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation |journal=Supplement to the Journal of the Royal Statistical Society |volume=8 |issue=1 |year=1946 |pages=128–138 |jstor=2983618 }}</ref> as the logarithm removes much of the asymmetry.<ref>{{Cite journal |title=Fixing the F Test for Equal Variances |first=Lewis H. |last=Shoemaker |journal=[[The American Statistician]] |volume=57 |issue=2 |year=2003 |pages=105–114 |jstor=30037243 }}</ref> Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:
* If ''X'' ~ ''χ''²(''k'') then <math>\scriptstyle\sqrt{2X}</math> is approximately normally distributed with mean <math>\scriptstyle\sqrt{2k-1}</math> and unit variance (result credited to [[R. A. Fisher]]).
* If ''X'' ~ ''χ''²(''k'') then <math>\scriptstyle\sqrt[3]{X/k}</math> is approximately normally distributed with mean <math>\scriptstyle 1-2/(9k)</math> and variance <math>\scriptstyle 2/(9k) .</math><ref>{{cite journal |last=Wilson |first=E. B. |last2=Hilferty |first2=M. M. |year=1931 |title=The distribution of chi-squared |journal=[[Proceedings of the National Academy of Sciences of the United States of America|Proc. Natl. Acad. Sci. USA]] |volume=17 |issue=12 |pages=684–688 |url=http://www.pnas.org/content/17/12/684.full.pdf+html }}</ref> This is known as the Wilson–Hilferty transformation.

==Relation to other distributions==
{{Refimprove section|date=September 2011}}

[[File:Chi on SAS.png|thumb|right|400px|Approximate formula for median compared with numerical quantile (top) as presented in [[SAS (software)|SAS Software]]. Difference between numerical quantile and approximate formula (bottom).]]
* As <math>k\to\infty</math>, <math> (\chi^2_k-k)/\sqrt{2k} ~ \xrightarrow{d}\ N(0,1) \,</math> ([[normal distribution]])

*<math> \chi_k^2 \sim {\chi'}^2_k(0)</math> ([[Noncentral chi-squared distribution]] with non-centrality parameter <math> \lambda = 0 </math>)

*If <math>X \sim \mathrm{F}(\nu_1, \nu_2)</math> then <math>Y = \lim_{\nu_2 \to \infty} \nu_1 X</math> has the chi-squared distribution <math>\chi^2_{\nu_{1}}</math>

*As a special case, if <math>X \sim \mathrm{F}(1, \nu_2)\,</math> then <math>Y = \lim_{\nu_2 \to \infty} X\,</math> has the chi-squared distribution <math>\chi^2_{1}</math>

*<math> \|\boldsymbol{N}_{i=1,...,k}{(0,1)}\|^2 \sim \chi^2_k </math> (The squared [[Norm (mathematics)|norm]] of '''k''' standard normally distributed variables is a chi-squared distribution with '''k''' [[degrees of freedom (statistics)|degrees of freedom]])

*If <math>X \sim {\chi}^2(\nu)\,</math> and <math>c>0 \,</math>, then <math>cX \sim {\Gamma}(k=\nu/2, \theta=2c)\,</math>. ([[gamma distribution]])

*If <math>X \sim \chi^2_k</math> then <math>\sqrt{X} \sim \chi_k</math> ([[chi distribution]])

*If <math>X \sim \chi^2 \left( 2 \right)</math>, then <math>X \sim \mathrm{Exp(1/2)}</math> is an [[exponential distribution]]. (See [[Gamma distribution]] for more.)

*If <math>X \sim \mathrm{Rayleigh}(1)\,</math> ([[Rayleigh distribution]]) then <math>X^2 \sim \chi^2(2)\,</math>

*If <math>X \sim \mathrm{Maxwell}(1)\,</math> ([[Maxwell distribution]]) then <math>X^2 \sim \chi^2(3)\,</math>

*If <math>X \sim \chi^2(\nu)</math> then <math>\tfrac{1}{X} \sim \mbox{Inv-}\chi^2(\nu)\, </math> ([[Inverse-chi-squared distribution]])

*The chi-squared distribution is a special case of type 3 [[Pearson distribution]]

* If <math>X \sim \chi^2(\nu_1)\,</math> and <math>Y \sim \chi^2(\nu_2)\,</math> are independent then <math>\tfrac{X}{X+Y} \sim {\rm Beta}(\tfrac{\nu_1}{2}, \tfrac{\nu_2}{2})\,</math> ([[beta distribution]])

*If <math> X \sim {\rm U}(0,1)\, </math> ([[Uniform distribution (continuous)|uniform distribution]]) then <math> -2\log{(X)} \sim \chi^2(2)\,</math>

* <math>\chi^2(6)\,</math> is a transformation of [[Laplace distribution]]

*If <math>X_i \sim \mathrm{Laplace}(\mu,\beta)\,</math> then <math>\sum_{i=1}^n{\frac{2 |X_i-\mu|}{\beta}} \sim \chi^2(2n)\,</math>

* chi-squared distribution is a transformation of [[Pareto distribution]]

* [[Student's t-distribution]] is a transformation of chi-squared distribution

* [[Student's t-distribution]] can be obtained from chi-squared distribution and [[normal distribution]]

* [[Noncentral beta distribution]] can be obtained as a transformation of chi-squared distribution and [[Noncentral chi-squared distribution]]

* [[Noncentral t-distribution]] can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with ''k'' degrees of freedom is defined as the sum of the squares of ''k'' independent [[standard normal distribution|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'' = (''Y''−''μ'')T''C''−1(''Y''−''μ'') 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 (linear algebra)|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-distribution|F-distributed]], ''Y'' ~ ''F''(''k''1,''k''2) if <math>\scriptstyle Y = \frac{X_1 / k_1}{X_2 / k_2}</math> where ''X''1 ~ ''χ''²(''k''1) and ''X''2  ~ ''χ''²(''k''2) are statistically independent.

* If ''X'' is chi-squared distributed, then <math>\scriptstyle\sqrt{X}</math> is [[chi distribution|chi distributed]].

* If {{nowrap|''X''1  ~  ''χ''2''k''1}} and {{nowrap|''X''2  ~  ''χ''2''k''2}} are statistically independent, then {{nowrap|''X''1 + ''X''2  ~ ''χ''2''k''1+''k''2}}. If ''X''1 and ''X''2 are not independent, then {{nowrap|''X''1 + ''X''2}} is not chi-squared distributed.

==Generalizations==
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.

===Chi-squared distributions===

====Noncentral chi-squared distribution====
{{Main|Noncentral chi-squared distribution}}
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====
{{Main|Generalized chi-squared distribution}}
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===
The chi-squared distribution ''X'' ~ '''''χ''²(''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'' ~ ''χ''²(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'' ~ ''χ''²(''k'') with even ''k'', then ''X'' is Erlang distributed with shape parameter ''k''/2 and scale parameter 1/2.

==Applications==
The chi-squared distribution has numerous applications in inferential [[statistics]], for instance in [[chi-squared test]]s and in estimating [[variance]]s. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a [[linear regression|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 variable]]s, 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.

*if ''X''1, ..., ''Xn'' are [[independent identically-distributed random variables|i.i.d.]] ''N''(''μ'', ''σ''2) [[random variable]]s, then <math>\sum_{i=1}^n(X_i - \bar X)^2 \sim \sigma^2 \chi^2_{n-1}</math> where <math>\bar X = \frac{1}{n} \sum_{i=1}^n X_i</math>.

*The box below shows probability distributions with name starting with '''chi''' for some [[statistic]]s based on {{nowrap|''Xi'' ∼ Normal(''μi'', ''σ''2''i''), ''i'' {{=}} 1, ⋯, ''k'', }} independent random variables:
<center>
{| class="wikitable" align="center"
|-
! Name !! Statistic
|-
| chi-squared distribution || <math>\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2</math>
|-
| [[noncentral chi-squared distribution]] || <math>\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2</math>
|-
| [[chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}</math>
|-
| [[noncentral chi distribution]] || <math>\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}</math>
|}
</center>

==Table of ''χ''2 value vs p-value==
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 ''χ''2 for the first 10 degrees of freedom.

A low p-value indicates greater [[Statistical significance|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.

{| class="wikitable"
|-
! Degrees of freedom (df)
!colspan=11| ''χ''2 value <ref>[http://www2.lv.psu.edu/jxm57/irp/chisquar.html 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</ref>
|-
| <div align="center"> 1
| 0.004
| 0.02
| 0.06
| 0.15
| 0.46
| 1.07
| 1.64
| 2.71
| 3.84
| 6.64
| 10.83
|-
| <div align="center"> 2
| 0.10
| 0.21
| 0.45
| 0.71
| 1.39
| 2.41
| 3.22
| 4.60
| 5.99
| 9.21
| 13.82
|-
| <div align="center"> 3
| 0.35
| 0.58
| 1.01
| 1.42
| 2.37
| 3.66
| 4.64
| 6.25
| 7.82
| 11.34
| 16.27
|-
| <div align="center"> 4
| 0.71
| 1.06
| 1.65
| 2.20
| 3.36
| 4.88
| 5.99
| 7.78
| 9.49
| 13.28
| 18.47
|-
| <div align="center"> 5
| 1.14
| 1.61
| 2.34
| 3.00
| 4.35
| 6.06
| 7.29
| 9.24
| 11.07
| 15.09
| 20.52
|-
| <div align="center"> 6
| 1.63
| 2.20
| 3.07
| 3.83
| 5.35
| 7.23
| 8.56
| 10.64
| 12.59
| 16.81
| 22.46
|-
| <div align="center"> 7
| 2.17
| 2.83
| 3.82
| 4.67
| 6.35
| 8.38
| 9.80
| 12.02
| 14.07
| 18.48
| 24.32
|-
| <div align="center"> 8
| 2.73
| 3.49
| 4.59
| 5.53
| 7.34
| 9.52
| 11.03
| 13.36
| 15.51
| 20.09
| 26.12
|-
| <div align="center"> 9
| 3.32
| 4.17
| 5.38
| 6.39
| 8.34
| 10.66
| 12.24
| 14.68
| 16.92
| 21.67
| 27.88
|-
| <div align="center"> 10
| 3.94
| 4.86
| 6.18
| 7.27
| 9.34
| 11.78
| 13.44
| 15.99
| 18.31
| 23.21
| 29.59
|-
! <div align="right"> P value (Probability)
| style="background: #ffa2aa" | 0.95
| style="background: #efaaaa" | 0.90
| style="background: #e8b2aa" | 0.80
| style="background: #dfbaaa" | 0.70
| style="background: #d8c2aa" | 0.50
| style="background: #cfcaaa" | 0.30
| style="background: #c8d2aa" | 0.20
| style="background: #bfdaaa" | 0.10
| style="background: #b8e2aa" | 0.05
| style="background: #afeaaa" | 0.01
| style="background: #a8faaa" | 0.001
|-
|}

==See also==
{{Portal|Statistics}}
{{Colbegin}}
*[[Cochran's theorem]]
* [[F-distribution]]
*[[Fisher's method]] for combining [[Statistical independence|independent]] tests of significance
* [[Gamma distribution]]
* [[Generalized chi-squared distribution]]
* [[Hotelling's T-squared distribution]]
* [[Pearson's chi-squared test]]
* [[Student's t-distribution]]
* [[Wilks' lambda distribution]]
* [[Wishart distribution]]
{{Colend}}

==References==
{{Reflist}}

==Further reading==
{{refbegin}}
* {{cite isbn|0471179124}}
* {{cite doi|10.1093/biomet/1.2.155}}
{{refend}}

==External links==
* {{springer|title=Chi-squared distribution|id=p/c022100}}
*[http://calculus-calculator.com/statistics/chi-squared-distribution-calculator.html Calculator for the pdf, cdf and quantiles of the chi-squared distribution]
*[http://jeff560.tripod.com/c.html Earliest Uses of Some of the Words of Mathematics: entry on Chi squared has a brief history]
*[http://www.stat.yale.edu/Courses/1997-98/101/chigf.htm Course notes on Chi-Squared Goodness of Fit Testing] from Yale University Stats 101 class.
*[http://demonstrations.wolfram.com/StatisticsAssociatedWithNormalSamples/ ''Mathematica'' demonstration showing the chi-squared sampling distribution of various statistics, e.g. Σ''x''², for a normal population]
*[http://www.jstor.org/stable/2348373 Simple algorithm for approximating cdf and inverse cdf for the chi-squared distribution with a pocket calculator]

{{ProbDistributions|continuous-semi-infinite}}
{{Common univariate probability distributions}}

{{DEFAULTSORT:Chi-Squared Distribution}}
[[Category:Continuous distributions]]
[[Category:Normal distribution]]
[[Category:Exponential family distributions]]
[[Category:Infinitely divisible probability distributions]]
[[Category:Probability distributions]]

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