Chi-square distribution
In probability theory and statistics, the chi-square distribution (also chi-squared or
distribution) is one of the most widely used theoretical probability distributions in inferential statistics, e.g., in statistical significance tests.[1][2][3][4] 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.
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. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.
Definition
If
are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable
![Q = \sum_{i=1}^k X_i^2](/2009-wikipedia_en_wp1-0.7_2009-05/I/ad761dd11361720588c0d14dc165e72a.png)
is distributed according to the chi-square distribution with
degrees of freedom. This is usually written
![Q\sim\chi^2_k.\,](/2009-wikipedia_en_wp1-0.7_2009-05/I/387a5d1a2500f40403b30fa519dda8de.png)
The chi-square distribution has one parameter:
- a positive integer that specifies the number of degrees of freedom (i.e. the number of
)
The chi-square distribution is a special case of the gamma distribution.
Characteristics
Probability density function
A probability density function of the chi-square distribution is
![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}](/2009-wikipedia_en_wp1-0.7_2009-05/I/6fbf6fd4750518ea4a273037631bc4ce.png)
where
denotes the Gamma function, which has closed-form values at the half-integers.
Cumulative distribution function
Its cumulative distribution function is:
![F(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)} = P(k/2, x/2)](/2009-wikipedia_en_wp1-0.7_2009-05/I/ce27f2b887b29201ae8e4c4d86aa0da3.png)
where
is the lower incomplete Gamma function and
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
![\chi(t;k)=(1-2it)^{-k/2}.\,](/2009-wikipedia_en_wp1-0.7_2009-05/I/1dd659d8838578914432d1a77f0ca820.png)
Expected value and variance
If
then
![\mathrm{E}(X)=k](/2009-wikipedia_en_wp1-0.7_2009-05/I/f5fbbbce2b1e7f05cd56fd32c79a4006.png)
![\mathrm{Var}(X)=2k](/2009-wikipedia_en_wp1-0.7_2009-05/I/2f2e0192f9ba07d9f6440c19d4cc3ada.png)
Median
The median of
is given approximately by
![k-\frac{2}{3}+\frac{4}{27k}-\frac{8}{729k^2}.](/2009-wikipedia_en_wp1-0.7_2009-05/I/3a157f03a30ff76fd9fe4b20cbe27dfc.png)
Information entropy
The information 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(k/2).](/2009-wikipedia_en_wp1-0.7_2009-05/I/32ef86a6f3421ac89534432f9a93dfd6.png)
where
is the Digamma function.
Derivation of the pdf for one degree of freedom
Let
where ![X \sim N(0,1)](/2009-wikipedia_en_wp1-0.7_2009-05/I/8502f9f731d3ff9caad07c4d1f7867be.png)
then ![P(Y<y) = P(X^2<y)=P(X<|\sqrt{y}|)=F_x(\sqrt{y})-F_x(-\sqrt{y})](/2009-wikipedia_en_wp1-0.7_2009-05/I/0283b31645eb5460415091fff9ad6659.png)
Then ![Y = X^2 \sim \chi^2(1)](/2009-wikipedia_en_wp1-0.7_2009-05/I/5fb388bff6bc2b1081ef433c6a34a89b.png)
Related distributions and 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.
- If
, then as
tends to infinity, the distribution of
tends to a normal distribution with mean
and variance
(convergence is slow as the skewness is
and the excess kurtosis is
)
- If
then
is approximately normally distributed with mean
and unit variance (result credited to R. A. Fisher).
- If
then
is approximately normally distributed with mean
and variance
(Wilson and Hilferty,1931)
is an exponential distribution if
(with 2 degrees of freedom).
is a chi-square distribution if
for
independent that are normally distributed.
- If
, where the
s are independent
random variables or
and
is an
idempotent matrix with rank
then the quadratic form
.
- If the
have nonzero means, then
is drawn from a noncentral chi-square distribution.
- The chi-square distribution
is a special case of the gamma distribution, in that
.
is an F-distribution if
where
and
are independent with their respective degrees of freedom.
is a chi-square distribution if
where
are independent and
.
- if
is chi-square distributed, then
is chi distributed.
- in particular, if
(chi-square with 2 degrees of freedom), then
is Rayleigh distributed.
- if
are i.i.d.
random variables, then
where
.
- if
, then ![\mathrm{log}(1 + e^{-X}) \sim \chi_2^2\,](/2009-wikipedia_en_wp1-0.7_2009-05/I/ce7b01e087b9fa89c3e1a93877370ef3.png)
- The box below shows probability distributions with name starting with chi for some statistics based on
independent random variables:
Name |
Statistic |
chi-square distribution |
![\sum_{i=1}^k \frac{\left(X_i-\mu_i\right)^2}{\sigma_i^2}](/2009-wikipedia_en_wp1-0.7_2009-05/I/cccebc4ae03f764b4d748d541e2e7a21.png) |
noncentral chi-square distribution |
![\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2](/2009-wikipedia_en_wp1-0.7_2009-05/I/cf5a4179baaaab60a26a30aac6e1e93e.png) |
chi distribution |
![\sqrt{\sum_{i=1}^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}](/2009-wikipedia_en_wp1-0.7_2009-05/I/d6624a2515a015566e49fe5120945a6e.png) |
noncentral chi distribution |
![\sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}](/2009-wikipedia_en_wp1-0.7_2009-05/I/e0a313a5b45e33f7fa6397496e7b17b6.png) |
See also
References
- ↑ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 0-486-61272-4 .
- ↑ NIST (2006). Engineering Statistics Handbook - Chi-Square Distribution
- ↑ 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.
- ↑ 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.
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