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In probability theory and statistics, the chi-square distribution (also chi-squared or χ2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-square distribution, a special case of the more general noncentral chi-square distribution. The chi-square distribution is used in the common chi-square 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, such as Friedman's analysis of variance by ranks.
If Z1, ..., Zk are independent, standard normal random variables, then the sum of their squares,
is distributed according to the chi-square distribution with k degrees of freedom. This is usually denoted as
The chi-square distribution has one parameter: a positive integer k that specifies the number of degrees of freedom (the number of random variables being summed, Zi s).
The chi-square distribution is used primarily in hypothesis testing, and to a lesser extent for confidence intervals for population variance when the underlying distribution is normal. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-square distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others:
It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis.
The primary reason for which the chi-square distribution is extensively used in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t-statistic in a t-test. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-square distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-square distribution could be used.
Suppose that
An additional reason that the chi-square distribution is widely used is that it turns up as the large sample distribution of generalized likelihood ratio tests (LRT).[1] LRT's have several desirable properties; in particular, simple LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma) and this leads also to optimality properties of generalised LRTs. However, the normal and chi-square approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chi-square approximation for a small sample size. Similarly, in analyses of contingency tables, the chi-square approximation will be poor for a small sample size, and it is preferable to use Fisher's exact test. Ramsey shows that the exact binomial test is always more powerful than the normal approximation.[2]
Lancaster shows the connections among the binomial, normal, and chi-square distributions, as follows.[3] De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. Specifically they showed the asymptotic normality of the random variable
where
Squaring both sides of the equation gives
Using
The expression on the right is of the form that Karl Pearson would generalize to the form:
where
In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large
The probability density function (pdf) of the chi-square distribution is
where
For derivations of the pdf in the cases of one, two and
Its cumulative distribution function is:
where
In a special case of
which can be easily derived by integrating
Tables of the chi-square cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.
Letting
The tail bound for the cases when
For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chi-square distribution.
If Z1, ..., Zk are independent identically distributed (i.i.d.), standard normal random variables, then
where
It follows from the definition of the chi-square distribution that the sum of independent chi-square variables is also chi-square distributed. Specifically, if
The sample mean of
Asymptotically, given that for a scale parameter
Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chi-square variable of degree
The differential entropy is given by
where ψ(x) is the Digamma function.
The chi-square distribution is the maximum entropy probability distribution for a random variate
The moments about zero of a chi-square distribution with
The cumulants are readily obtained by a (formal) power series expansion of the logarithm of the characteristic function:
The chi-squared distribution exhibits strong concentration around its mean. The standard Laurent-Massart [7] bounds are:
By the central limit theorem, because the chi-square distribution is the sum of
The sampling distribution of
A chi-square variable with
If
The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-square distribution called the noncentral chi-square distribution.
If
If
The chi-square distribution is also naturally related to other distributions arising from the Gaussian. In particular,
The chi-square 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.
If
The noncentral chi-square distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.
The generalized chi-square 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.
The chi-square distribution
Because the exponential distribution is also a special case of the gamma distribution, we also have that if
The Erlang distribution is also a special case of the gamma distribution and thus we also have that if
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, each divided by their respective degrees of freedom.
Following are some of the most common situations in which the chi-square distribution arises from a Gaussian-distributed sample.
Name | Statistic |
---|---|
chi-square distribution | |
noncentral chi-square distribution | |
chi distribution | |
noncentral chi distribution |
The chi-square distribution is also often encountered in magnetic resonance imaging.[15]
The p-value is the probability of observing a test statistic at least as extreme in a chi-square 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. A low p-value, below the chosen significance level, indicates statistical significance, i.e., sufficient evidence to reject the null hypothesis. A significance level of 0.05 is often used as the cutoff between significant and non-significant results.
The table below gives a number of p-values matching to
Degrees of freedom (df) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.004 | 0.02 | 0.06 | 0.15 | 0.46 | 1.07 | 1.64 | 2.71 | 3.84 | 6.63 | 10.83 |
2 | 0.10 | 0.21 | 0.45 | 0.71 | 1.39 | 2.41 | 3.22 | 4.61 | 5.99 | 9.21 | 13.82 |
3 | 0.35 | 0.58 | 1.01 | 1.42 | 2.37 | 3.66 | 4.64 | 6.25 | 7.81 | 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 |
These values can be calculated evaluating the quantile function (also known as “inverse CDF” or “ICDF”) of the chi-square distribution;[17] e. g., the χ2 ICDF for p = 0.05 and df = 7 yields 2.1673 ≈ 2.17 as in the table above, noticing that 1 - p is the p-value from the table.
This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875–6,[18][19] where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmert'sche ("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-square test, published in 1900, with computed table of values published in (Elderton 1902), collected in (Pearson 1914). The name "chi-square" 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).[20] The idea of a family of "chi-square distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.[18]