In probability theory and statistics, the chisquare distribution (also chisquared or χ^{2}distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chisquare 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.^{[2]}^{[3]}^{[4]}^{[5]} When it is being distinguished from the more general noncentral chisquare distribution, this distribution is sometimes called the central chisquare distribution.
The chisquare distribution is used in the common chisquare 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.
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✪ An Introduction to the ChiSquare Distribution

✪ ChiSquared Distribution

✪ An Introduction to the ChiSquare Distribution

✪ ChiSquared Distribution
Transcription
In this video, we'll just talk a little bit about what the chisquare distribution is, sometimes called the chisquared distribution. And then in the next few videos, we'll actually use it to really test how well theoretical distributions explain observed ones, or how good a fit observed results are for theoretical distributions. So let's just think about it a little bit. So let's say I have some random variables. And each of them are independent, standard, normal, normally distributed random variables. So let me just remind you what that means. So let's say I have the random variable X. If X is normally distributed, we could write that X is a normal random variable with a mean of 0 and a variance of 1. Or you could say that the expected value of X, is equal to 0, or in that the variance of our random variable X is equal to 1. Or just to visualize it is that, when we take an instantiation of this very variable, we're sampling from a normal distribution, a standardized normal distribution that looks like this. Mean of 0 and then a variance of 1, which would also mean, of course, a standard deviation of 1. So that could be the standard deviation, or the variance, or the standard deviation, that would be equal to 1. So a chisquare distribution, if you just take one of these random variables and let me define it this way. Let me define a new random variable. Let me define a new random variable Q that is equal to you're essentially sampling from this the standard normal distribution and then squaring whatever number you got. So it is equal to this random variable X squared. The distribution for this random variable right here is going to be an example of the chisquare distribution. Actually what we're going to see in this video is that the chisquare, or the chisquared distribution is actually a set of distributions depending on how many sums you have. Right now, we only have one random variable that we're squaring. So this is just one of the examples. And we'll talk more about them in a second. So this right here, this we could write that Q is a chisquared distributed random variable. Or that we could use this notation right here. Q is we could write it like this. So this isn't an X anymore. This is the Greek letter chi, although it looks a lot like a curvy X. So it's a member of chisquared. And since we're only taking one sum over here we're only taking the sum of one independent, normally distributed, standard or normally distributed variable, we say that this only has 1 degree of freedom. And we write that over here. So this right here is our degree of freedom. We have 1 degree of freedom right over there. So let's call this Q1. Let's say I have another random variable. Let's call this Q let me do it in a different color. Let me do Q2 in blue. Let's say I have another random variable, Q2, that is defined as let's say I have one independent, standard, normally distributed variable. I'll call that X1. And I square it. And then I have another independent, standard, normally distributed variable, X2. And I square it. So you could imagine both of these guys have distributions like this. And they're independent. So get to sample Q2, you essentially sample X1 from this distribution, square that value, sample X2 from the same distribution, essentially, square that value, and then add the two. And you're going to get Q2. This over here here we would write so this is Q1. Q2 here, Q2 we would write is a chisquared, distributed random variable with 2 degrees of freedom. Right here. 2 degrees of freedom. And just to visualize kind of the set of chisquared distributions, let's look at this over here. So this, I got this off of Wikipedia. This shows us some of the probability density functions for some of the chisquare distributions. This first one over here, for k of equal to 1, that's the degrees of freedom. So this is essentially our Q1. This is our probability density function for Q1. And notice it really spikes close to 0. And that makes sense. Because if you are sampling just once from this standard normal distribution, there's a very high likelihood that you're going to get something pretty close to 0. And then if you square something close to 0 remember, these are decimals, they're going to be less than 1, pretty close to 0 it's going to become even smaller. So you have a high probability of getting a very small value. You have high probabilities of getting values less than some threshold, this right here, less than, I guess, this is 1 right here. So the less than 1/2. And you have a very low probability of getting a large number. I mean, to get a 4, you would have to sample a 2 from this distribution. And we know that 2 is actually it's 2 variances or 2 standard deviations from the mean. So it's less likely. And actually that's to get a 4. So to get even larger numbers are going to be even less likely. So that's why you see this shape over here. Now when you have 2 degrees of freedom, it moderates a little bit. This is the shape, this blue line right here is the shape of Q2. And notice you're a little bit less likely to get values close to 0 and a little bit more likely to get numbers further out. But it still is kind of shifted or heavily weighted towards small numbers. And then if we had another random variable, another chisquared distributed random variable so then we have, let's say, Q3. And let's define it as the sum of 3 of these independent variables, each of them that have a standard normal distribution. So X1, X2 squared plus X3 squared. Then all of a sudden, our Q3 this is Q2 right here has a chisquared distribution with 3 degrees of freedom. And so this guy right over here that will be this green line. Maybe I should have done this in green. This will be this green line over here. And then notice, now it's starting to become a little bit more likely that you'd get values in this range over here because you're taking the sum. Each of these are going to be pretty small values, but you're taking the sum. So it starts to shift it a little over to the right. And so the more degrees of freedom you have, the further this lump starts to move to the right and, to some degree, the more symmetric it gets. And what's interesting about this, I guess it's different than almost every other distribution we've looked at, although we've looked at others that have this property as well, is that you can't have a value below 0 because we're always just squaring these values. Each of these guys can have values below 0. They're normally distributed. They could have negative values. But since we're squaring and taking the sum of squares, this is always going to be positive. And the place that this is going to be useful and we're going to see in the next few videos is in measuring essentially error from an expected value. And if you took take this total error, you can figure out the probability of getting that total error if you assume some parameters. And we'll talk more about it in the next video. Now with that said, I just want to show you how to read a chisquared distribution table. So if I were to ask you, if this is our distribution let me pick this blue one right here. So over here, we have 2 degrees of freedom because we're adding 2 of these guys right here. If I were to ask you, what is the probability of Q2 being greater than or, let me put it this way. What is the probability of Q2 being greater than 2.41? And I'm picking that value for a reason. So I want the probability of Q2 being greater than 2.41. What I want to do is I'll look at a chisquare table like this. Q2 is a version of chisquared with 2 degrees of freedom. So I look at this row right here under 2 degrees of freedom. And I want the probability of getting a value above 2.41. And I picked 2.41 because it's actually at this table. And so most of these chisquared the reason why we have these weird numbers like this instead of whole numbers or easytoread fractions is it is actually driven by the p value. It's driven by the probability of getting something larger than that value. So normally you would look at the other way. You'd say, OK, if I want to say, what chisquared value for 2 degrees of freedom, there's a 30% chance of getting something larger than that? Then I would look up 2.41. But I'm doing it the other way just for the sake of this video. So if I want the probability of this random variable right here being greater than 2.41, or its p value, we read it right here. It is 30%. And just to visualize it on this chart, this chisquare distribution this was Q2, the blue one, over here 2.41 is going to sit let's see. This is 3. This is 2.5. So 2.41 is going to be someplace right around here. So essentially, what that table is telling us is, this entire area under this blue line right here, what is that? And that right there is going to be 30% of well, it's going to be 0.3. Or you could view it as 30% of the entire area under this curve, because obviously all the probabilities have to add up to 1. So that's our intro to the chisquare distribution. In the next video, we're actually going to use it to make some, or to test some, inferences.
Contents
Definitions
If Z_{1}, ..., Z_{k} are independent, standard normal random variables, then the sum of their squares,
is distributed according to the chisquare distribution with k degrees of freedom. This is usually denoted as
The chisquare distribution has one parameter: a positive integer k that specifies the number of degrees of freedom (the number of Z_{i} s).
Introduction
The chisquare 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 chisquare distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others:
 Chisquare test of independence in contingency tables
 Chisquare test of goodness of fit of observed data to hypothetical distributions
 Likelihoodratio test for nested models
 Logrank test in survival analysis
 Cochran–Mantel–Haenszel test for stratified contingency tables
It is also a component of the definition of the tdistribution and the Fdistribution used in ttests, analysis of variance, and regression analysis.
The primary reason that the chisquare distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the tstatistic in a ttest. 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 chisquare distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chisquare distribution could be used.
Suppose that is a random variable sampled from the standard normal distribution, where the mean equals to and the variance equals to : . Now, consider the random variable . The distribution of the random variable is an example of a chisquare distribution: The subscript 1 indicates that this particular chisquare distribution is constructed from only 1 standard normal distribution. A chisquare distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chisquare distribution. Just as extreme values of the normal distribution have low probability (and give small pvalues), extreme values of the chisquare distribution have low probability.
An additional reason that the chisquare distribution is widely used is that it is a member of the class of likelihood ratio tests (LRT).^{[6]} LRT's have several desirable properties; in particular, LRT's commonly provide the highest power to reject the null hypothesis (Neyman–Pearson lemma). However, the normal and chisquare approximations are only valid asymptotically. For this reason, it is preferable to use the t distribution rather than the normal approximation or the chisquare approximation for a small sample size. Similarly, in analyses of contingency tables, the chisquare 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.^{[7]}
Lancaster shows the connections among the binomial, normal, and chisquare distributions, as follows.^{[8]} 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 is the observed number of successes in trials, where the probability of success is , and .
Squaring both sides of the equation gives
Using , , and , this equation simplifies to
The expression on the right is of the form that Karl Pearson would generalize to the form:
where
 = Pearson's cumulative test statistic, which asymptotically approaches a distribution.
 = the number of observations of type .
 = the expected (theoretical) frequency of type , asserted by the null hypothesis that the fraction of type in the population is
 = the number of cells in the table.
In the case of a binomial outcome (flipping a coin), the binomial distribution may be approximated by a normal distribution (for sufficiently large ). Because the square of a standard normal distribution is the chisquare distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by the normal or the chisquare distribution. However, many problems involve more than the two possible outcomes of a binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the normal approximation to the binomial, Pearson sought for and found a multivariate normal approximation to the multinomial distribution. Pearson showed that the chisquare distribution, the sum of multiple normal distributions, was such an approximation to the multinomial distribution ^{[8]}
Probability density function
The probability density function (pdf) of the chisquare distribution is
where denotes the gamma function, which has closedform values for integer .
For derivations of the pdf in the cases of one, two and degrees of freedom, see Proofs related to chisquare distribution.
Cumulative distribution function
Its cumulative distribution function is:
where is the lower incomplete gamma function and is the regularized gamma function.
In a special case of = 2 this function has a simple form:^{[citation needed]}
and the integer recurrence of the gamma function makes it easy to compute for other small even .
Tables of the chisquare cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.
Letting , Chernoff bounds on the lower and upper tails of the CDF may be obtained.^{[9]} For the cases when (which include all of the cases when this CDF is less than half):
The tail bound for the cases when , similarly, is
For another approximation for the CDF modeled after the cube of a Gaussian, see under Noncentral chisquare distribution.
Properties
Additivity
It follows from the definition of the chisquare distribution that the sum of independent chisquare variables is also chisquare distributed. Specifically, if are independent chisquare variables with , degrees of freedom, respectively, then is chisquare distributed with degrees of freedom.
Sample mean
The sample mean of i.i.d. chisquared variables of degree is distributed according to a gamma distribution with shape and scale parameters:
Asymptotically, given that for a scale parameter going to infinity, a Gamma distribution converges towards a normal distribution with expectation and variance , the sample mean converges towards:
Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chisquare variable of degree the expectation is , and its variance (and hence the variance of the sample mean being ).
Entropy
The differential entropy is given by
where ψ(x) is the Digamma function.
The chisquare distribution is the maximum entropy probability distribution for a random variate for which and are fixed. Since the chisquare 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 momentgenerating function of the sufficient statistic.
Noncentral moments
The moments about zero of a chisquare distribution with degrees of freedom are given by^{[10]}^{[11]}
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 chisquare distribution is the sum of independent random variables with finite mean and variance, it converges to a normal distribution for large . For many practical purposes, for the distribution is sufficiently close to a normal distribution for the difference to be ignored.^{[12]} Specifically, if , then as 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 .
The sampling distribution of converges to normality much faster than the sampling distribution of ,^{[13]} as the logarithm removes much of the asymmetry.^{[14]} Other functions of the chisquare distribution converge more rapidly to a normal distribution. Some examples are:
 If then is approximately normally distributed with mean and unit variance (1922, by R. A. Fisher, see (18.23), p. 426 of.^{[4]}
 If then is approximately normally distributed with mean and variance ^{[15]} This is known as the Wilson–Hilferty transformation, see (18.24), p. 426 of.^{[4]}
Related distributions
 As , (normal distribution)
 (noncentral chisquare distribution with noncentrality parameter )
 If then has the chisquare distribution
 As a special case, if then has the chisquare distribution
 (The squared norm of k standard normally distributed variables is a chisquare distribution with k degrees of freedom)
 If and , then . (gamma distribution)
 If then (chi distribution)
 If , then is an exponential distribution. (See gamma distribution for more.)
 If , then is an Erlang distribution.
 If , then
 If (Rayleigh distribution) then
 If (Maxwell distribution) then
 If then (Inversechisquare distribution)
 The chisquare distribution is a special case of type 3 Pearson distribution
 If and are independent then (beta distribution)
 If (uniform distribution) then
 is a transformation of Laplace distribution
 If then
 If follows the generalized normal distribution (version 1) with parameters then ^{[16]}
 chisquare distribution is a transformation of Pareto distribution
 Student's tdistribution is a transformation of chisquare distribution
 Student's tdistribution can be obtained from chisquare distribution and normal distribution
 Noncentral beta distribution can be obtained as a transformation of chisquare distribution and Noncentral chisquare distribution
 Noncentral tdistribution can be obtained from normal distribution and chisquare distribution
A chisquare variable with degrees of freedom is defined as the sum of the squares of independent standard normal random variables.
If is a dimensional Gaussian random vector with mean vector and rank covariance matrix , then is chisquare distributed with degrees of freedom.
The sum of squares of statistically independent unitvariance Gaussian variables which do not have mean zero yields a generalization of the chisquare distribution called the noncentral chisquare distribution.
If is a vector of i.i.d. standard normal random variables and is a symmetric, idempotent matrix with rank , then the quadratic form is chisquare distributed with degrees of freedom.
If is a positivesemidefinite covariance matrix with strictly positive diagonal entries, then for and a random vector independent of such that and it holds that
^{[14]}
The chisquare distribution is also naturally related to other distributions arising from the Gaussian. In particular,
 is Fdistributed, if , where and are statistically independent.
 If and are statistically independent, then . If and are not independent, then is not chisquare distributed.
Generalizations
The chisquare 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.
Linear combination
If are chi square random variables and , then a closed expression for the distribution of is not known. It may be, however, approximated efficiently using the property of characteristic functions of chisquare random variables.^{[17]}
Chisquare distributions
Noncentral chisquare distribution
The noncentral chisquare distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.
Generalized chisquare distribution
The generalized chisquare 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 chisquare distribution is a special case of the gamma distribution, in that using the rate parameterization of the gamma distribution (or 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 , then is an exponential distribution.
The Erlang distribution is also a special case of the gamma distribution and thus we also have that if with even , then is Erlang distributed with shape parameter and scale parameter .
Occurrence and applications
The chisquare distribution has numerous applications in inferential statistics, for instance in chisquare 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 chisquare distribution arises from a Gaussiandistributed sample.
 if are i.i.d. random variables, then where .
 The box below shows some statistics based on independent random variables that have probability distributions related to the chisquare distribution:
Name  Statistic 

chisquare distribution  
noncentral chisquare distribution  
chi distribution  
noncentral chi distribution 
The chisquare distribution is also often encountered in magnetic resonance imaging.^{[18]}
Computational methods
Table of χ^{2} values vs pvalues
The pvalue is the probability of observing a test statistic at least as extreme in a chisquare 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. A low pvalue, 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 nonsignificant results.
The table below gives a number of pvalues matching to for the first 10 degrees of freedom.
Degrees of freedom (df)  value^{[19]}  

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 chisquare distribution;^{[20]} e. g., the χ^{2} ICDF for p = 0.05 and df = 7 yields 14.06714 ≈ 14.07 as in the table above.
History
This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875–6,^{[21]}^{[22]} 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 chisquare test, published in 1900, with computed table of values published in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII). The name "chisquare" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing −½χ^{2} for what would appear in modern notation as −½x^{T}Σ^{−1}x (Σ being the covariance matrix).^{[23]} The idea of a family of "chisquare distributions", however, is not due to Pearson but arose as a further development due to Fisher in the 1920s.^{[21]}
See also
 Chi distribution
 Cochran's theorem
 Fdistribution
 Fisher's method for combining independent tests of significance
 Gamma distribution
 Generalized chisquare distribution
 Hotelling's Tsquare distribution
 Noncentral chisquare distribution
 Pearson's chisquare test
 Reduced chisquared statistic
 Student's tdistribution
 Wilks's lambda distribution
 Wishart distribution
References
 ^ M.A. Sanders. "Characteristic function of the central chisquare distribution" (PDF). Archived from the original (PDF) on 20110715. Retrieved 20090306.
 ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 940. ISBN 9780486612720. LCCN 6460036. MR 0167642. LCCN 6512253.
 ^ NIST (2006). Engineering Statistics Handbook – ChiSquared Distribution
 ^ ^{a} ^{b} ^{c} Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "ChiSquare Distributions including Chi and Rayleigh". Continuous Univariate Distributions. 1 (Second ed.). John Wiley and Sons. pp. 415–493. ISBN 9780471584957.
 ^ Mood, Alexander; Graybill, Franklin A.; Boes, Duane C. (1974). Introduction to the Theory of Statistics (Third ed.). McGrawHill. pp. 241–246. ISBN 9780070428645.
 ^ Westfall, Peter H. (2013). Understanding Advanced Statistical Methods. Boca Raton, FL: CRC Press. ISBN 9781466512108.
 ^ Ramsey, PH (1988). "Evaluating the Normal Approximation to the Binomial Test". Journal of Educational Statistics. 13 (2): 173–82. doi:10.2307/1164752. JSTOR 1164752.
 ^ ^{a} ^{b} Lancaster, H.O. (1969), The Chisquared Distribution, Wiley
 ^ Dasgupta, Sanjoy D. A.; Gupta, Anupam K. (January 2003). "An Elementary Proof of a Theorem of Johnson and Lindenstrauss" (PDF). Random Structures and Algorithms. 22 (1): 60–65. doi:10.1002/rsa.10073. Retrieved 20120501.
 ^ 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 (1978). Statistics for experimenters. Wiley. p. 118. ISBN 9780471093152.
 ^ Bartlett, M. S.; Kendall, D. G. (1946). "The Statistical Analysis of VarianceHeterogeneity and the Logarithmic Transformation". Supplement to the Journal of the Royal Statistical Society. 8 (1): 128–138. doi:10.2307/2983618. JSTOR 2983618.
 ^ ^{a} ^{b} Pillai, Natesh S. (2016). "An unexpected encounter with Cauchy and Lévy". Annals of Statistics. 44 (5): 2089–2097. arXiv:1505.01957. doi:10.1214/15aos1407.
 ^ Wilson, E. B.; Hilferty, M. M. (1931). "The distribution of chisquared". Proc. Natl. Acad. Sci. USA. 17 (12): 684–688. Bibcode:1931PNAS...17..684W. doi:10.1073/pnas.17.12.684. PMC 1076144. PMID 16577411.
 ^ Bäckström, T.; Fischer, J. (January 2018). "Fast Randomization for Distributed LowBitrate Coding of Speech and Audio". IEEE/ACM Transactions on Audio, Speech, and Language Processing. 26 (1): 19–30. doi:10.1109/TASLP.2017.2757601.
 ^ Bausch, J. (2013). "On the Efficient Calculation of a Linear Combination of ChiSquare Random Variables with an Application in Counting String Vacua". J. Phys. A: Math. Theor. 46 (50): 505202. arXiv:1208.2691. Bibcode:2013JPhA...46X5202B. doi:10.1088/17518113/46/50/505202.
 ^ den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", Physica Medica, [1]
 ^ 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. Two values have been corrected, 7.82 with 7.81 and 4.60 with 4.61
 ^ R Tutorial: Chisquared Distribution
 ^ ^{a} ^{b} Hald 1998, pp. 633–692, 27. Sampling Distributions under Normality.
 ^ 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, pp. 102–219
 ^ R. L. Plackett, Karl Pearson and the ChiSquared Test, International Statistical Review, 1983, 61f. See also Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics.
Further reading
 Hald, Anders (1998). A history of mathematical statistics from 1750 to 1930. New York: Wiley. ISBN 9780471179122.
 Elderton, William Palin (1902). "Tables for Testing the Goodness of Fit of Theory to Observation". Biometrika. 1 (2): 155–163. doi:10.1093/biomet/1.2.155.
 Hazewinkel, Michiel, ed. (2001) [1994], "Chisquared distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
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
 Values of the Chisquared distribution