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Chi-squared distribution

From Wikipedia, the free encyclopedia

chi-square
Probability density function
Chi-square pdf.svg
Cumulative distribution function
Chi-square cdf.svg
Notation or
Parameters (known as "degrees of freedom")
Support if , otherwise
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF       [1]
PGF

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.[2][3][4][5] When it is being distinguished from the more general noncentral chi-square distribution, this distribution is sometimes called the central 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.

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Transcription

In this video, we'll just talk a little bit about what the chi-square distribution is, sometimes called the chi-squared 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 chi-square 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 chi-square distribution. Actually what we're going to see in this video is that the chi-square, or the chi-squared 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 chi-squared 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 chi-squared. 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 chi-squared, distributed random variable with 2 degrees of freedom. Right here. 2 degrees of freedom. And just to visualize kind of the set of chi-squared 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 chi-square 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 chi-squared 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 chi-squared 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 chi-squared 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 chi-square table like this. Q2 is a version of chi-squared 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 chi-squared-- the reason why we have these weird numbers like this instead of whole numbers or easy-to-read 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 chi-squared 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 chi-square 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 chi-square distribution. In the next video, we're actually going to use it to make some, or to test some, inferences.

Contents

Definitions

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 Zi s).

Introduction

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 that the chi-square distribution is used extensively 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 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 chi-square distribution: The subscript 1 indicates that this particular chi-square distribution is constructed from only 1 standard normal distribution. A chi-square 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 chi-square distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-square distribution have low probability.

An additional reason that the chi-square 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 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.[7]

Lancaster shows the connections among the binomial, normal, and chi-square 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 chi-square 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 chi-square 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 chi-square 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 chi-square distribution is

where denotes the gamma function, which has closed-form values for integer .

For derivations of the pdf in the cases of one, two and degrees of freedom, see Proofs related to chi-square distribution.

Cumulative distribution function

Chernoff bound for the CDF and tail (1-CDF) of a chi-square random variable with ten degrees of freedom ( k {\displaystyle k}  = 10)
Chernoff bound for the CDF and tail (1-CDF) of a chi-square random variable with ten degrees of freedom ( = 10)

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 chi-square 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 chi-square distribution.

Properties

Additivity

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 are independent chi-square variables with , degrees of freedom, respectively, then is chi-square distributed with degrees of freedom.

Sample mean

The sample mean of i.i.d. chi-squared 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 chi-square 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 chi-square distribution is the maximum entropy probability distribution for a random variate for which and are fixed. Since the chi-square 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 moment-generating function of the sufficient statistic.

Noncentral moments

The moments about zero of a chi-square 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 chi-square 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 chi-square 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

Approximate formula for median compared with numerical quantile (top). Difference between numerical quantile and approximate formula (bottom).
Approximate formula for median compared with numerical quantile (top). Difference between numerical quantile and approximate formula (bottom).
  • As , (normal distribution)
  • (noncentral chi-square distribution with non-centrality parameter )
  • If then has the chi-square distribution
  • As a special case, if then has the chi-square distribution
  • (The squared norm of k standard normally distributed variables is a chi-square 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 (Inverse-chi-square distribution)
  • The chi-square 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]
  • chi-square distribution is a transformation of Pareto distribution
  • Student's t-distribution is a transformation of chi-square distribution
  • Student's t-distribution can be obtained from chi-square distribution and normal distribution
  • Noncentral beta distribution can be obtained as a transformation of chi-square distribution and Noncentral chi-square distribution
  • Noncentral t-distribution can be obtained from normal distribution and chi-square distribution

A chi-square 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 chi-square distributed with 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-square distribution called the noncentral chi-square 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 chi-square distributed with degrees of freedom.

If is a positive-semidefinite covariance matrix with strictly positive diagonal entries, then for and a random -vector independent of such that and it holds that

[14]

The chi-square distribution is also naturally related to other distributions arising from the Gaussian. In particular,

  • is F-distributed, if , where and are statistically independent.
  • If and are statistically independent, then . If and are not independent, then is not chi-square distributed.

Generalizations

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.

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 chi-square random variables.[17]

Chi-square distributions

Noncentral chi-square distribution

The noncentral chi-square distribution is obtained from the sum of the squares of independent Gaussian random variables having unit variance and nonzero means.

Generalized chi-square distribution

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.

Gamma, exponential, and related distributions

The chi-square 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 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.

  • 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 chi-square distribution:
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.[18]

Computational methods

Table of χ2 values vs p-values

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 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 chi-square distribution;[20] e. g., the χ2 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 chi-square 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 "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).[23] 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.[21]

See also

References

  1. ^ M.A. Sanders. "Characteristic function of the central chi-square distribution" (PDF). Archived from the original (PDF) on 2011-07-15. Retrieved 2009-03-06.
  2. ^ 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 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  3. ^ NIST (2006). Engineering Statistics Handbook – Chi-Squared Distribution
  4. ^ a b c Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "Chi-Square Distributions including Chi and Rayleigh". Continuous Univariate Distributions. 1 (Second ed.). John Wiley and Sons. pp. 415–493. ISBN 978-0-471-58495-7.
  5. ^ Mood, Alexander; Graybill, Franklin A.; Boes, Duane C. (1974). Introduction to the Theory of Statistics (Third ed.). McGraw-Hill. pp. 241–246. ISBN 978-0-07-042864-5.
  6. ^ Westfall, Peter H. (2013). Understanding Advanced Statistical Methods. Boca Raton, FL: CRC Press. ISBN 978-1-4665-1210-8.
  7. ^ 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.
  8. ^ a b Lancaster, H.O. (1969), The Chi-squared Distribution, Wiley
  9. ^ 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 2012-05-01.
  10. ^ Chi-squared distribution, from MathWorld, retrieved Feb. 11, 2009
  11. ^ M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN 978-0-387-34657-1
  12. ^ Box, Hunter and Hunter (1978). Statistics for experimenters. Wiley. p. 118. ISBN 978-0471093152.
  13. ^ Bartlett, M. S.; Kendall, D. G. (1946). "The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation". Supplement to the Journal of the Royal Statistical Society. 8 (1): 128–138. doi:10.2307/2983618. JSTOR 2983618.
  14. ^ a b Pillai, Natesh S. (2016). "An unexpected encounter with Cauchy and Lévy". Annals of Statistics. 44 (5): 2089–2097. arXiv:1505.01957. doi:10.1214/15-aos1407.
  15. ^ Wilson, E. B.; Hilferty, M. M. (1931). "The distribution of chi-squared". Proc. Natl. Acad. Sci. USA. 17 (12): 684–688. Bibcode:1931PNAS...17..684W. doi:10.1073/pnas.17.12.684. PMC 1076144. PMID 16577411.
  16. ^ Bäckström, T.; Fischer, J. (January 2018). "Fast Randomization for Distributed Low-Bitrate Coding of Speech and Audio". IEEE/ACM Transactions on Audio, Speech, and Language Processing. 26 (1): 19–30. doi:10.1109/TASLP.2017.2757601.
  17. ^ Bausch, J. (2013). "On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua". J. Phys. A: Math. Theor. 46 (50): 505202. arXiv:1208.2691. Bibcode:2013JPhA...46X5202B. doi:10.1088/1751-8113/46/50/505202.
  18. ^ den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", Physica Medica, [1]
  19. ^ 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. Two values have been corrected, 7.82 with 7.81 and 4.60 with 4.61
  20. ^ R Tutorial: Chi-squared Distribution
  21. ^ a b Hald 1998, pp. 633–692, 27. Sampling Distributions under Normality.
  22. ^ 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
  23. ^ R. L. Plackett, Karl Pearson and the Chi-Squared Test, International Statistical Review, 1983, 61f. See also Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics.

Further reading

External links

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