Chi-Square Distribution

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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.

chi-square distribution normal distribution chi-square tests

References

Westfall, Peter H. (2013). Understanding Advanced Statistical Methods. Boca Raton, FL: CRC Press. ISBN 978-1-4665-1210-8.
Ramsey, PH (1988). "Evaluating the Normal Approximation to the Binomial Test". Journal of Educational Statistics 13 (2): 173–82. doi:10.2307/1164752. https://dx.doi.org/10.2307%2F1164752
Lancaster, H.O. (1969), The Chi-squared Distribution, Wiley
Dasgupta, Sanjoy D. A.; Gupta, Anupam K. (January 2003). "An Elementary Proof of a Theorem of Johnson and Lindenstrauss". Random Structures and Algorithms 22 (1): 60–65. doi:10.1002/rsa.10073. http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf. Retrieved 2012-05-01.
Chi-squared distribution, from MathWorld, retrieved Feb. 11, 2009 http://mathworld.wolfram.com/Chi-SquaredDistribution.html
M. K. Simon, Probability Distributions Involving Gaussian Random Variables, New York: Springer, 2002, eq. (2.35), ISBN:978-0-387-34657-1
https://projecteuclid.org/journals/annals-of-statistics/volume-28/issue-5/Adaptive-estimation-of-a-quadratic-functional-by-model--selection/10.1214/aos/1015957395.full, Lemma 1, retrieved May 1, 2021
Box, Hunter and Hunter (1978). Statistics for experimenters. Wiley. p. 118. ISBN 978-0471093152. https://archive.org/details/statisticsforexp00geor/page/118.
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. https://dx.doi.org/10.2307%2F2983618
Pillai, Natesh S. (2016). "An unexpected encounter with Cauchy and Lévy". Annals of Statistics 44 (5): 2089–2097. doi:10.1214/15-aos1407. https://dx.doi.org/10.1214%2F15-aos1407
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.
Wilson, E. B.; Hilferty, M. M. (1931). "The distribution of chi-squared". Proc. Natl. Acad. Sci. USA 17 (12): 684–688. doi:10.1073/pnas.17.12.684. PMID 16577411. Bibcode: 1931PNAS...17..684W. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1076144
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. https://aaltodoc.aalto.fi/handle/123456789/33466.
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. doi:10.1088/1751-8113/46/50/505202. Bibcode: 2013JPhA...46X5202B. https://dx.doi.org/10.1088%2F1751-8113%2F46%2F50%2F505202
den Dekker A. J., Sijbers J., (2014) "Data distributions in magnetic resonance images: a review", Physica Medica, [1]
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 http://www2.lv.psu.edu/jxm57/irp/chisquar.html
R Tutorial: Chi-squared Distribution http://www.r-tutor.com/elementary-statistics/probability-distributions/chi-squared-distribution
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 http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN599415665_0021&DMDID=DMDLOG_0018
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. https://www.jstor.org/stable/1402731?seq=3

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