Chi Square Distribution  Definition
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ChiSquare (C2) Distribution Definition
In probability theory and statistics, the Chisquared distribution also referred as chisquare or X2distribution, with k degrees of freedom, is the distribution of a sum of squares of k independent standard regular normal variables. Chidistribution is a unique case of a gamma distribution and is among the most broadly applied probability distribution in inferential statistics. It is used commonly in hypothesis evaluation or development of an acceptable range of deviation.
A Little More on What is ChiSquare (C2) Distribution
The chisquared is applied in the regular chisquared tests for goodness of fit of a witnessed distribution to a hypothetical one. More specifically, it measures the independence of the two methods of a grouping of qualitative information and confidence range approximation for population standard deviation of the normal distribution from a representative standard deviation. Other mathematical studies such as Friedman's analysis of variance by ranks apply chisquare distribution. The chisquared distribution is most commonly employed in hypothesis testing. Despite popular distributions, for instance, normal distribution and the exponential distributions, chisquare distribution is rarely applied in direct modeling of ordinary occurrences. It results in the following hypothesis evaluation:
 Chisquared test of independence in contingency tables
 Chisquared test of goodness of fit of observed data to hypothetical distributions
 Likelihoodratio test for nested models
 Logrank test in survival analysis
 CochranMantelHaenszel test for stratified contingency tables
Besides the above applications, chisquared distribution is a part of the definition of tdistribution and Fdistribution useful in ttests which are an analysis of variance and regression analysis. The major reason for the extensive use of chisquare in postulate evaluation is its association to the normal distribution. Many hypothesis tests use test statistics, for example, tstatistic in a ttest. For these ttests, as the sample size, n, increases the sample distribution of the test statistic moves to the normal distribution in a central limit theorem concept. As a result of test statistics being asymptotically normally distributed, given that the sample size is large enough, the distribution applied for hypothesis testing may be estimated by a normal distribution. The process of testing hypotheses using a normal distribution is well understood and is relatively easy. The simplest chisquared distribution is the square of the standard normal distribution. In case of testing a hypothesis using a normal distribution, a chisquare distribution may be used. Additionally, Chisquared distribution is generally applied is that it belongs to a class of likelihood ratio tests (LRT). LRTs possess favorable characteristics specifically; it provides the high power in the null hypothesis rejection. On the other hand, Normal and chisquared estimations are invalid asymptotically, and this preference is given to a tdistribution instead of normal estimation or chisquared approximation for small sample size. Ramsey indicated that exact binomial test is normally powerful than a normal approximation.
The ChiSquare Statistic
Assume we perform the following statistical experiment. We choose a random sample of n from a normal population, with a standard deviation equal to . Standard deviation is found to be s. with this information we can define a statistic referred to as chisquare using this equation 2 = [ ( n  1 ) * s2 ] / 2 The distribution of the chisquare statistic is referred to as the chisquare distribution. The chisquare distribution is given by the following probability density function: Y = Y0 * ( 2 ) ( v/2  1 ) * e2 / 2 Where Y0 is a constant that depends on the number of degrees of freedom, 2 is the chisquare statistic, v = n  1 is the number of degrees of freedom, and e is a constant equal to the base of the natural logarithm system (estimated 2.71828). Y0 is defined so that the area under the chisquare curve is equal to 1.
References for Chi Square (c2) Distribution
 http://www.businessdictionary.com/definition/chisquarec2distribution.html
 https://en.wikipedia.org/wiki/Chisquared_distribution
 https://thelawdictionary.org/chisquarec2distribution/
 https://stattrek.com/probabilitydistributions/chisquare.aspx
Academic Research for Chi Square (c2) Distribution
 The relation of control charts to the analysis of variance and chisquare tests, Scheffe, H. (1947). Journal of the American Statistical Association, 42(239), 425431. This paper shows some established connections by simple and intuitive paths among these statistical methods: Shewhart control charts, analysis of variance, and chisquare tests.
 Remarks on a multivariate transformation, Rosenblatt, M. (1952). The Annals of mathematical statistics, 23(3), 470472. This paper points out and discusses a simple transformation of an absolute continuous kvariate distribution into a uniform distribution on the kdimensional hypercube. One can illustrate that random vector Z=TX is evenly distributed on the kdimensional hypercube.
 Chisquare test for continuous distributions with shift and scale parameters, Nikulin, M. S. (1974). Theory of Probability & Its Applications, 18(3), 559568. This paper examines the verification problem of the null hypothesis that the distribution function of independent similar distributed random variables belonging to a family of continuous function depending on the shift factor and the scalar factor in the given distribution function. Dividing the line into kintervals by the points and grouping over these intervals, a frequency vector is obtained and probability vector.
 Bounds on normal approximations to Student's and the chisquare distributions, Wallace, D. L. (1959). The Annals of Mathematical Statistics, 30(4), 11211130. The paper addresses the conversion of upper tail values of t or chisquare variates with n degrees of freedom to normal deviates. The main purpose of this paper is to develop bounds on the deviation from the actual normal deviates to the extent of absolute deviation is bounded by an12cn 12 evenly throughout the tail.
 Bivariate distributions of some ratios of independent noncentral chisquare random variables, Hawkins, D. L., & Han, C. P. (1986). Communications in StatisticsTheory and Methods, 15(1), 261277. The paper illustrates the examination of threepaired ratios of bivariate distribution of independent noncentral chisquared random variables. These ratios emerge from the problem of calculating the combined power of simultaneous in balanced Ftests in balanced ANOVA and ANCOVA.
 On the choice of the number of class intervals in the application of the chisquare test, Mann, H. B., & Wald, A. (1942). The Annals of Mathematical Statistics, 13(3), 306317. The paper states that to verify whether a sample has been taken from a population with a particular probability distribution, the range of the variable is divided into some class range and the statistic calculated. Under the null hypothesis, it is clear that the statistic; has asymptotically the chisquare distribution with k1 degrees of freedom when each population number is large. When a choice is made regarding the number of class intervals, it is normally possible to get the alternative hypothesis with class probabilities similar to the class probabilities under the null hypothesis.
 Density functions of the bivariate chisquare distribution, Gunst, R. F., & Webster, J. T. (1973). Journal of statistical computation and simulation, 2(3), 275288. This paper was intended to provide an experimental approach in solving simultaneous testing and approximation challenges faced by experimenters. Bivariate Chisquare which enables direct computer programming was introduced. An estimation that decreases the general dependency type to specific form is proposed, supported by strong theoretical reasoning. The last function of the Bivariate Chisquare is calculating the density function of a linear combination of private Chisquare irregular variables.
 On the limiting power function of the frequency chisquare test, Mitra, S. K. (1958). The Annals of Mathematical Statistics, 29(4), 12211233. The paper states that many authors have studied the power function of the frequency X2test by obtaining a large sample of the expression of simple goodness of fit X2test. As a result of challenges in finding the power function of the frequency X2test regularly, a suggestion of the derivation of its Pitman limiting factor and illustration was given in the simple goodness fits case. The asymptotic power concept has been applied in various areas including the nonparametric conclusion, and it proved its usefulness in comparison of various consistent tests or alternative designs for study.
 A family of transformed chisquare distributions, Rahman, M. S., & Gupta, R. P. (1992). Communications in StatisticsTheory and Methods, 22(1), 135146. The paper defines a family of Transformed Chisquare distributions, a special class of exponential family of distribution. Outward terminologies for the minimum variance unbiased predictors with less variance of a function of a factor of this family are given. The sign is and the power function for different hypothesis tests for the elements of this family are also found.
 Monitoring process means and variability with one noncentral chisquare chart, Costa, A. F. B., & Rahim, M. A. (2004). Journal of Applied Statistics, 31(10), 11711183. Conventionally, an Xchart is useful in controlling the process average and Rchart to regulate process variance. These charts are insensitive to small changes in the process factors. A better option to these charts is exponential weighted moving average (AWMA) control chart for controlling the process average and variation ability that is highly effective in detecting small process changes. Besides, the EWMA control chart based on noncentered Chisquare data is more effective in identifying the average variability.
 On chisquare goodnessoffit tests for continuous distributions, Watson, G. S. (1958). Journal of the Royal Statistical Society. Series B (Methodological), 4472. The paper suggests that Given that in a Chisquared goodnessoffit test the unknown factors are approximated from the probability of the continuous observations before clustering. The impacts on the distribution of test criterion are studied.