Chi Square Distribution  Explained
What is a ChiSquare Distribution?
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What is a ChiSquare (C2) Distribution?
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.
How is the ChiSquare (C2) Distribution Used?
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.