Chi Square Distribution - Explained
What is a Chi-Square Distribution?
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Table of ContentsWhat is a Chi-Square (C2) Distribution?How is the Chi-Square (C2) Distribution Used? The Chi-Square StatisticAcademic Research for Chi Square (c2) Distribution
What is a Chi-Square (C2) Distribution?
In probability theory and statistics, the Chi-squared distribution also referred as chi-square or X2-distribution, with k degrees of freedom, is the distribution of a sum of squares of k independent standard regular normal variables. Chi-distribution 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.
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How is the Chi-Square (C2) Distribution Used?
The chi-squared is applied in the regular chi-squared 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 chi-square distribution. The chi-squared distribution is most commonly employed in hypothesis testing. Despite popular distributions, for instance, normal distribution and the exponential distributions, chi-square distribution is rarely applied in direct modeling of ordinary occurrences. It results in the following hypothesis evaluation:
- Chi-squared test of independence in contingency tables
- Chi-squared test of goodness of fit of observed data to hypothetical distributions
- Likelihood-ratio test for nested models
- Log-rank test in survival analysis
- CochranMantelHaenszel test for stratified contingency tables
Besides the above applications, chi-squared distribution is a part of the definition of t-distribution and F-distribution useful in t-tests which are an analysis of variance and regression analysis. The major reason for the extensive use of chi-square in postulate evaluation is its association to the normal distribution. Many hypothesis tests use test statistics, for example, t-statistic in a t-test. For these t-tests, 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 chi-squared distribution is the square of the standard normal distribution. In case of testing a hypothesis using a normal distribution, a chi-square distribution may be used. Additionally, Chi-squared 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 chi-squared estimations are invalid asymptotically, and this preference is given to a t-distribution instead of normal estimation or chi-squared approximation for small sample size. Ramsey indicated that exact binomial test is normally powerful than a normal approximation.
The Chi-Square 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 chi-square using this equation 2 = [ ( n - 1 ) * s2 ] / 2 The distribution of the chi-square statistic is referred to as the chi-square distribution. The chi-square distribution is given by the following probability density function: Y = Y0 * ( 2 ) ( v/2 - 1 ) * e-2 / 2 Where Y0 is a constant that depends on the number of degrees of freedom, 2 is the chi-square 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 chi-square curve is equal to 1.