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What is CHI-SQUARE Test?

Chi-Square tests are appropriate for nominal-level measurements in which the samples are counts of items arranged in categories. For example college students can be categorized according to whether they are freshmen, sophomores, juniors’ or seniors. If a sample of college students is taken and the number of students in each of the four categories is recorded, then those data are nominal-level measurements that can be analyzed with a chi-square test.

Chi-Square tests evaluate whether the observed frequencies in the various categories differ significantly from the frequencies that are expected under a specified set of assumptions. For such a test the chi-square test statistic: 

X2 = KΣi = 1 [(oi - Ei)2/Ei

Where oi is the observed number of items in category i and Ei is the expected number of items in category I if the null hypothesis is true. The summation is overall k categories of the data. Here the test statistics is based on the square of the differences, so that the sample values are always positive. 

When the observed frequencies are similar to the expected frequencies, then the sampling distribution of the test statistics is approximately a chi-square distribution, which is a family of distribution with the single parameter, degrees of freedom v. For small degrees of freedom (v<10), then X2 distribution is skewed to the right. But, as the degree of freedom increase, the distribution approaches a normal distribution. 

In the chi-square test, the degrees of freedom are set by the number of categories rather than by sample size. Sample size is important; however it is helpful in determining closely the chi-square distribution of the test statistic; the larger the sample size; the most common requirement is that all expected frequencies should be at least 5.
Chi-Square tests have some assumptions that are as follows:

• Independent sampling
• A mutually exclusive category that is each item fits one and only one category.
• An exhaustive list of categories includes all items in the sample. 

The tests are performed on frequencies they are not appropriate for percentages or proportions.

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