The χ² test is one of the most popular statistical inference procedures today. It is applicable to a very large number of problems in practice which can be summed up under the following heads.

1. χ² test as a test of independence. With the help of χ² test we can find out whether two or more attributes are associated or not. Suppose we have N observations classified according   some attributes we may ask whether the attributes are related or independent. Thus, we can find out whether quinine is effective in controlling fever or not, whether there is any association between marriage and failure, or eye color of husband and wife. In order to test whether or not the attributes are associated we take the null hypothesis what there is no association in the attributes under study or, in other words, the two attributes are independent. If the calculated value of χ² is less. Then the table value at a certain level of significance (generally 5% level). We say that the results of the experiment provide no evidence for doubting the hypotheses or, in other words the hypothesis that the attributes are not associated holds good. 

2. χ² test as a lest of goodness of fit, χ² test is very popularly known as test of goodness of fit for the reason that it enables us to ascertain how appropriately the theoretical distributions such as binomial. Poisson, normal, etc. fit empirical distributions, le those obtained from sample data

Precision can be secured by applying the χ² test.

1. Null and alternative hypotheses are established, and a significance level is selected fo rejection of the null hypotheses. 

2. A random sample of observations is drawn from a relevant statistical population.

3. A set of expected or theoretical frequencies is derived under the assumption that the null hypothesis is true. The generally takes the form of assuming the particular probability distribution is applicable to the statistical population under consideration.

4. The observed frequencies are compared with the expected or theoretical frequencies.

5. If the calculated value of χ² is less than the table value at a certain level of significance (generally 5% level) and for certain degrees of freedom the fit is considered to be good, the divergence between the actual and expected frequencies is attributed to fluctuations of simple sampling. On the other hand, if the calculated value of χ² is greater than table value. The fit is considered to be poor. It cannot be attributed to fluctuations of simple sampling rather it is due to the inadequacy of the theory to fit the observer facts.

It should be borne in mind that in repeated sampling too good a fit is just as likely as too bad a fit. When the computed chi-square value is too close to zero, we should suspect the possibility that two sets of frequencies have been manipulated in order to force them to agree and, therefore, the design of our experiment should be thoroughly checked.

For determining degrees of freedom for a chi-square goodness-of-fit test, we should first count the number of classes (symbolized is going to be made, to the total, we have to impose further restrictions the degrees of freedom would be reduced by the number of restrictions  imposed. For example, for normal distribution, we have seven classes of observed frequencies k = 7 and u =7 - 1 = 6. If we have to use the sample mean as an estimate of the population with only 5. And if we have to use the sample standard deviation to estimate the population standard deviation, we will have to subtract one more degree the (K -1 ) rule and then subtract an additional degree of freedom for each population parameter that has to be estimated from the sample data.

3. χ² test as a test of homogeneity the χ2  test of homogeneity is an extension of the chi-square test of independence. Tests of homogeneity are designed to determine whether two or more independent random samples are drawn from the same population or from different populations. Instead of one sample as we use with independence problem we shall not have two or more samples. For example we may be interested in finding out whether or not university students of various levels, under graduate quirked by their professors too much work, right amount of work or too little work. We shall take the hypothesis that the three samples come from the same population: that is, the three classifications are home plenteous in so far as the opinion of three different groups of students about the amount of work required by their professors is concerned. This also means there exists no difference in opinion among the three classes of people on the issue. 

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