Chi-Square Test

The various tests of significance such as t, F, and Z are based on the assumption that samples are drawn from normally distributed populations. These tests are known parametric tests because they require assumptions about the population parameters. There may be situations in which it is not possible to make any rigid assumption about the distribution of the population from which samples are drawn. This has resulted in the development of non-parametric tests. These tests are distribution-free and do not make any assumptions about the population parameters. The Chi-square test of independence and goodness of fit is a prominent example of the non-parametric tests. Here, we are limiting our discussion to the Chi-square test.

Chi-Square Defined:

The Chi-square test is one of the simplest and most commonly used non-parametric tests in statistical work. The Greek Letter x2 is used to denote this test. The quantity x2 describes the magnitude of discrepancy theory and observation.

Degrees of Freedom:

Determine the degrees of freedom in making a comparison between the calculated value of x2 and the table value. Therefore, it is very important to understand what do we mean by degree of freedom. it means the number of classes to which values can be assigned arbitrarily or at will without violating the restriction placed.

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Chi-Square Distribution:

The sampling distribution of the Chi-square statistic, X2 can be closely approximated by a continuous curve known as chi-square distribution. This distribution has only one parameter v, the number of degrees of freedom. The probability function of Chi-square depends upon degrees of freedom v. As v changes, the probability function of Chi-square also changes. For very small numbers of degrees of freedom, the Chi-square distribution is severally skewed to the right. As the degree of freedom increases, the curve, rapidly becomes more symmetrical until the number reaches large values, at this point this distribution can be approximated by the normal distribution.

Conditions for the Validity of Chi-square Test:

The Chi-square test statistics can be used only if the following conditions are satisfied:

  • N the total number of frequencies, should be reasonably large, say greater than 50.
  • The sample observations should be independent. This implies that no individual item should be included twice or more in the sample.
  • The constraints on the cell frequencies. if any should be linear.
  • No theoretical cell frequency should be small. If expected frequencies are small then the value of x2 would be overestimated. This will result rejection of many null hypothesis. Small is a relative term. Preferably each theoretical frequency should be larger than 10, but in any case not less than 5. If any theoretical frequency is less than 5 then we cannot apply x2 test as such. In that case, we use the technique of polishing which consists of adding the frequency which is less than 5 with preceding or succeeding frequency so, that the resulting sum is greater than 5 and adjust the degree of freedom accordingly.
  • The given distribution should not be replaced by relative frequencies or proportions but data should be given in original units.


Application of Chi-Square Test:

The Chi-square distribution has a number of applications. The most important applications are enumerated below:

  • Chi-square test of goodness of fit.
  • Chi-square test for independence of attributes ad.
  • Chi-square test as a test of Homogeneity.