Chi-Square Goodness of Fit Test

The Chi-Square goodness of fit test is a non-parametric test that determines how significantly the observed value of a given phenomenon differs from the expected value. In the Chi-Square goodness of fit test, we use the term “goodness of fit” to compare the observed sample distribution with the expected probability distribution.  This test determines how well theoretical distribution (such as normal, binomial, or Poisson) fits the empirical distribution. Researchers divide sample data into intervals. Then, they compare the number of points in each interval with the expected number of points.

Procedure for Chi-Square Goodness of Fit Test:

• Set up the hypothesis:

A. Null hypothesis: The null hypothesis assumes that there is no significant difference between the observed and the expected value.

B. Alternative hypothesis: The alternative hypothesis assumes that there is a significant difference between the observed and the expected value.

• Compute the value of Chi-Square goodness of fit test using the following formula:

Where, = Chi-Square goodness of fit test O= observed value E= expected value

Degree of freedom: The degree of freedom depends on the distribution of the sample.  The following table shows the distribution and an associated degree of freedom:

Type of distribution	No of constraints	Degree of freedom
Binominal distribution	1	n-1
Poisson distribution	2	n-2
Normal distribution	3	n-3

Hypothesis testing: Hypothesis testing is similar to other tests, such as the t-test or ANOVA. Then, the calculated value of the Chi-Square goodness of fit test is compared with the table value. If the calculated value is greater than the table value, we will reject the null hypothesis and conclude that there is a significant difference between the observed and the expected frequency.  If the calculated value is less than the table value, we will accept the null hypothesis and conclude that there is no significant difference between the observed and expected value.

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