Determining Degrees of Freedom

One of the prerequisites for using chi square test is that we should calculate the number of degrees of freedom for the contingency table. To understand degrees of freedom, let us look at this example. We know that 3 + 4 + 5 = 12, 2 + 6 + 4 = 12 and 6 + 1 + 5 = 12. If we express these equations in the manner shown below, what value will the blanks take so that we get a total of twelve in all the equations.

3 + 4 + ___ = 12, 2 + 6 + ___ = 12, 6 + 1 + ___ = 12

The possible values are 5, 4 and 5. Here the point to note is that after deciding what the first two numbers are going to be we lose the independence to assign the number at our discretion which the third value should take. It becomes mandatory that the number should be 5, 4 and 5 but no other number. In this case the number of degrees of freedom which we enjoy is two, as we have discretion to decide only the first two numbers.

For the contingency tables, it is calculated as follows:

Number of Degrees of Freedom = (Number of rows - 1)  (Number of columns - 1)

The number of rows and the number of columns in our example are 2 and 4. Therefore the number of degrees of freedom is given by

                   (2 - 1)(4 - 1) = 1 x 3 = 3

We use this value to look up the value of the chi square statistic. The value in the table is found at the intersection point of number of degrees of freedom (calculated from the contingency table) and the corresponding level of significance. This value denotes the area to the right end of the tail. In our problem, at α = 10% and 3 degrees of freedom the area under the right tail is 0.10 and the chi square statistic is 6.251. Since the sample chi square statistic does not fall in the acceptance region, we reject the null hypothesis.