Demystifying Degrees of Freedom in Chi-Square Tests: A Comprehensive Guide

Chi-square tests are invaluable tools for analyzing categorical data, offering insights into goodness-of-fit, independence, and homogeneity. In this blog post, we'll discuss the formulae for calculating degrees of freedom in different chi-square tests, accompanied by practical examples.

Formula for Degrees of Freedom in Different Chi-Square Tests

Goodness-of-Fit Test:
$\text{Degrees of Freedom (d.f.)}=�-1$ Here, $�$ represents the number of observations or categories. Let's illustrate this with a hypothetical scenario:

Color	# of pieces in a bag	Expected # of pieces
Red	28	25
Yellow	23	25
Blue	18	25
Green	31	25

The table above is a hypothetical breakdown of the color and number of pieces of skittles that one can expect in a bag with 100 pieces of candy. When utilizing the good- ness-of-fit test our degrees of freedom will be based on the category we are observing which in this case is candy color, therefore d.f. = 4 –1 = 3. Once we have the degrees of freedom, we can use the significance level (α) along with the d.f to correctly deduce the Chi-square critical value from a Chi-square distribution table. 

Test of Independence and Test of Homogeneity:
$\text{Degrees of Freedom (d.f.)}=(�1-1)\times (�2-1)$ Here, $�1$ represents the number of observations for the first variable, and $�2$ represents the number of observations for the second variable.

Consider a scenario where we want to explore if gender correlates with favorite color:

	Blue	Yellow	Pink	Orange	Total
Male	24	32	28	26	110
Female	15	40	16	39	110
Total	39	72	44	65	N = 220

 

The table above shows how to calculate the degrees of freedom when using the test of independence or homogeneity. In the scenario below, we want to figure out if gender correlates with favorite color. In calculating the degrees of freedom some may write the equation as d.f.= (rows-1) x (columns-1). Below our first/row variable is gender and the second/column variable is the color, inputting the information into the equation we get: d.f = (2-1) x (4-1) = 3

Author: Blessing

References: Chi-Square Test of Independence. www.jmp.com. https://www.jmp.com/en_us/statistics-knowledge-portal/chi-square-test/chi-square-test-of-independence.html 

Chi-Square Goodness of Fit Test. www.jmp.com. https://www.jmp.com/en_us/statistics-knowledge-portal/chi-square-test/chi-square-goodness-of-fit-test.html