• Categorical variables (or nominal variables) are variables whose values are qualitative labels / names / categories. For example, the outcome of flipping a coin is either head or tail; the response to the question "What is your gender?" can be among {female, male, non-binary, prefer not to say}; a choice of background color of presentation slides can be white, black, yellow, blue, green, etc.

• Categorical variables are said to have binary outcomes if there are only TWO possible outcomes, e.g., outcome of a coin flip: {head, tail}, answer to a multiple-choice exam question: {correct, incorrect}, result of a penalty kick in football: {score, miss}. For a fixed number of observations of binary outcomes (e.g., 10 coin flips), we can conduct statistical tests on the underlying proportion or probability of one outcome (e.g., to test whether p(head) = 0.5). Such tests are called tests of proportions.

• For categorical variables with 3 or more outcomes, we can conduct statistical tests on whether the distribution of counts for different outcomes "fit" a certain underlying outcome distribution. For this purpose, we can use the goodness-of-fit chi-square test.

• Chi-square test can also be used for testing whether the association between two categorical variables exists or not for example, if there's any association between gender (male or female) and favourite colour (e.g., red/yellow/green, etc.). If there's no association between two categorical variables, the allocation of each combination should be even (or weighted even). If there's an association, the actual, observed allocation of each combination will deviate from the even one. The chi-square test of such difference is the test of independence.

The goodness of fit test is applied when you have one categorical variable with two or more values from a single population.

The null hypothesis (H0 ) assumes that there is no significant difference between the observed and the expected value. The alternative hypothesis (H1) assumes that there is a significant difference between the observed and the expected value.

Conclusion/ Interpretation (APA format):

Students' preferences for living hostel was not equally distributed in the population, X2 (3, N = 1000) = 266, p < .001.

The test of independence compares two sets of data in a contingency table to see if there is an association. It can only assess the associations, but cannot provide any inferences about causation.

The data must meet the following requirements:

• Large random sample size

  • The expected frequency of each category must be at least 5

• Two categorical variables

• Two or more groups for each variable

• Independence of observations

  • There is no relationship between the subjects in each group.

  • The categorical variables are not "paired" in any way (e.g. pre-test/post-test observations)

The null hypothesis (H0 ) assumes that two variables are not associated with each other. The alternative hypothesis (H1) assumes that two variables are associated with each other.

Expected counts = Column totals X Row totals / Grand total for each of the cells

Conclusion/ Interpretation (APA format):

There is no association between faculty and relationship status, X2 (2, N = 1000) = 1.01, p = .603.