# 14: Chi-square

• 14.1: Categories and Frequency Tables
Our data for the χ2 test are categorical, specifically nominal, variables. Recall that nominal variables have no specified order and can only be described by their names and the frequencies with which they occur in the dataset. Thus, unlike our other variables that we have tested, we cannot describe our data for the χ2 test using means and standard deviations. Instead, we will use frequencies tables.
• 14.2: Goodness-of-Fit
The first of our two χ² tests assesses one categorical variable against a null hypothesis of equally sized frequencies. Equal frequency distributions are what we would expect to get if categorization was completely random. We could, in theory, also test against a specific distribution of category sizes if we have a good reason to (e.g. we have a solid foundation of how the regular population is distributed), but this is less common, so we will not deal with it in this text.
• 14.3: χ² Statistic
The calculations for our test statistic in χ² tests combine our information from our observed frequencies ( O ) and our expected frequencies ( E ) for each level of our categorical variable. For each cell (category) we find the difference between the observed and expected values, square them, and divide by the expected values. We then sum this value across cells for our test statistic.
• 14.4: Pineapple on Pizza
There is a very passionate and on-going debate on whether or not pineapple should go on pizza. Being the objective, rational data analysts that we are, we will collect empirical data to see if we can settle this debate once and for all. We gather data from a group of adults asking for a simple Yes/No answer.
• 14.5: Contingency Tables for Two Variables
The goodness-of-fit test is a useful tool for assessing a single categorical variable. However, what is more common is wanting to know if two categorical variables are related to one another. This type of analysis is similar to a correlation, the only difference being that we are working with nominal data, which violates the assumptions of traditional correlation coefficients. This is where the χ² test for independence comes in handy.
• 14.6: Test for Independence
The χ² test performed on contingency tables is known as the test for independence. In this analysis, we are looking to see if the values of each categorical variable (that is, the frequency of their levels) is related to or independent of the values of the other categorical variable. Because we are still doing a χ² test, which is nonparametric, we still do not have mathematical versions of our hypotheses. The actual interpretations of the hypotheses are quite simple.
• 14.7: College Sports
• 14.E: Chi-square (Exercises)