• A two-way table of counts organizes data about two categorical variables. Values of the row variable label the rows that run across the table, and values of the column variable label the columns that run down the table. Two-way tables are often used to summarize large amounts of data by grouping outcomes into categories.
• The joint distribution of the row and column variables is found by dividing the count in each cell by the total number of observations.
• The row totals and column totals in a two-way table give the marginal distributions of the two variables separately. It is clearer to present these distributions as percents of the table total. Marginal distributions do not give any information about the relationship between the variables.
• To find the conditional distribution of the row variable for one specific value of the column variable, look only at that one column in the table. Find each entry in the column as a percent of the column total.
• There is a conditional distribution of the row variable for each column in the table. Comparing these conditional distributions is one way to describe the association between the row and the column variables. It is particularly useful when the column variable is the explanatory variable. When the row variable is explanatory, find the conditional distribution of the column variable for each row and compare these distributions.
• Bar graphs are a flexible means of presenting categorical data. There is no single best way to describe an association between two categorical variables.
• We present data on three categorical variables in a three-way table, printed as separate two-way tables for each level of the third variable. A comparison between two variables that holds for each level of a third variable can be changed or even reversed when the data are aggregated by summing over all levels of the third variable. Simpson’s paradox refers to the reversal of a comparison by aggregation. It is an example of the potential effect of lurking variables on an observed association.