IV. DISPLAYING CATEGORICAL DATA  

Introduction

A picture is said to be worth a thousand words.  Tables and graphs, properly designed, can provide clear pictures of patterns contained in many thousands of pieces of information.   In this topic, we will describe several ways of displaying information about categorical variables in tabular and graphic form.  In later topics, ways of displaying information about continuous variables will be explained.

Frequency Tables

A frequency table (or frequency distribution) displays numbers and percentages for each value of a variable.  It is useful for categorical variables (that is, those with values falling into a relatively small number of discrete categories, such as party affiliation, religious affiliation, or region of a country) rather than for continuous variables (such as age in years or income in dollars). 

The following frequency distribution shows the number of seats in the U.S. House of Representatives by region of the country:

The first column in the table provides a label for each category of the variable.  The second and third columns show, respectively, the number and percent of cases in each category for all cases.  The fourth column shows the percent in each category after eliminating cases for which we do not have information (missing data).  Since we know the location of every congressional district, the fourth column is identical to the third in this case.   The last column shows the cumulative percentages as one goes from the first to the last category.  Note that this last column makes sense only if the values of the variable can be meaningfully ranked.  In other words, cumulative frequencies assume at least ordinal level measurement.  The numbers in this column make no sense in this example, since it wouldn't be meaningful to say that 42.1 percent of seats "are held by the Midwest or less." 

Contingency Tables

A contingency table (also called a crosstabulation, or crosstab for short) displays the relationship between one categorical variable and another.  It is called a “contingency table” because it allows us to examine a hypothesis that the values of one variable are contingent (dependent) upon those of another.

The following crosstabulation shows the relationship between region and party affiation in the House of Representatives:

There are several important things to notice about the way in which the table has been set up:

• The table consists of a matrix of rows and columns.  Since there are two rows and four columns, it would be referred to as a “two by four” table.  A table with four rows and three columns would be referred to as a “four by three” table.  (Note that the number of rows is always given first.)  The values of one of the variables are placed in the rows and those of the other in the columns.   Usually, as in this case, the dependent variable is the row variable and the independent variable is the column variable.  There is, however, no requirement that this be the case.  If it fits the available space better to put the dependent variable in the columns, go ahead and do so.
• The number of cases in each cell of the table has been converted to a percent.  In this example Democrats dominate in the Northeast and hold a narrower edge in the West.; the Midwest is quite evenly split, while the GOP holds most Southern seats.  By converting numbers of cases to percents, we facilitate making comparisons among regions, despite fairly large differences in the number of seats each holds, since each totals to the same 100%.
• In this instance, we have percentaged down the columns rather than across the rows.  Always percentage in the direction of the independent variable.  This is necessary because we are testing the hypothesis that different categories of the independent variable will tend to have different values for the dependent variable. 
• The table includes information on the number of cases (N) on which the percentages are based.   This gives us a rough idea of the reliability of the percentages.[1]  If there were only a few cases in one of the categories, we would view the findings for this category with skepticism.

Do not let all the trees get in the way of seeing the forest.  In interpreting a crosstab, it is crucial to look for the overall pattern.  In this case, the table shows that there are substantial regional differences in party strength..  Look first for the overall pattern, and don’t allow yourself to get bogged down in the pursuit of trivia.