Two-way tables

EXAMPLE 1 Racial differences in six-year graduation rates

One measurement of student success is the percentage completing a bachelor’s degree within six years. The National Center for Education Statistics publishes reports. The following two-way table is representative of the percentages of students completing a bachelor’s degree within six years for four racial/ethnic groups:

Race/ethnicity White Black Hispanic Asian Total
Graduated 7380 1185 1503 824 10,892
Did not graduate 4620 1815 1497 376 8308
Total 12,000 3000 3000 1200 19,200

How should we evaluate the information in this table?

Graduation status (completing a bachelor’s degree within six years or not) and race of students are both categorical variables. That is, they place individuals into categories but do not have numerical values that allow us to describe relationships by scatterplots, correlation, or regression lines. To display relationships between two categorical variables, use a two-way table like the table of graduation status and race of applicants. Graduation status is the row variable because each row in the table describes one of the possible admission decisions for a student. Race is the column variable because each column describes one of the racial/ethnic groups. The entries in the table are the counts of students in each graduation status–by–race class.

How can we best grasp the information contained in this table? First, look at the distribution of each variable separately. The distribution of a categorical variable says how often each outcome occurred. The “Total” column at the right of the table contains the totals for each of the rows. These row totals give the distribution of graduation status for all students, for all racial/ethnic groups combined. The “Total” row at the bottom of the table gives the distribution of race for students, with both categories of graduation status (“Graduated” or “Did not graduate”) combined. It is often clearer to present these distributions using percentages. We might report the distribution of race as

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The two-way table contains more information than the two distributions of graduation status alone and race alone. The nature of the relationship between graduation status and race cannot be deduced from the separate distributions but requires the full table. To describe relationships among categorical variables, calculate appropriate percentages from the counts given.

EXAMPLE 2 Racial differences in six-year graduation rates

Because there are only two categories of graduation status, we can see the relationship between race and graduation status by comparing the percentages of those who completed a bachelor’s degree within six years for each race:

More than 60% of the White students and almost 70% of the Asian students completed a bachelor’s degree within six years, but only about 40% of Black students and 50% of Hispanic students completed a bachelor’s degree within six years.

In working with two-way tables, you must calculate lots of percentages. Here’s a tip to help decide what fraction gives the percentage you want. Ask, “What group represents the total that I want a percentage of ?” The count for that group is the denominator of the fraction that leads to the percentage. In Example 2, we wanted the percentage of each racial/ethnic group who completed a bachelor’s degree within six years, so the counts of each race form the denominators.

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