ACT® · Math: Statistics and Probability — Describing Data
Two-Way Frequency Tables and Categorical Association
A two-way table sorts data by two categories at once, so you can compute percentages within a row, within a column, or over the whole table. Comparing those percentages tells you whether the two categories are actually linked.
A two-way table lists one categorical variable down the rows and another across the columns, with counts filling the grid. Say 400 students are split by grade (freshman, sophomore, junior, senior) and by whether they play a sport. Neither list alone tells you whether juniors play sports at a different rate than seniors do. The table does, because each cell shows exactly how many students share both traits at once.
Every cell can be read three ways. Divide a cell by its row total and you get the percent of that row's category with that column's trait: the percent of juniors who play a sport. Divide by the column total instead and you flip the question: the percent of sport-players who are juniors. Divide by the grand total and you get the share of the entire surveyed group in that cell. Same number, three totally different meanings depending on what you divide by.
To check association, compute the same conditional percentage across every row (or every column) and compare. If the percent of students playing a sport is roughly the same for freshmen, sophomores, juniors and seniors, grade and sport are not associated. If the percents swing noticeably, they are. This is the same logic that sits under scatterplots and residuals: there you look for a pattern between two numerical variables, here you look for a pattern between two categorical ones. A third variable can also split a table apart and change or erase an association that looked real in the combined data.
Key terms
- Two-way table
- A grid showing counts for combinations of two categorical variables at once.
- Row total
- The sum of all counts across one row, used to find what percent of that category has each trait.
- Column total
- The sum of all counts down one column, used to flip the question and find what percent of that trait falls in each category.
- Conditional percentage
- A percent calculated within one row or column rather than over the whole table.
- Association
- A pattern where the conditional percentages change noticeably across categories, showing the two variables are linked.
Watch it explained
- How it worksHow do row totals, column totals and the grand total pull three different percentages out of one table?
- How it worksWalk me through deciding whether two categorical variables in a two-way table are associated.
- Worked exampleFind the percent of juniors who play a sport when 120 juniors are surveyed and 78 of them play one.
- Common mistakeWhy isn't a big count in one cell of a two-way table evidence that the two variables are related?
- Going furtherWhat happens to an association when a two-way table is broken apart by a third variable?
Check yourself
End of topic.