Key takeaway: A two-way table is a grid that sorts one group of people or objects by two categorical variables at once, with counts in the cells and totals along the margins.
Grade 88.SP.A.4Statistics and Data
Definition of Two-Way Table
A two-way table (also called a contingency table) counts how many individuals fall into each combination of two categories. One variable labels the rows and the other labels the columns, so a survey of students can show grade level against the way they travel to school.
The inner numbers are joint frequencies. The totals at the right and the bottom are marginal frequencies, and the number in the bottom right corner is the grand total. Reading across a row or down a column lets you compare groups and look for an association between the two variables.
Formula and notation
Each cell holds a count. Totals check themselves:
\[\text{row totals} = \text{column totals} = \text{grand total}\]
To compare groups fairly, divide a cell by its row total or column total, not by the grand total, and write the result as a percent. A two-way table with percents is also called a relative frequency table.
Key facts
- The grand total in the corner equals both the sum of row totals and the sum of column totals.
- Marginal totals describe one variable at a time; inner cells describe both together.
- Dividing by a row or column total gives percentages for that group alone.
- Differences between row percentages hint at an association between the variables.
Two-Way Table example
A school surveys 80 students about how they get to school. Using the table above, find the percent of 8th graders who walk.
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Look at the 8th grade row: 21 walk, 19 take the bus, and the row total is 40.
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Divide the cell by the row total: \(\frac{21}{40} = 0.525\).
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Convert to a percent: \(0.525 = 52.5\%\).
About 52.5% of the 8th graders walk to school.
Common mistake
Watch out
Asked for the percent of 8th graders who walk, Lena computes \(\frac{21}{35} = 60\%\).
Why it is wrong
She divided by the number of walkers (the column total, 35), which answers a different question: what percent of walkers are 8th graders. The question names the 8th graders as the group, so use their row total: \(\frac{21}{40} = 52.5\%\).
Related terms and lessons
- Grade 8 · Two-Way TablesLessonWorksheetQuiz
- Algebra 1 · Two-Way Tables, Scatter Plots and CorrelationLessonWorksheetQuiz
Browse the glossary: A–Z list · Grade 8 vocabulary · Statistics and Data