Some data are categories, not numbers: yes or no, 7th or 8th grade. A two-way table counts how many fall into each pair of categories. Reading it uses the same thinking as compound probability.
Why it works
Counts can mislead when the groups have different sizes. If 30 of 50 seventh graders and 30 of 100 eighth graders play a sport, the counts match but the rates are very different: 60% and 30%. Comparing relative frequencies, the share within each group, puts the groups on equal footing.
Key idea
Add row and column totals. To compare groups, divide each count by its row total (or column total) to get relative frequencies. If the groups’ percents differ a lot, there is an association between the two variables.
| Plays a sport | Does not | Total | |
|---|---|---|---|
| 7th grade | 30 | 20 | 50 |
| 8th grade | 12 | 28 | 40 |
| Total | 42 | 48 | 90 |
Worked examples
Worked example: Use the totals
In the table, how many students in all play a sport, and how many 8th graders were surveyed?
- The column total for “Plays a sport” is $30 + 12 = 42$.
- The row total for 8th grade is $12 + 28 = 40$.
Answer: 42 and 40
Worked example: Relative frequencies
What percent of 7th graders play a sport? Of 8th graders?
- 7th grade: $\frac{30}{50} = 60\%$.
- 8th grade: $\frac{12}{40} = 30\%$.
Answer: 60% and 30%
Worked example: Is there an association?
Is grade level associated with playing a sport in this survey?
- 60% of 7th graders play, but only 30% of 8th graders.
- The percents are far apart, so yes: in this group, 7th graders were more likely to play a sport.
Answer: Yes
Common mistake
Comparing counts instead of percents. “30 seventh graders play but only 12 eighth graders” sounds convincing, but the groups have different sizes. Always divide by the group total before comparing.