You have read two-way tables of counts. Counts can mislead when the groups have different sizes, so we turn them into relative frequencies, fractions or percents of a whole. The key question is always: percent of what?
| Soccer | Basketball | Total | |
|---|---|---|---|
| 9th grade | $30$ | $10$ | $40$ |
| 10th grade | $24$ | $36$ | $60$ |
| Total | $54$ | $46$ | $100$ |
- Joint: a cell over the grand total. $\frac{30}{100} = 30\%$ of all students are 9th graders who prefer soccer.
- Marginal: a total over the grand total. $\frac{40}{100} = 40\%$ of the students are 9th graders.
- Conditional: a cell over its row or column total. Of the 9th graders, $\frac{30}{40} = 75\%$ prefer soccer.
Why it works
There are $30$ 9th graders and $24$ 10th graders who prefer soccer, so it may look as if both grades like soccer about equally. But the groups differ in size. Within each grade, $75\%$ of 9th graders prefer soccer and only $\frac{24}{60} = 40\%$ of 10th graders do. Comparing conditional percents puts the groups on the same footing, and a big difference between them is evidence of an association between grade and sport.
Key idea
Name the whole before dividing: “of all students” uses the grand total, “of the 9th graders” uses that row's total, “of the soccer fans” uses that column's total. Compare groups with conditional percents.
Worked examples
Worked example: A column percent
Of the students who prefer basketball, what percent are 10th graders?
- The whole is the basketball column: $46$ students.
- $\frac{36}{46} \approx 0.783$.
Answer: About $78\%$.
Worked example: An association
Is there an association between grade and preferring soccer?
- Row percents: 9th grade $\frac{30}{40} = 75\%$; 10th grade $\frac{24}{60} = 40\%$.
- A $35$-point difference is large.
Answer: Yes: 9th graders in this survey are much more likely to prefer soccer.
Common mistake
Dividing by the grand total when the question asks about one group. “What percent of 9th graders prefer soccer?” is $\frac{30}{40}$, not $\frac{30}{100}$. Check by reading the question for the words after “of”: they name the whole.