Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Every summary statistic for two categorical variables is a relative frequency: a count ÷ a total.
- Joint: one cell as a share of all individuals (“A and B”).
- Marginal: a row or column total as a share of all individuals (one variable on its own).
- Conditional: a cell as a share of one row or one column (“of the A, what proportion are B”).
- A conditional distribution is the set of conditional relative frequencies within one group; it adds to 1.
- Variables are associated if the conditional distributions differ across groups.
Key relationships
| Statistic | Numerator | Denominator | Example wording |
|---|---|---|---|
| Joint | cell count | grand total | “What proportion of all customers are under 30 and use the app?” |
| Marginal | row or column total | grand total | “What proportion of all customers use the app?” |
| Conditional (row) | cell count | that row’s total | “Of the under-30s, what proportion use the app?” |
| Conditional (column) | cell count | that column’s total | “Of the app users, what proportion are under 30?” |
| Link | joint = marginal × conditional | 0.40 × 0.50 = 0.20 | |
| No association | conditional distributions all equal (and equal to the marginal distribution) |
Assumptions and conventions
- Each individual is in exactly one cell, so the joint relative frequencies add to 1.
- When one variable is explanatory, condition on it: compare the response within each explanatory group.
- Small differences between sample proportions can arise by chance. Describe the size of a difference; a formal test comes later in the course.
Mistakes to avoid
- Swapping the direction of a conditional relative frequency.
- Dividing by the grand total when the question restricts to a group.
- Judging association from counts or joint values instead of conditional proportions.
- Calling the largest share “most” when it is under 0.50.
- Claiming cause from observational data.
Quick self-check
Use this table of 100 fictional people: Group A has 30 Yes and 20 No; Group B has 10 Yes and 40 No.
- What proportion of all people are in Group A and said Yes? (30 ÷ 100 = 0.30, a joint relative frequency)
- What proportion of all people said Yes? (40 ÷ 100 = 0.40, a marginal relative frequency)
- Of the Group A people, what proportion said Yes? Of the Yes answers, what proportion came from Group A? (30 ÷ 50 = 0.60; 30 ÷ 40 = 0.75)
- Are group and answer associated? (Yes: 0.60 of Group A said Yes against 10 ÷ 50 = 0.20 of Group B)
Next: practice questions.