Algebra
Data and statistics
Two-way tables
17 practice questions
2 video lessons
Theory + worked examples
Theory
A two-way frequency table shows counts for two categories with row and column totals:
- Joint: a cell over the grand total.
- Marginal: a row/column total over the grand total.
- Conditional: a cell over its row or column total.
The margins (totals) give the marginal frequencies.
A two-way table with totals.
Reading the table.
From the counts:
\[P(\text{joint})=\dfrac{\text{cell}}{\text{grand total}},\quad P(\text{cond.})=\dfrac{\text{cell}}{\text{row/col total}}\]
Divide by the correct total for each type.
How to read a two-way table
- Find the relevant cell or total.
- Joint/marginal: divide by the grand total.
- Conditional: divide by the row or column total.
- Check the totals add up.
Example 1 — Joint frequency
From the table (\(40\) students), find \(P(\text{boy and yes})\).
Solution
Cell over grand total.
| \(\dfrac{12}{40}\) | \(=\) | \(0.30\) |
Example 2 — Marginal
Find \(P(\text{yes})\).
Solution
Column total over grand total.
| \(\dfrac{27}{40}\) | \(=\) | \(0.675\) |
Example 3 — Conditional
Find \(P(\text{yes}\mid\text{girl})\).
Solution
Condition on the \(20\) girls.
| \(\dfrac{15}{20}\) | \(=\) | \(0.75\) |
Example 4 — Totals check
Why must the row and column totals match?
Solution
They both add to the same grand total \(40\).
Common pitfalls
Joint uses the grand total; conditional uses a row/column total.
Condition on the correct margin.
Totals must add up to the grand total.
Frequently asked questions
What is a two-way table?
A table of counts by two categories with totals.
What is a joint frequency?
A single cell count, over the grand total for probability.
What is a marginal frequency?
A row or column total.
What is a conditional frequency?
A cell over its row or column total.
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