MISSION · Data and probability
Two-Way Tables
Where the rows and columns meet, and what the totals tell you
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
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LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
A two-way table sorts the same group of people by two characteristics at the same time, so every cell answers a question about both.
Reading one
| Walks | Cycles | Bus | Total | |
|---|---|---|---|---|
| Year 7 | 32 | 18 | 25 | 75 |
| Year 8 | 28 | 12 | 35 | 75 |
| Total | 60 | 30 | 60 | 150 |
Each inner cell is a count of people who are both things: 18 students who are in Year 7 and cycle.
The outer row and column are the totals, and they answer single-characteristic questions. 30 students cycle altogether, regardless of year. 75 students are in Year 7, regardless of transport.
The grand total in the corner is everybody, and it must equal the sum of the row totals and the sum of the column totals. That’s the first check to run on any table you’re given — if the margins don’t agree, something is wrong before you start.
Filling in the gaps
Exam questions usually give you a table with holes. Every missing value is recoverable, because each row and each column must add to its total.
If a row shows 32 and 18 against a total of 75, the third entry is \(75 - 50 = 25\). Work around the table filling whatever has only one unknown left, and it unlocks in a few steps.
Probability from a table
Three different questions, three different denominators — and this is where the marks are lost.
P(cycles) — out of everybody: \(\frac{30}{150} = 0.2\).
P(Year 7 and cycles) — out of everybody, but only the cell: \(\frac{18}{150} = 0.12\).
P(cycles given Year 7) — out of Year 7 only, so the row total is the denominator: \(\frac{18}{75} = 0.24\).
The word given shrinks the group you’re counting within. Notice the last two use the same 18 and produce different answers entirely, because they’re answering different questions.
Spotting a relationship
Compare the proportions across rows rather than the raw counts.
Year 7: \(\frac{25}{75} = 33\%\) take the bus. Year 8: \(\frac{35}{75} = 47\%\). The percentages differ, which suggests year group and transport are related — older students travel further, perhaps.
If the proportions had matched, the two characteristics would be independent, and knowing one would tell you nothing about the other. Comparing raw counts instead only works when the row totals happen to be equal, which they usually aren’t.
Common mix-ups
- Using the grand total when the question says “given”.
- Comparing counts across rows of different sizes. Convert to proportions.
- Adding a row and column total together, which counts the overlapping cell twice.
- Forgetting the margins must reconcile.
Check yourself
Using the table above, find P(walks given Year 8) and P(Year 8 and walks).
Given Year 8, the denominator is that row: \(\frac{28}{75} \approx 0.37\). For “and”, the denominator is everybody: \(\frac{28}{150} \approx 0.19\).
LEVEL 3 · REFLECT
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