What a two way table shows
A two way table arranges a group of people or objects by two categories at the same time. One category runs across the columns, the other down the rows, and each inner cell counts how many fall into that particular combination.
- Two way table
- A grid that cross-tabulates two categories. The margins (the final row and column) hold the totals for each category, and the bottom-right corner holds the grand total.
Suppose \(50\) students are asked their year group and whether they walk or ride to school. A two way table captures both facts about each student in one place, which a single list could never do so neatly.
Completing the totals
Many questions hand you a table with gaps and ask you to fill it in first. The trick is that every row and every column must add to its total, so you can always work backwards from the numbers you do have.
Here is a completed table for the \(50\) students:
| Walk | Ride | Total | |
|---|---|---|---|
| Year 4 | 12 | 8 | 20 |
| Year 5 | 18 | 12 | 30 |
| Total | 30 | 20 | 50 |
Check it: the row totals \(20 + 30 = 50\), the column totals \(30 + 20 = 50\), and both routes lead to the same grand total. When the margins agree in both directions, the table is consistent.
Fill the gaps from the totals
If a cell is missing, subtract the known cells in that row or column from its total. There is nearly always a row or column with only one gap, so start there and the rest follow.
Reading probabilities
With the table complete, a probability is the matching cell (or total) divided by the relevant total. For a student chosen at random from all \(50\), the denominator is the grand total.
A student is chosen at random from the \(50\). Find (a) the probability they walk, and (b) the probability they are a Year 4 student who rides.
Watch the denominator
The one thing to stay alert to is which total sits on the bottom. "From the whole group" uses the grand total. But a question that says "given they are in Year 5" restricts you to that row, so the Year 5 total becomes the denominator instead.
For example, the probability a student rides given they are in Year 5 uses only the Year 5 row: \(\dfrac{12}{30} = \dfrac{2}{5}\), not \(\dfrac{12}{50}\). The wording "given" or "of the Year 5 students" is your signal to switch the denominator.
Where this is assessed
Completing the table and computing probabilities is Criterion A (Knowing and understanding). Presenting the grid with headings and correct totals is Criterion C (Communicating). Collecting real survey data and interpreting what the probabilities suggest is Criterion D (Applying in real-life contexts).
Check yourself
Use the \(50\)-student table above for each of these.
1. Find the probability a randomly chosen student is in Year 5. +
The Year 5 row totals \(30\) students out of \(50\). \(P(\text{Year 5}) = \dfrac{30}{50} = \) \(\dfrac{3}{5}\).
2. Find the probability a student walks and is in Year 4. +
The Year 4 and Walk cell holds \(12\) students. \(P(\text{Year 4 and walk}) = \dfrac{12}{50} = \) \(\dfrac{6}{25}\).
3. Given a student walks, find the probability they are in Year 4. +
"Given they walk" restricts to the Walk column, which totals \(30\). Of those, \(12\) are in Year 4. So the probability is \(\dfrac{12}{30} = \) \(\dfrac{2}{5}\).
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