How to Use Two-Way Tables: KS3 Maths

A two-way table sorts data into rows and columns simultaneously, showing how two characteristics relate. To read one, find the row for the first category and the column for the second — where they intersect is your value. To complete a table, use the fact that every row and column must add up to its total.

What is a two-way table?

A two-way table (also called a contingency table) organises data by two variables simultaneously. For example, it might show how many students in each year group prefer each type of music, or how many survey respondents are in each gender–sport combination.

Each cell contains the count for that specific combination of row category and column category. The right-hand column and bottom row contain totals.

How do you read a two-way table?

Here is an example. 60 students were asked whether they preferred football or tennis, split by gender.

| Football | Tennis | Total
Boys | 18 | 7 | 25
Girls | 14 | 21 | 35
Total | 32 | 28 | 60

To find any value:

"How many girls prefer tennis?" — find the Girls row and the Tennis column: 21.

"How many students prefer football?" — read the Football total: 32.

"How many boys are there in total?" — read the Boys total: 25.

How do you complete a two-way table with missing values?

Use subtraction within rows and columns. Each row must sum to its row total; each column must sum to its column total.

Worked example — fill in the gaps:

| Cat | Dog | Total
Year 7 | 12 | ? | 30
Year 8 | ? | 15 | ?
Total | 27 | ? | 60

Step 1: Year 7 dog owners = 30 − 12 = 18.
Step 2: Total dog owners = 18 + 15 = 33.
Step 3: Total cat owners already given: 27. Check: 27 + 33 = 60. ✓
Step 4: Year 8 cat owners = 27 − 12 = 15.
Step 5: Year 8 total = 60 − 30 = 30. Check: 15 + 15 = 30. ✓

Completed table:

| Cat | Dog | Total
Year 7 | 12 | 18 | 30
Year 8 | 15 | 15 | 30
Total | 27 | 33 | 60

How do you find fractions and proportions from a two-way table?

Express the target cell as a fraction of the appropriate total. Decide carefully whether the question asks for a fraction of the overall total, a row total, or a column total.

Using the pet table above:

Fraction of all students who are Year 7 cat owners = 12/60 = 1/5.

Fraction of Year 8 students who prefer dogs = 15/30 = 1/2.

Fraction of dog owners who are in Year 7 = 18/33 = 6/11.

How do you use a two-way table for probability?

If a student is picked at random, the two-way table gives you the probability directly.

Example: From the football/tennis table, one student is chosen at random.

P(girl who prefers tennis) = 21/60 = 7/20.

P(prefers football) = 32/60 = 8/15.

P(boy | prefers tennis) — read as "given that they prefer tennis, probability it is a boy" = 7/28 = 1/4.

The last example is a conditional probability: the denominator changes to the column total (28 tennis-preference students), not the grand total (60).

What mistakes do students commonly make?

Using the wrong total as the denominator. Always re-read the question. "What fraction of girls…?" uses the girls total (35), not the grand total (60).

Not checking row and column sums. After completing a table, verify that every row sums to its row total and every column sums to its column total. A single arithmetic error cascades.

Confusing "and" with "given that". P(girl AND prefers tennis) uses the grand total. P(prefers tennis GIVEN girl) uses only the girls total.

Frequently asked questions

Why are two-way tables useful in statistics?

They let you spot patterns between two categorical variables at a glance. For instance, if far more girls than boys prefer tennis, the table reveals the imbalance immediately without any further calculation. This makes them a quick tool for identifying potential associations before conducting formal statistical tests.

How do two-way tables connect to Venn diagrams?

Both represent the same information in different forms. A two-way table with two row categories and two column categories maps directly to a two-region-overlap Venn diagram. The intersection cell of the table corresponds to the overlapping region of the Venn diagram. Being able to switch between the two representations is a useful KS3 skill.

Can a two-way table have more than two categories per variable?

Yes. A table might have four year groups as rows and three subjects as columns. The same fill-in method applies: every row sums to its row total and every column sums to its column total. Work from the cells with the most given information outward to the unknown cells.

How do I know if the table totals are consistent?

The grand total (bottom-right corner) must equal both the sum of all row totals and the sum of all column totals. If row totals sum to 60 and column totals sum to 58, there is an error somewhere. This cross-check takes five seconds and catches almost all mistakes.

For Socratic statistics and data handling practice, see aitutors.me.