Two-Way Tables –

Two-way tables are a staple of GCSE Maths papers at both Foundation and Higher tier across AQA, Edexcel and OCR. They organise data into rows and columns so you can read off values, fill in missing entries and calculate probabilities. Whether the question asks you to complete a table, find a total or work out the probability of a randomly chosen item, the approach is the same — use the row and column totals to work backwards. This guide covers the method clearly, provides worked examples at every level and flags the errors examiners see most often. For a full overview of every topic, see our complete GCSE Maths topics list.

What Is a Two-Way Table?

A two-way table displays data that has been classified by two categories at the same time. One category runs across the top (columns) and the other runs down the side (rows). Each cell shows the frequency for that combination of categories, and the margins show the row and column totals. The bottom-right cell holds the grand total.

Key Formulas

Step-by-Step Method

1. Read the table carefully — identify which category is on the rows and which is on the columns.
2. Use row and column totals to find any missing values: subtract the known cells from the total.
3. Check your work — every row must add to its row total, every column must add to its column total, and all row totals must add to the grand total.
4. To find a probability, divide the relevant frequency by the grand total.

Worked Example 1 — Foundation Level

Question: 120 students were asked whether they prefer football, tennis or swimming. The results are shown below.

Complete the table.

Working:

Boys swimming = 60 − 28 − 14 = 18.

Girls football = 60 − 22 − 18 = 20.

Swimming total = 18 + 18 = 36.

Check: 48 + 36 + 36 = 120. Correct.

Answer: Boys swimming = 18, Girls football = 20, Swimming total = 36.

Worked Example 2 — Higher Level

Question: Using the completed table above, a student is chosen at random. Find (a) P(the student is a girl who prefers tennis), (b) P(the student prefers football).

Working:

(a) Girls who prefer tennis = 22. Grand total = 120.

P(girl and tennis) = 22/120 = 11/60.

(b) Football total = 48.

P(football) = 48/120 = 2/5.

Answer: (a) 11/60 (b) 2/5.

Worked Example 3 — Exam Style

Question: A survey of 200 employees records whether they travel by car, bus or bicycle and whether they work full-time or part-time.

(a) Complete the table. (b) One employee is chosen at random. Find the probability that this person is a part-time worker who travels by car.

Working:

Full-time bicycle = 130 − 68 − 42 = 20. Part-time car = 70 − 28 − 22 = 20. Car total = 68 + 20 = 88. Bicycle total = 20 + 22 = 42. Check: 88 + 70 + 42 = 200. Correct.

(b) P(part-time and car) = 20/200 = 1/10.

Answer: (a) Full-time bicycle = 20, Part-time car = 20, Car total = 88, Bicycle total = 42. (b) 1/10.

Common Mistakes

• Adding across then down and getting different totals. Always cross-check: row totals summed must equal column totals summed and both must equal the grand total.
• Confusing a joint frequency with a marginal total. The cell in the body of the table gives the count for a specific combination; the margin gives the overall category count.
• Giving a probability as a ratio or "1 in 5". Always express probability as a fraction, decimal or percentage.

Exam Tips

• Work systematically: fill in the easiest missing values first, then use totals to find the rest.
• If the question says "a person is chosen at random", you must divide by the grand total, not a row or column total (unless told otherwise).
• Two-way table questions often lead into probability — be ready to simplify your fraction.
• For harder probability from tables, see probability from two-way tables.

Practice Questions

Q1 (Foundation): 80 people were asked whether they prefer tea or coffee. 35 are male; 20 males prefer tea; 30 females prefer coffee. Complete the table and find the total who prefer tea.

Q2 (Foundation): Using Q1, find the probability that a randomly chosen person is a female who prefers tea.

Q3 (Higher): 150 students study French, Spanish or both. 90 study French, 85 study Spanish. How many study both? (Hint: use a two-way table or addition principle.)

Practise two-way tables and more for free on GCSEMathsAI.

Related Topics

• Probability from Two-Way Tables
• Probability Basics and Relative Frequency
• Bar Charts, Pie Charts and Pictograms

Summary

A two-way table organises data by two categories using rows and columns. Row and column totals let you find missing values by subtraction. To calculate a probability from a two-way table, divide the relevant cell frequency by the grand total. Always cross-check that every row and column adds up correctly, and express probabilities as simplified fractions, decimals or percentages. Two-way tables appear frequently on Foundation and Higher papers and often lead into probability questions.