Two-Way Tables

What is a two-way table?

A two-way table sorts data by two features at once. Instead of just counting how many students like each sport, you can also split them by, say, gender at the same time. One feature goes across the columns, the other goes down the rows, and totals sit on the edges.

This is more powerful than a simple frequency table, because it lets you ask questions about combinations: How many boys chose football? Did more girls than boys choose swimming?

Reading the parts

Here is a survey of 40 students asked which after-school club they joined.

	Art	Sport	Music	Total
Boys	5	9	4	18
Girls	8	7	7	22
Total	13	16	11	40

Every number tells a precise story:

• The 18 is a row total: all the boys.
• The 16 is a column total: everyone who chose Sport.
• The 7 where the Girls row meets the Music column means 7 girls chose Music.
• The bottom-right 40 is the grand total: every student.

The golden rule: the row totals add to the grand total (18 + 22 = 40) and the column totals add to the grand total (13 + 16 + 11 = 40). Both checks must give 40.

Worked example: filling in the gaps

Exam questions love to leave blanks. Suppose you are given this incomplete table for 50 people travelling to work.

	Walk	Bus	Total
Adults	12	?	30
Children	?	6	?
Total	26	?	50

Work through it step by step using the totals.

1. Adults, Bus: the Adults row total is 30, and 12 walk, so 30 − 12 = 18 take the bus.
2. Children, Walk: the Walk column total is 26, and 12 are adults, so 26 − 12 = 14 children walk.
3. Children total: 14 walk + 6 bus = 20 children. (Check: 30 adults + 20 children = 50. ✓)
4. Bus column total: 18 + 6 = 24. (Check: 26 walk + 24 bus = 50. ✓)

Completed table:

	Walk	Bus	Total
Adults	12	18	30
Children	14	6	20
Total	26	24	50

Pulling out a probability

Because a two-way table holds every count, it is perfect for probability. Using the completed travel table, pick one of the 50 people at random.

• P(takes the bus) = bus total ÷ grand total = 24/50 = 12/25.
• P(is a child who walks) = 14/50 = 7/25.
• P(is an adult) = 30/50 = 3/5.

Always read the right total off the table — the denominator is usually the grand total unless the question narrows it down.

Activity: survey your class

1. Choose two features, such as year group (Year 7 or Year 8) and favourite drink (water, juice, milk).
2. Draw the empty grid with rows, columns and Total cells.
3. Tally each person into exactly one cell.
4. Fill in the totals and check the row totals and column totals both reach your grand total.
5. Ask a probability question, such as "What is the chance a random person is a Year 7 who prefers juice?"

Why this matters

Two-way tables organise messy real data so you can compare groups fairly and calculate probabilities with confidence — skills used in surveys, science experiments and medical trials. They connect closely to probability basics and to sorting ideas you met in Venn diagrams and sets.