Maths
What is a two-way table, and how do you complete and read one?
Quick answer
A two-way table sorts data by two features at once, with rows for one and columns for the other. Rows and columns add up to totals, so you can fill in gaps and find probabilities.
In one sentence
A two-way table sorts things by two features at once, one along the rows and one down the columns, so every box counts the things that have both features.
Explained
Imagine asking your class two questions: "Are you a boy or a girl?" and "Which sport do you like best?" A simple list or a tally chart can show one answer at a time. A two-way table shows both together. The rows are for one feature (boy, girl) and the columns for the other (football, swimming, tennis). Each box, called a cell, holds the number of people who match its row and its column.
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Young children meet the same idea as a sorting grid. Put shapes down the side (circle, square, triangle) and colours across the top (red, blue). The red square goes in the box where the square row meets the red column. In maths lessons the boxes usually hold numbers instead of objects.
Most two-way tables also have a total column and a total row:
- the cells in a row add up to the row total;
- the cells in a column add up to the column total;
- the row totals and the column totals both add up to the grand total, in the bottom-right corner.
These rules let you complete a table with gaps. Look for a row or column with only one missing number, fill it in, and repeat. Each new number usually lets you fill another gap.
Two-way tables are also a quick way to find a probability. If you pick one person at random, the probability of a feature is the number with that feature divided by the total you are choosing from. Be careful which total that is: "a girl who chose tennis" uses the grand total, but "of the girls, who chose tennis?" uses only the girls' total. A two-way table and a Venn diagram can often show the same information in two different ways.
Examples
A class survey of 55 students:
| Football | Swimming | Tennis | Total | |
|---|---|---|---|---|
| Boys | 14 | 6 | 5 | 25 |
| Girls | 9 | 11 | 10 | 30 |
| Total | 23 | 17 | 15 | 55 |
- How many girls chose swimming? Read across the Girls row to the Swimming column: 11.
- How many students chose tennis? The Tennis column total: 15.
- The probability that a random student is a girl who chose tennis: 10/55 = 2/11.
- Of the students who chose swimming, what fraction are girls? 11 out of 17: 11/17.
Worked example (completing a table): 40 students, 22 of them girls. 15 students walk to school, and 9 of the walkers are girls. How many boys do not walk?
| Walk | Don't walk | Total | |
|---|---|---|---|
| Girls | 9 | 22 − 9 = 13 | 22 |
| Boys | 15 − 9 = 6 | 18 − 6 = 12 | 40 − 22 = 18 |
| Total | 15 | 40 − 15 = 25 | 40 |
So 12 boys do not walk. Check: the Don't walk column is 13 + 12 = 25. ✓
Common mistakes
- Mixing up rows and columns. Fix: put a finger on the row label and another on the column label, and slide them together to the cell.
- Using the wrong total for a probability. Fix: ask "out of whom?". Out of everyone, use the grand total; out of the boys, use the boys' total.
- Comparing counts instead of fractions. If 12 of 40 girls walk and 10 of 20 boys walk, more girls walk, but a boy is more likely to walk (1/2 against 3/10).
- Forgetting to check. Fix: when the table is full, the row totals and the column totals must both add up to the grand total.
Step by step
Step by step
- Draw the gridOne feature down the side, the other across the top, plus a Total row and a Total column.
- Fill in what you knowPut every number from the question into its cell, including the grand total.
- Find a single gapLook for a row or column with only one empty cell, and work it out by subtracting from the total.
- RepeatEach new number opens up another gap. Keep going until the table is full.
- Check and answerRows and columns must add up. Then read off the cell the question asks for.
For a probability, write "number that fit ÷ total you are choosing from" and simplify the fraction.
Practice questions
Two-way tables
Try each one, then check your answer.
- A row has a total of 28. Its other two cells are 9 and 11. Find the missing cell.
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28 − 9 − 11 = 8 - A column has a total of 45 and two cells. One cell is 17. What is the other?
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45 − 17 = 28 - The row totals in a table are 25 and 30. What is the grand total?
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25 + 30 = 55 - In the class survey table above, how many boys chose football?
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Boys row, Football column: 14 - In the class survey, how many students chose swimming?
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Swimming column total: 17 - What is the probability that a random student from the survey chose football?
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23 out of 55: 23/55 - What is the probability that a random student is a boy who chose tennis?
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5/55 = 1/11 - What fraction of the boys chose football?
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14 out of the 25 boys: 14/25 (56%) - Of the students who chose swimming, what fraction are boys?
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6 out of 17: 6/17 - At a pet show there are 50 people: 30 adults and 20 children. 28 people prefer dogs and the rest prefer cats. 12 children prefer dogs. How many adults prefer cats?
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Adults who prefer dogs: 28 − 12 = 16. Adults who prefer cats: 30 − 16 = 14 - In question 10, what is the probability that a random person is a child who prefers cats?
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Children who prefer cats: 20 − 12 = 8, so 8/50 = 4/25 - In a class of 32, 18 have a pet. There are 16 girls and 10 of them have a pet. How many boys have no pet?
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Boys: 32 − 16 = 16. Boys with a pet: 18 − 10 = 8. Boys with no pet: 16 − 8 = 8 - 120 students choose French or Spanish. 70 choose French, and 40 of the French students are girls. Half of the Spanish students are girls. How many girls are there?
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Spanish: 120 − 70 = 50, so 25 girls. Girls: 40 + 25 = 65 - In the class survey, is a boy or a girl more likely to choose football?
Show answer
Boys: 14/25 = 0.56. Girls: 9/30 = 0.3. A boy is more likely.
Word problems
Two-way tables in real life
Try each one, then check your answer.
- A café served 60 customers: 36 had coffee and the rest had tea. 15 coffee drinkers and 8 tea drinkers also bought cake. How many customers did not buy cake?
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Cake: 15 + 8 = 23. No cake: 60 − 23 = 37 - In question 1, what is the probability that a random customer had tea without cake?
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Tea: 60 − 36 = 24; tea without cake: 24 − 8 = 16. 16/60 = 4/15 - 90 students go on a school trip: 50 from Year 7 and 40 from Year 8. 55 choose the zoo and the rest the museum. 20 Year 8 students choose the museum. How many Year 7 students choose the zoo?
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Year 8 zoo: 40 − 20 = 20. Year 7 zoo: 55 − 20 = 35 - In a survey, 12 of the 20 adults asked want a new park. The town has 3,000 adults. Estimate how many want the park.
Show answer
12/20 = 0.6, and 0.6 × 3,000 = 1,800 adults - 12 of 40 girls and 10 of 20 boys walk to school. Sam says, "More girls walk, so girls are more likely to walk." Is Sam right?
Show answer
No. 12/40 = 30% of girls walk, but 10/20 = 50% of boys walk, so a boy is more likely to walk.
Questions people ask
What do two-way tables show?
They show how two features are linked in the same group, such as gender and favourite sport, or age group and how people travel. You can see each combination, the totals for each feature, and the grand total.
How do you complete a two-way table?
Find a row or column with only one missing number and subtract the known numbers from its total. Each number you fill in usually lets you find another one. Check at the end that every row and column adds up.
How do you do probability with two-way tables?
Probability = the number in the cell (or total) you want ÷ the total you are choosing from. For a random person from the whole table, divide by the grand total. If the question says "given that" or "of the boys", divide by that row or column total instead.
What is a two-way frequency table?
It is the usual kind, where each cell holds a count (a frequency). A two-way relative frequency table shows each cell as a fraction or percentage of a total instead, which makes groups of different sizes easier to compare.
Is a two-way table the same as a contingency table?
Yes. Statisticians often call it a contingency table or a cross-tabulation. In a spreadsheet you can build one from a list of answers with a PivotTable.
What is a two-way table of specification?
That is a different tool with a similar name. Teachers use it to plan a test: topics down the side, types of question across the top, and the number of questions in each cell.
Quick quiz
Two-way tables
Check what you know with 4 quick questions.
Question 1 of 4
In a two-way table with row and column totals, where is the grand total?
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For grown-ups
Related topics
Probability
Probability measures how likely something is to happen, on a scale from 0 (impossible) to 1 (certain). A fair coin landing heads has a probability of 1/2.
Venn diagrams
A Venn diagram uses overlapping circles to sort things into groups. The overlap shows what belongs to both groups, and the outside shows what belongs to neither.
Tally charts
A tally chart is a table that counts things with little marks. Each mark is one, and every fifth mark crosses the four before it, so you can count in fives.
Line graphs
A line graph shows how something changes over time. Points are plotted on a grid and joined by a line, so you can see rises, falls and patterns at a glance.
Pie charts
A pie chart is a circle divided into slices that show parts of a whole. A bigger slice means a bigger share, and all the slices together make 100 percent, or 360 degrees.
Conditional probability
Conditional probability is the chance of an event A when you already know event B has happened. It is written P(A|B) and found with P(A|B) = P(A and B) ÷ P(B).