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Frequency Trees and Two-Way Tables

Completing frequency trees and two-way tables and using them to find probabilities.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Complete a two-way table from given totals.
  2. 2Draw and complete a frequency tree.
  3. 3Use a two-way table or frequency tree to work out probabilities.
  4. 4Use probabilities and frequencies together to estimate numbers (expected frequencies).

Organising counts

A two-way table sorts people or objects by two features at once, such as year group and how they travel to school. A frequency tree does the same job with branches, using numbers of people rather than probabilities. Both turn a wordy problem into a picture where missing numbers can be found by simple addition and subtraction. The golden rule is that rows add up to their totals and columns add up to theirs, so every blank can be found from the ones around it.

Two-way tables

Each cell is the number of items that have both features, and the totals are on the right and at the bottom.

  • Rows and columns

    Each row adds up to its total on the right, and each column adds up to its total at the bottom.

  • The grand total

    The total of the totals, in the bottom right corner, is how many there are altogether, and it is the same whichever way you add.

  • Find a blank

    Look for a row or a column with only one blank. Subtract the known numbers from the total.

  • Check

    Add the row totals and the column totals, and both should give the grand total.

Completing a two-way table

In a survey of 100 students, 50 are in Year 10 and 50 are in Year 11. 44 travel by bus, and 13 cycle. Of the Year 10 students, 24 go by bus and 18 walk. Of the Year 11 students, 25 walk. Work out how many Year 11 students cycle.

Show the solutionHide the solution
  1. 1 Year 10 cycling \(50 - 24 - 18 = 8\).
  2. 2 Total cycling The cycling column totals 13, so Year 11 cycling \(= 13 - 8 = 5\).
  3. 3 Check with the Year 11 row Year 11 bus is \(44 - 24 = 20\), and \(20 + 25 + 5 = 50\). Correct.
  4. 4 Check the grand total \(50 + 50 = 100\).

Answer5

Building a frequency tree

Start at the left with the total and split it in stages.

  • Use the information in order

    The first split is by the first feature, such as girls and boys.

  • Subtract for the other branch

    If 60 of 100 are girls, then \(100 - 60 = 40\) are boys.

  • Use percentages or fractions

    "40% of the girls walk" means \(0.4 \times 60 = 24\) walk and \(60 - 24 = 36\) take the bus.

  • Check each pair

    Each pair of branches must add to the number on the left.

A frequency tree from percentages

200 people go to a gym. 60% are women. 25% of the women and 40% of the men go in the morning. Work out how many people go in the morning.

Show the solutionHide the solution
  1. 1 Women and men \(60\% \text{ of } 200 = 120\) women, so \(200 - 120 = 80\) men.
  2. 2 Women in the morning \(25\% \text{ of } 120 = 30\).
  3. 3 Men in the morning \(40\% \text{ of } 80 = 32\).
  4. 4 Add \(30 + 32 = 62\), so 62 people go in the morning.

Answer62

Probability from tables and trees

The total is on the bottom, and the number in the cell, or the sum of cells, is on the top.

  • One cell

    In the survey of 100 students, \(P(\text{Year 10 and bus}) = \dfrac{24}{100} = \dfrac{6}{25}\).

  • A row or a column

    \(P(\text{cycles}) = \dfrac{13}{100}\), because the cycling column total is 13.

  • Combining cells

    \(P(\text{Year 11 and walks or cycles}) = \dfrac{25 + 5}{100} = \dfrac{3}{10}\).

  • Estimating numbers

    If the probability is \(\dfrac{13}{100}\), then out of 500 students you would expect \(\dfrac{13}{100} \times 500 = 65\) to cycle.

Test yourself

  1. 1

    Where do the totals go in a two-way table?

    Show answerHide answer

    At the end of each row and the bottom of each column.

  2. 2

    How do you find a missing value in a row?

    Show answerHide answer

    Subtract the known values from the row total.

  3. 3

    What do the numbers on a pair of branches in a frequency tree add up to?

    Show answerHide answer

    The number they came from.

  4. 4

    What is 40% of 80?

    Show answerHide answer

    32.

  5. 5

    How do you find the probability from a two-way table?

    Show answerHide answer

    The number in the cell, divided by the grand total.

Exam technique: tables and trees

Fill in everything you can before you answer the question.

  • Complete the whole table

    Fill every blank, and use the totals to check.

  • Read the row and column headings

    Many errors come from taking a number from the wrong row.

  • Put the denominator right

    The total on the bottom of a probability is the whole group, not a part of it.

  • Show subtractions

    \(50 - 24 - 18 = 8\) is a method mark.

Summary and exam focus

  • A two-way table sorts data by two features, with row and column totals.
  • Find missing numbers by adding or subtracting from totals, and check the grand total.
  • A frequency tree splits a total in stages, and each pair of branches adds to the number before it.
  • Probabilities from a table are the cell or sum of cells divided by the grand total.

Exam focus

In a survey, 50 people were asked whether they have a cat or a dog. 28 have a cat, 20 have a dog, and 6 have both. Draw a two-way table to show this, and work out how many have neither. (4 marks) (4 marks)

With 6 owning both, 22 own a cat only and 14 own a dog only. That is \(22 + 14 + 6 = 42\) with at least one pet, so \(50 - 42 = 8\) have neither. Check by adding the table back up to 50.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Two-way table
A table that shows how many items have each combination of two features.
Frequency tree
A tree diagram with the number of items on each branch.
Frequency
The number of times something happens or the number of items in a group.
Row
A horizontal line of cells in a table.
Column
A vertical line of cells in a table.
Total
The sum of a row, a column or the whole table.
Grand total
The total of everything in the table.
Survey
A set of questions asked to a group of people to collect data.
Expected frequency
The number of times you expect something to happen, found as probability times the number of trials.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 3 marks Easier

There are 60 people at a gym. 36 of them are women. 9 of the women and 6 of the men use the pool. (a) Work out the number of men. (1 mark) (b) Work out the number of people who do not use the pool. (2 marks)

Mark scheme — 3 marks available

  • (a) 24 — B1
  • (b) \(9 + 6 = 15\) — M1
  • (b) 45 — A1

Model answer

(a) \(60 - 36 = 24\). (b) \(9 + 6 = 15\) people use the pool, so \(60 - 15 = 45\) do not.

2. Exam question Work out 4 marks Easier

The two-way table shows information about how 100 students travel to school. Some of the numbers are missing. (a) Complete the two-way table. (3 marks) (b) A student is chosen at random. Work out the probability that the student is in Year 11 and walks. (1 mark)

A two-way table of how Year 10 and Year 11 students travel to school, with four missing values.

Mark scheme — 4 marks available

  • (a) At least two correct values — B1
  • (a) At least three correct values — B1
  • (a) All four correct: 8, 20, 50 and 43 — B1
  • (b) \(\dfrac{25}{100}\) or \(\dfrac{1}{4}\) — B1

Model answer

(a) Year 10 cycle: \(50 - 24 - 18 = 8\). Year 11 bus: \(44 - 24 = 20\). Year 11 total: 50. Total who walk: \(18 + 25 = 43\). (b) \(\dfrac{25}{100} = \dfrac{1}{4}\).

3. Exam question Work out 3 marks Core

A survey asks 200 people if they own a dog. 120 of the people are female, and the rest are male. 30% of the females and 25% of the males own a dog. Work out the total number of people who own a dog.

Mark scheme — 3 marks available

  • \(0.3 \times 120 = 36\) — M1
  • \(0.25 \times 80 = 20\) — M1
  • 56 — A1

Model answer

Females: \(0.3 \times 120 = 36\). There are 80 males, and \(0.25 \times 80 = 20\). The total is \(36 + 20 = 56\).

4. Exam question Work out 3 marks Core

In a school, 60% of the students are girls. 20% of the girls and 40% of the boys play hockey. A student is chosen at random. Work out the probability that the student plays hockey.

Mark scheme — 3 marks available

  • \(0.6 \times 0.2\) or \(0.4 \times 0.4\) — M1
  • \(0.6 \times 0.2 + 0.4 \times 0.4\) — M1
  • \(0.28\) — A1

Model answer

\(0.6 \times 0.2 + 0.4 \times 0.4 = 0.12 + 0.16 = 0.28\).

5. Exam question Work out 4 marks Core

There are 150 students in Year 11. 90 of them are boys. 35 of the boys and 25 of the girls are left-handed. (a) Work out the number of girls. (1 mark) (b) Work out the number of students who are right-handed. (2 marks) (c) A student is chosen at random. Work out the probability that the student is a left-handed girl. (1 mark)

Mark scheme — 4 marks available

  • (a) 60 — B1
  • (b) \(55\) and \(35\) seen, or \(150 - 35 - 25\) — M1
  • (b) 90 — A1
  • (c) \(\dfrac{1}{6}\) or \(\dfrac{25}{150}\) — B1

Model answer

(a) \(150 - 90 = 60\). (b) Right-handed boys \(90 - 35 = 55\), right-handed girls \(60 - 25 = 35\), so \(55 + 35 = 90\). (c) \(\dfrac{25}{150} = \dfrac{1}{6}\).

6. Exam question Work out 4 marks Stretch

There are 80 pupils on a trip. The ratio of boys to girls is \(3 : 5\). 40% of the boys and 20% of the girls bring a packed lunch. Work out the fraction of all the pupils who bring a packed lunch. Give your answer in its simplest form.

Mark scheme — 4 marks available

  • 30 boys and 50 girls — M1
  • \(0.4 \times 30 = 12\) and \(0.2 \times 50 = 10\) — M1
  • 22 — A1
  • \(\dfrac{11}{40}\) — A1

Model answer

Boys: \(80 \div 8 \times 3 = 30\). Girls: 50. Packed lunches: \(0.4 \times 30 = 12\) and \(0.2 \times 50 = 10\), which is 22. The fraction is \(\dfrac{22}{80} = \dfrac{11}{40}\).

7. Multiple choice 1 mark Easier

A frequency tree starts with 100 pupils, and 60 are girls. How many are boys?

  1. A \(40\) Correct
  2. B \(60\)
  3. C \(160\)
  4. D \(100\)

Why: \(100 - 60 = 40\).

8. Multiple choice 1 mark Easier

40% of 60 girls walk to school. How many girls walk?

  1. A \(36\)
  2. B \(40\)
  3. C \(20\)
  4. D \(24\) Correct

Why: \(0.4 \times 60 = 24\).

9. Multiple choice 1 mark Core

In a two-way table the Year 10 row total is 50. 24 take the bus and 18 walk. How many cycle?

  1. A \(42\)
  2. B \(6\)
  3. C \(8\) Correct
  4. D \(18\)

Why: \(50 - 24 - 18 = 8\).

10. Multiple choice 1 mark Core

In a survey of 100 students, 24 are in Year 10 and take the bus. What is the probability that a student is in Year 10 and takes the bus?

  1. A \(\dfrac{1}{24}\)
  2. B \(\dfrac{6}{25}\) Correct
  3. C \(\dfrac{24}{50}\)
  4. D \(\dfrac{1}{4}\)

Why: \(\dfrac{24}{100} = \dfrac{6}{25}\).

11. Multiple choice 1 mark Core

13 out of 100 students cycle to school. How many of 500 students would you expect to cycle?

  1. A \(65\) Correct
  2. B \(13\)
  3. C \(130\)
  4. D \(6.5\)

Why: \(\dfrac{13}{100} \times 500 = 65\).

12. Multiple choice 1 mark Easier

What is 25% of 120?

  1. A \(25\)
  2. B \(48\)
  3. C \(95\)
  4. D \(30\) Correct

Why: \(0.25 \times 120 = 30\).

13. Multiple choice 1 mark Stretch

200 people go to a gym. 60% are women. 25% of the women and 40% of the men go in the morning. How many people go in the morning?

  1. A \(54\)
  2. B \(60\)
  3. C \(62\) Correct
  4. D \(70\)

Why: 120 women and 80 men. \(30 + 32 = 62\).

14. Multiple choice 1 mark Stretch

50 people were asked about pets. 28 have a cat, 20 have a dog and 6 have both. How many have neither?

  1. A \(6\)
  2. B \(8\) Correct
  3. C \(14\)
  4. D \(22\)

Why: At least one: \(22 + 6 + 14 = 42\). \(50 - 42 = 8\).

15. Multiple choice 1 mark Easier

In a frequency tree, the branches after “60 girls” show 24 who walk and some who take the bus. How many take the bus?

  1. A \(36\) Correct
  2. B \(24\)
  3. C \(84\)
  4. D \(60\)

Why: The branches add up to the number they came from: \(60 - 24 = 36\).