Two-Way Tables and Frequency Trees
A two-way table organises data that has been sorted by two different categories at once, for example year group and favourite subject, into rows and columns, with a row and column of totals so that every count can be checked by adding across or down. A frequency tree shows the same kind of two-stage categorical data as a diagram instead of a table: a starting total splits into branches for the first category, and each of those branches splits again into the second category, with every frequency, an actual count rather than a probability, written on its branch.
Before you start
Make sure you're comfortable with these topics first:
Method
- In a two-way table, read the row and column headings carefully first to know exactly what each cell, row and column represents.
- Use the fact that every row must add up to its row total, and every column must add up to its column total, to find any missing values.
- The total of all the row totals must equal the total of all the column totals, which equals the grand total in the corner - use this as a final check.
- In a frequency tree, start with the overall total and work outwards: each pair of branches from a point must add up to the frequency at that point.
- To find a missing frequency on a tree, subtract the known branch from the total it splits from.
- To find a probability from a completed two-way table or frequency tree, divide the relevant frequency by the appropriate total (the grand total, or a row/column total for a conditional probability).
Worked example
90 students in Year 8 were asked whether they walk or get the bus to school. 34 of the 48 boys walk to school. 23 girls get the bus to school. (a) Complete a two-way table for this information. (b) Find the probability that a student picked at random from this group walks to school.
- Set up a table with rows Walk and Bus, and columns Boys, Girls and Total. Since there are 48 boys out of 90 students, there are 90 - 48 = 42 girls in total.
- 34 of the 48 boys walk, so boys who get the bus = 48 - 34 = 14.
- 23 girls get the bus, so girls who walk = 42 - 23 = 19.
- Total who walk = 34 (boys) + 19 (girls) = 53. Total who get the bus = 14 (boys) + 23 (girls) = 37. Check: 53 + 37 = 90, which matches the grand total.
- P(walks) = 53/90 (this does not simplify further, since 53 is a prime number).
Practice questions
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Q1A two-way table shows that 15 boys like football and 9 boys do not like football. How many boys were surveyed in total?Show answer
Answer: 24 (15 + 9)
Q2In a survey of 50 people, 22 are vegetarian. How many people are not vegetarian?Show answer
Answer: 28 (50 - 22)
Q3A frequency tree shows that 60 people were surveyed. Of these, 36 own a pet. Of the 36 who own a pet, 21 own a dog. How many of the pet owners own a different kind of pet (not a dog)?Show answer
Answer: 15 (36 - 21)
Q4A frequency tree shows that 85 visitors came to a museum on Saturday. 52 of them were adults. How many were children (not adults)?Show answer
Answer: 33 (85 - 52)
Q5A two-way table has a row total of 40 for Year 9 and a row total of 35 for Year 10, with no other rows. What is the grand total?Show answer
Answer: 75 (40 + 35)
Q6In a two-way table, the column for Tea has a total of 27, made up of 12 men and some women. How many women prefer tea?Show answer
Answer: 15 (27 - 12)
Q7A frequency tree splits 120 students into Studies French (72 students) and Does not study French. Of the 72 who study French, 45 are girls. How many of the French students are boys?Show answer
Answer: 27 (72 - 45)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
150 tourists visiting a castle were asked whether they arrived by car or by coach, and whether they were UK residents or overseas visitors. The results include: 54 UK residents arrived by car, 38 overseas visitors arrived by car, and 41 overseas visitors arrived by coach. (a) Use these figures, and the grand total of 150, to find how many UK residents arrived by coach. (b) Find the probability that a tourist chosen at random from this group is a UK resident who arrived by coach.
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A frequency tree shows that 200 employees at a company were surveyed about how they travel to work. 140 of the employees drive to work; of these, 91 drive alone and the rest car-share. Of the 60 employees who do not drive to work, 38 walk and the rest cycle. (a) Find the number of employees who car-share and the number who cycle. (b) Find the probability that an employee chosen at random from the 200 neither drives alone nor walks.
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A frequency tree begins with 500 concert tickets sold, split into Adult and Child tickets. 320 are adult tickets. Each branch then splits into Standing and Seated. Of the adult tickets, 210 are seated. Of the child tickets, 65 are standing. (a) Find the number of child tickets sold in total, the number of adult tickets that are standing, and the number of child tickets that are seated. (b) Find the probability that a ticket picked at random is a seated child ticket.
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Free printable worksheet
Want more practice on paper? Download the two-way tables and frequency trees worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 25 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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