Frequency Trees

This lesson covers frequency trees — a method for organising and displaying data involving two characteristics or events. Frequency trees are part of the AQA GCSE Mathematics specification and are a common exam topic at both Foundation and Higher tier.

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What Is a Frequency Tree?

A frequency tree is a diagram that splits a group of items (usually people) into categories based on two characteristics. Unlike a probability tree (which shows probabilities on branches), a frequency tree shows actual counts (frequencies).

graph LR
    Total[Total: 80 students] -->|Male| M[Male: 45]
    Total -->|Female| F[Female: 35]
    M -->|Pass| MP[Pass: 30]
    M -->|Fail| MF[Fail: 15]
    F -->|Pass| FP[Pass: 25]
    F -->|Fail| FF[Fail: 10]

At each stage, the numbers on the branches must add up to the number at the node they came from.

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Drawing a Frequency Tree

Worked Example 1

A school surveys 120 Year 11 students about whether they walk to school.

• 70 are boys, 50 are girls.
• 28 boys walk to school.
• 35 girls walk to school.

Draw a frequency tree and find the number of students who do not walk to school.

graph LR
    T[120 students] -->|Boys| B[70]
    T -->|Girls| G[50]
    B -->|Walk| BW[28]
    B -->|Not walk| BN[42]
    G -->|Walk| GW[35]
    G -->|Not walk| GN[15]

• Boys who do not walk = 70 - 28 = 42
• Girls who do not walk = 50 - 35 = 15
• Total who do not walk = 42 + 15 = 57

Exam Tip: Always check that the numbers at each split add up correctly. For example, 28 + 42 = 70 (boys) and 35 + 15 = 50 (girls). This is a quick way to spot errors.

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Extracting Probabilities from a Frequency Tree

Once you have completed a frequency tree, you can calculate probabilities by dividing the relevant frequency by the appropriate total.

Worked Example 2

Using the data from Worked Example 1:

(a) A student is chosen at random. What is the probability that the student walks to school?

Total who walk = 28 + 35 = 63

P(walks) = 63/120 = 21/40

(b) A boy is chosen at random. What is the probability that he walks to school?

We are told the student is a boy, so we only consider the 70 boys.

P(walks | boy) = 28/70 = 2/5

(c) A student who does not walk to school is chosen at random. What is the probability that the student is a girl?

We consider only the 57 who do not walk.

P(girl | does not walk) = 15/57 = 5/19

Exam Tip: Pay close attention to the wording. "A student is chosen at random" means the denominator is the total (120). "A boy is chosen at random" means the denominator is the number of boys (70). The phrasing determines which total you use.

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Frequency Trees from Two-Way Tables

Frequency trees and two-way tables contain the same information in different formats. You should be able to convert between them.

Worked Example 3

A two-way table shows the results of a survey about pet ownership:

	Cat	No cat	Total
Dog	12	28	40
No dog	18	42	60
Total	30	70	100

Convert this to a frequency tree:

graph LR
    T[100 people] -->|Dog| D[40]
    T -->|No dog| ND[60]
    D -->|Cat| DC[12]
    D -->|No cat| DNC[28]
    ND -->|Cat| NDC[18]
    ND -->|No cat| NDNC[42]

From the frequency tree:

• P(cat and dog) = 12/100 = 3/25
• P(cat | dog) = 12/40 = 3/10
• P(no pet at all) = 42/100 = 21/50

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Completing a Frequency Tree with Missing Information

Exam questions often provide partial information and ask you to complete the tree.

Worked Example 4

200 people were asked whether they are left-handed or right-handed, and whether they wear glasses.

The following information is given:

• 30 people are left-handed
• 50 people wear glasses
• 8 left-handed people wear glasses

Complete the frequency tree.

graph LR
    T[200 people] -->|Left| L[30]
    T -->|Right| R[170]
    L -->|Glasses| LG[8]
    L -->|No glasses| LNG[22]
    R -->|Glasses| RG[42]
    R -->|No glasses| RNG[128]

Working:

• Right-handed = 200 - 30 = 170
• Left-handed, no glasses = 30 - 8 = 22
• Right-handed, glasses = 50 - 8 = 42
• Right-handed, no glasses = 170 - 42 = 128

Check: 8 + 22 + 42 + 128 = 200 (correct)

Exam Tip: When completing a frequency tree with missing values, work through the tree one branch at a time. Use subtraction to find each missing value. Always verify your answers add up at every node.

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When to Use a Frequency Tree vs a Probability Tree

Feature	Frequency Tree	Probability Tree
What is shown on branches	Counts (frequencies)	Probabilities
Numbers at each split	Add up to the parent total	Add up to 1
Best used for	Sorting data, finding conditional probabilities	Calculating combined probabilities
Key phrase in question	"Complete the frequency tree"	"Draw a tree diagram showing probabilities"

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Worked Example 5 — Full Exam-Style Question

A factory produces 500 items per day. Each item is either standard or premium. Each item passes or fails a quality check.

• 350 items are standard
• 40 standard items fail the quality check
• 20 premium items fail the quality check

(a) Complete the frequency tree.

Branch	Count
Standard	350
Premium	500 - 350 = 150
Standard, fail	40
Standard, pass	350 - 40 = 310
Premium, fail	20
Premium, pass	150 - 20 = 130

(b) An item is chosen at random. Find the probability that it passed the quality check.

Total passed = 310 + 130 = 440

P(pass) = 440/500 = 22/25

(c) An item that failed the quality check is chosen at random. Find the probability that it is premium.

Total failed = 40 + 20 = 60

P(premium | fail) = 20/60 = 1/3

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Summary