Probability and statistics · GCSE Maths
Frequency trees
GCSE Maths frequency trees: two-stage totals split on branches, missing values from row and column constraints, and link to two-way tables.
A frequency tree shows counts, not fractions. Branch totals add to the parent. The whole tree sums to the total number of people or items.
The important bits
What you need to know
- 1
A frequency tree records how many items fall into categories across two or more stages — counts, not probabilities.
- 2
Each branch split must sum to the number on the parent branch. If 40 people split into 25 and 15, 25 + 15 = 40.
- 3
The entire tree sums to the total sample. Every terminal branch is a complete pathway count.
- 4
Missing values are found by subtraction: if 60 total and one branch is 22, the sibling branch is 38.
- 5
Two-stage trees often mirror two-way tables: rows and columns give the same counts in a different layout.
- 6
From a frequency tree you can estimate probabilities: P(red) = red count ÷ total, if the item was chosen fairly from the sample.
- 7
Three-stage trees multiply structure but still add at each split. Check every fork sums correctly.
- 8
Label branches clearly (e.g. male/female then glasses/not) so pathway counts map to the question’s wording.
Quotations worth analysing
Short evidence. Real method.
“Branches sum to the parent”
If male is 35 and female is 25, the parent must be 60. A missing female branch is 60 − 35, not guessed from nowhere.
“Counts, not probabilities on the branches”
Frequency trees show how many people. Probability trees show fractions. Convert only when the question asks for P(...).
“Same information as a two-way table”
Swap between tree and table to find missing cells. The total in the corner is the root of the tree.
Go deeper
Building from partial information
80 students: 50 pass maths, 30 fail. Of the 50 who pass, 30 also pass science. Of the 30 who fail maths, 10 pass science. Tree: root 80 → pass 50, fail 30. From pass 50 → pass sci 30, fail sci 20. From fail 30 → pass sci 10, fail sci 20. Check leaves: 30 + 20 + 10 + 20 = 80. “Pass both” = 30. “Pass exactly one” = 20 + 10 = 30. If only told 50 pass maths and 40 pass science total, fill the tree using the intersection given or from a table. Subtraction is the tool: sibling branch = parent − known branch.
Go deeper
Link to two-way tables
Rows: pass/fail maths. Columns: pass/fail science. Cell counts match tree pathways. Marginal totals: pass maths row sums to 50, fail maths to 30; pass science column to 40. Interior cells 30, 20, 10, 20. If a table gives row total 50 and one cell 30, the other cell in that row is 20 — that is the tree split under “pass maths”. Exam questions flip representation: given table, draw tree; given tree, complete table. Totals must agree both ways.
Go deeper
Estimating probability from frequencies
From the tree above, P(pass maths) = 50/80 = 5/8. P(pass science | pass maths) = 30/50 = 3/5 — conditional, only the pass-maths branch. P(both) = 30/80 = 3/8. These are estimates from data, not theoretical fairness. Larger samples give stabler estimates. If the question says “a student is chosen at random from the group”, divide pathway count by 80. Frequency tree first, probability second — do not put fractions on the tree unless the question is a hybrid diagram.
See the idea in action
60 adults: 36 own a car, 24 do not. Of car owners, 20 have insurance. Of non-car owners, 8 have insurance. Complete the tree and find how many have insurance. Root 60 → car 36, no car 24. Car 36 → insured 20, not insured 16 (36 − 20). No car 24 → insured 8, not insured 16 (24 − 8). Total insured = 20 + 8 = 28. Check leaves: 20 + 16 + 8 + 16 = 60.
Exam technique
Turn knowledge into marks
Fill in obvious subtractions before hunting harder cells. Every parent equals the sum of its children. Total all terminal branches to verify the sample size.
Common mistakes
Do not give these marks away
- 01
Putting probabilities on a frequency tree when the question gave counts.
- 02
Branches that do not sum to the parent — arithmetic slip on the sibling branch.
- 03
Forgetting conditional paths: “pass science given pass maths” uses 50 as denominator, not 80.
A frequency tree root is 100. One branch splits into 35 and 65. The 35 branch splits into 20 and 15. How many on the 15 leaf?
A15
B20
C35
D65
Show the answer
15. The split of 35 is 20 + 15. The question asks for the 15 leaf directly. 20 is the sibling leaf under 35.
Quick questions
If this is the bit you searched
What is the difference between a frequency tree and a probability tree?
Frequency trees show counts of people or items. Probability trees show fractions or probabilities on branches. Convert with ÷ total when needed.
How do I find a missing value?
Subtract from the parent: sibling = parent − known child. Row and column totals in a table work the same way.
Can a frequency tree have three stages?
Yes. Each stage splits counts; terminal leaves are full pathways. All leaves sum to the root total.
How do I get probability from counts?
Divide the pathway count by the total sample for P(event). For conditional probability, divide by the restricted branch total.