Frequency trees GCSE maths questions ask you to organise raw counted data into a branching diagram that splits a total into groups, then splits each group again. Unlike a probability tree, every number on a frequency tree is an actual count, not a fraction — so all branches from a point must add up to the total that feeds into it.
What is a frequency tree?
A frequency tree is a diagram that splits a total number of items into two or more categories, then splits each of those categories again into sub-categories, using actual counts (frequencies) at every branch. It looks similar to a probability tree diagram, but every value written on a frequency tree is a whole number of people or items — never a probability or fraction.
Frequency trees are used to organise data collected from a survey or experiment, such as splitting a group of students first by gender and then by whether they play a sport, or splitting a batch of products first by pass/fail and then by which fault caused a failure.
How is a frequency tree different from a probability tree diagram?
The two diagrams look almost identical, but they answer different questions:
| Feature | Frequency tree | Probability tree diagram |
|---|---|---|
| Values shown | Actual counts (e.g. 45 people) | Probabilities (e.g. 0.3, or 3/10) |
| Branches from a node sum to | The total at that node | 1 (always) |
| Used for | Organising real, counted data | Calculating chances of combined events |
| Typical question | "Complete the frequency tree" | "Find the probability that..." |
A useful check: on a frequency tree, the numbers on branches leaving any point must add up exactly to the number that arrived at that point. On a probability tree, the numbers on branches leaving any point must add up to 1.
How do you draw and complete a frequency tree?
Follow these steps whenever you're given raw data (in a sentence, table, or partially completed diagram) and asked to build or complete a frequency tree:
- Write the overall total at the start of the tree — this is the total number of people or items in the whole sample.
- Draw the first set of branches, splitting the total into its first-level categories (for example, male and female), labelling each branch with its category name and frequency.
- Check the first-level branches sum to the total. If one branch's frequency is missing, subtract the known branch(es) from the total to find it.
- Draw the second set of branches from each first-level category, splitting it further (for example, "plays sport" and "does not play sport"), again labelling each with its frequency.
- Check each second-level pair sums to its parent branch. Use subtraction to fill in any missing values, exactly as in step 3.
- Read off answers by tracing the correct path through the tree and, if the question asks for a probability, dividing the frequency at the end of that path by the appropriate total.
Worked example: completing a frequency tree from raw data
150 students were surveyed about whether they walk or take the bus to school. Of the 150 students, 80 are in Year 7 and the rest are in Year 8. Of the Year 7 students, 55 walk to school. Of the Year 8 students, 30 take the bus. Complete the frequency tree and find how many students in total walk to school.
- Overall total: 150 students.
- First-level split (year group): Year 7 = 80 (given). Year 8 = 150 − 80 = 70.
- Year 7 branch: Walk = 55 (given). Bus = 80 − 55 = 25.
- Year 8 branch: Bus = 30 (given). Walk = 70 − 30 = 40.
- Total who walk: Year 7 walkers (55) + Year 8 walkers (40) = 95 students.
Every branch checks out: 80 + 70 = 150, 55 + 25 = 80, and 40 + 30 = 70 — confirming the completed tree is internally consistent.
How do you use a frequency tree to find a probability?
Once a frequency tree is complete, you can answer probability questions by dividing the relevant frequency by the appropriate total, treating the frequencies exactly like a frequency table.
Worked example (continued): Using the tree above, find the probability that a randomly chosen student from the survey is in Year 8 and takes the bus.
- Number of Year 8 students who take the bus: 30 (read directly from the tree)
- Total number of students surveyed: 150
- Probability: $\dfrac{30}{150} = \dfrac{1}{5} = 0.2$
For a conditional probability — for example, "given that a student is in Year 8, find the probability they take the bus" — divide by the Year 8 total (70) instead of the overall total (150): $30 \div 70 = \dfrac{3}{7}$.
What mistakes commonly lose marks on frequency tree questions?
The most common mistake is treating the branch values as probabilities and trying to make them sum to 1, when frequency trees always use whole-number counts that sum to the parent total. A second mistake is dividing by the wrong total when calculating a probability — using the overall total (150) when the question actually asks for a conditional probability based on a subgroup (70), or vice versa. A third mistake is arithmetic slips when subtracting to find a missing branch — always double-check that both levels of the tree still sum correctly once every branch is filled in.
Frequently asked questions
Do frequency tree branches always have to split into exactly two categories?
No — while two-way splits (yes/no, male/female) are the most common at GCSE, a frequency tree can split into three or more categories at each level, as long as every branch leaving a point sums to the frequency at that point. The method for completing missing values by subtraction works the same way regardless of how many branches there are.
How do I know whether to add or subtract when filling in a frequency tree?
If you know the total at a point and all but one of the branches leaving it, subtract the known branches from the total to find the missing one. If you're instead combining two completed branches to answer a question (like a total that walks to school from two year groups), add them together.
Can a frequency tree have more than two levels?
Yes, though GCSE questions typically use two levels (an overall split, then a further split of each category). A three-level tree works identically — you just check that branches sum to their parent frequency at every level, from the very first split through to the last.
Is a frequency tree the same as a two-way table?
They present the same kind of data differently. A two-way table shows the same combined frequencies in a grid, while a frequency tree shows them as a branching diagram. Some GCSE questions ask you to convert data from one format to the other, and the row/column totals in a two-way table correspond exactly to the branch totals in a frequency tree.
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