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To calculate relative frequency, divide the number of times an outcome occurs by the total number of trials. Free GCSE Maths notes and 11 FAQs.
Recording and Analysing Probability Experiments is part of Probability Foundations in GCSE AQA Higher Maths.
5 min
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GCSE AQA Higher
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To calculate relative frequency, divide the number of times an outcome occurs by the total number of trials. Free GCSE Maths notes and 11 FAQs.
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Use this lesson to stay inside Probability Foundations while connecting the detail back to the wider Probability topic.
Read the core explanations from Recording and Analysing Probability Experiments before testing yourself from memory.
You flip a coin and get heads five times in a row. Is the coin biased? You cannot answer that from theory alone — you need data. Recording and analysing probability experiments is how mathematicians (and scientists, insurers, and game designers) move from guesswork to evidence.
This lesson covers two core tools: frequency tables and frequency trees. Both let you organise raw experimental results so you can calculate relative frequencies, spot patterns, and draw conclusions that hold up under exam scrutiny.
When you run a probability experiment — rolling a die, spinning a spinner, surveying students — every trial produces an outcome. A frequency table is the most direct way to record how often each outcome occurs.
Here is a typical setup: Mohammed rolls a six-sided die 60 times and records each result. The table below shows his data.
| Outcome | Tally | Frequency |
|---|---|---|
| 1 | |||| |||| | 9 |
| 2 | |||| ||||| | 10 |
| 3 | |||| |||| || | 12 |
| 4 | |||| |||| | 9 |
| 5 | |||| |||| | | 11 |
| 6 | |||| |||| | 9 |
| Total | 60 |
Notice that the frequencies do not all equal 10, even though a fair die gives each face a theoretical probability of $\frac{1}{6}$. That is completely normal — experimental results vary. The question is: how do we turn these counts into a probability estimate?
Raw frequencies tell you how many times something happened. Relative frequency converts that count into a proportion, giving you an experimental probability you can compare against the theoretical value.
Using Mohammed's die data, the relative frequency of rolling a 3 is $\frac{12}{60} = 0.2$. The theoretical probability is $\frac{1}{6} \approx 0.167$. They are close but not identical — that gap is expected variation, not evidence of a biased die (60 trials is not enough to be certain either way).
Worked example: Calculating Relative Frequency from a Table
Aisha spins a four-section spinner 40 times. The outcomes are: Red 14, Blue 8, Green 11, Yellow 7. Calculate the relative frequency of each colour. [4 marks]
Answer: Red: 0.35, Blue: 0.2, Green: 0.275, Yellow: 0.175
Frequency tables work well for a single category. But what happens when your data splits into two groups at once — for example, students who are left- or right-handed and whether they prefer maths or English? That is where a frequency tree becomes the cleaner tool.
A frequency tree looks similar to a probability tree, but there is a key difference: the branches carry frequencies (whole number counts), not probabilities. You can convert to relative frequencies afterwards, but the tree itself records raw data. [graph]
Reading a frequency tree is straightforward once you know the rule: to find the frequency of a combined outcome, follow the branches and read the final count. To find a relative frequency, divide that count by the overall total.
Worked example: Building and Reading a Frequency Tree
In a survey of 80 people, 55 own a car and 25 do not. Of those who own a car, 40 drive to work and 15 do not. Of those who do not own a car, 5 drive to work (borrowed vehicle) and 20 do not. Construct a frequency tree and find the relative frequency of a person who owns a car and drives to work.
Answer: Relative frequency of owning a car and driving to work = $\frac{40}{80} = 0.5$
Recording data is only half the task. The higher-tier specification also requires you to analyse frequency data — which means comparing relative frequencies to theoretical probabilities, identifying patterns, and commenting on what the results suggest.
The most important analytical idea at GCSE is this: the more trials you carry out, the closer the relative frequency gets to the true probability. A coin flipped 10 times might give 7 heads (relative frequency 0.7), but after 1000 flips the relative frequency will be much closer to 0.5.
This convergence is what makes experimental probability useful. When you are asked to describe results in an exam, you need to comment on whether the relative frequencies are close to the theoretical values and whether the number of trials is sufficient to draw a reliable conclusion.
Exam questions on this topic often combine building or completing a frequency tree with calculating a relative frequency and then making an analytical comment. Here is a full worked example that mirrors Higher-tier exam style.
Worked example: Exam-Style: Completing a Frequency Tree and Analysing Results
A school records whether 150 students walk or cycle to school, and whether they arrive on time or late. 90 students walk. Of those who walk, 72 arrive on time. 60 students cycle. Of those who cycle, 6 arrive late. Complete the frequency tree and find the relative frequency of a student arriving on time. [4 marks]
Answer: Relative frequency of arriving on time = $\frac{126}{150} = 0.84$
Before moving to exam technique, work through these problems independently. Each one targets a specific skill from this lesson — check your method against the solutions once you have attempted them.
The skills in this lesson appear across both calculator-allowed questions and questions without a calculator. Relative frequency calculations are usually straightforward arithmetic, but the analysis questions — where you must describe or explain — are where marks are most commonly dropped.
When a question asks you to describe results from a frequency table or tree, you need two things: a numerical comparison (quote the relative frequency and the theoretical probability) and a conclusion (state whether the results support or challenge the assumption of fairness). A bare statement like "the die is biased" without figures will not earn the mark.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-21.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Divide the number of times an outcome occurs by the total number of trials. If a spinner lands on Red 14 times in 40 spins, the relative frequency of Red is 14/40 = 0.35, and all the relative frequencies should sum to 1.
Fill in missing branches by subtracting from the total at the branch before. If 90 of 150 students walk and 72 of them arrive on time, then 60 cycle and 18 who walk are late; with 6 who cycle late, 72 + 54 = 126 arrive on time, a relative frequency of 126/150 = 0.84.
A frequency tree is a branching diagram that records frequency data across two or more categories, with whole-number counts on the branches. The counts at each level must sum to the total at the branch before.
A frequency tree carries frequencies, whole-number counts, on its branches, while a probability tree carries probabilities. A frequency tree can be converted to relative frequencies afterwards by dividing by the overall total.
Record each trial with a tally in a frequency table, then count the tallies to get the frequency of each outcome. Check that the frequencies sum to the total number of trials; for example, a die rolled 60 times might show a 3 twelve times.
The relative frequency of rolling a 3 is 12/60 = 0.2. The theoretical probability is 1/6 ≈ 0.167, so the two are close but not identical, which is expected variation with only 60 trials.
Relative frequencies always add up to 1, which makes a useful check. For a spinner with Red 0.35, Blue 0.2, Green 0.275 and Yellow 0.175, the total is 0.35 + 0.2 + 0.275 + 0.175 = 1.0.
More trials give a better estimate because the relative frequency gets closer to the true probability as the number of trials increases. A coin flipped 10 times might give 7 heads, 0.7, but after 1000 flips the relative frequency will be much closer to 0.5.
Frequency is the number of times a particular outcome occurs in an experiment, recorded as a whole number count for each category. A frequency table records these counts so they can be turned into relative frequencies.
Uneven results do not on their own mean a die is biased, because experimental results vary. A fair die rolled 60 times gave frequencies such as 9, 10 and 12 rather than all 10, and 60 trials is not enough to be certain either way.
A comment on experimental results needs a numerical comparison, quoting the relative frequency and the theoretical probability, and a conclusion about whether the results support fairness. A bare statement like 'the die is biased' without figures will not earn the mark.