Relative frequency is an estimate of probability from an experiment: relative frequency = number of times the outcome happened ÷ total number of trials. It is also called experimental probability. The more trials there are, the more reliable the estimate, and comparing the relative frequency with the theoretical probability shows whether something, such as a dice, might be biased.
Relative frequency
- Relative frequency = frequency of outcome ÷ total trials
- More trials give a more reliable estimate
- Biased: relative frequency far from the theoretical probability after many trials
- Estimate = relative frequency × number of future trials
Calculating relative frequency
Divide the number of times the outcome happened by the total number of trials. A spinner that lands on blue 33 times in 120 spins has a relative frequency of 33/120 = 0.275 for blue. It is an estimate of the probability, based on what actually happened.
More trials, better estimate
A small number of trials can give a misleading result just by chance. As the number of trials increases, the relative frequency tends to settle down close to the true probability. That is why an estimate from 1000 trials is far more reliable than one from 10.
When to use relative frequency
Some probabilities cannot be worked out from equally likely outcomes, such as the chance of a drawing pin landing point-up or of a machine producing a faulty item. For these, relative frequency from an experiment or from past data is the only way to estimate the probability.
Judging fairness
Compare the relative frequency with the probability you would expect for a fair object. A fair die should show a six about 1/6 ≈ 0.167 of the time. If, after 300 rolls, a six has appeared 82 times, the relative frequency is 0.273, which is much higher, so the die is probably biased.
Making predictions
Use the relative frequency to estimate future outcomes: multiply it by the number of future trials. With a relative frequency of 0.275 for blue, about 0.275 × 400 = 110 blues would be expected in 400 spins.
Worked examples
Example 1. A spinner lands on blue 33 times in 120 spins. Find the relative frequency of blue.
- 33 ÷ 120.
Answer: 0.275.
Example 2. A drawing pin is dropped 200 times and lands point-up 130 times. Find the relative frequency of landing point-up.
- 130 ÷ 200.
Answer: 0.65.
Example 3. A die is rolled 300 times and shows a six 82 times. Is the die likely to be fair? Explain.
- Relative frequency of six: 82 ÷ 300 = 0.273…
- For a fair die, P(six) = 1/6 ≈ 0.167.
Answer: Probably not: after many rolls, 0.273 is much higher than 0.167.
Example 4. Using a relative frequency of 0.275 for blue, estimate the number of blues in 400 spins.
- 0.275 × 400.
Answer: 110.
Example 5. Two students test a coin. One gets 7 heads in 10 flips, the other 507 heads in 1000 flips. Whose result gives the better estimate of P(heads)?
- The second student did many more trials.
Answer: The second student's, giving 0.507, because more trials are more reliable.
Common mistakes
- Dividing by the wrong total. Divide by the total number of trials.
- Judging bias from a few trials. Small samples can vary a lot by chance.
- Confusing relative frequency with frequency. Relative frequency is a fraction or decimal of the total.
- Expecting the relative frequency to match exactly. It is an estimate; it will usually differ slightly.
- Not comparing with the theoretical probability. To judge fairness, compare with what a fair object would give.
How relative frequency is examined
Relative frequency is Foundation content in the Edexcel 1MA1 GCSE and the 4MA1 International GCSE, usually alongside expected frequency. Questions ask you to:
- work out a relative frequency from results
- say which estimate is more reliable, and why
- decide whether a dice, coin or spinner is biased
- use a relative frequency to make a prediction
The relative frequency worksheet has questions with answers, and the probability revision engine gives more practice.
Predictions use expected frequency; theoretical probabilities come from basic probability.
Frequently asked questions
What is relative frequency?
An estimate of probability from an experiment: the number of times an outcome happened divided by the total number of trials.
Why do more trials give a better estimate?
Because random variation evens out over many trials, so the result gets closer to the true probability.
How can relative frequency show bias?
If, after many trials, it is far from the expected probability for a fair object.