Relative Frequency
Relative frequency is the probability you observe from an experiment: the number of times an outcome happens divided by the number of trials. This NSW Year 9 Core maths subtopic revises it from Stage 4 (NESA outcome MA4‑PRO‑C‑01) and compares it with theoretical probability.
You will find relative frequencies as decimals and fractions, read frequency tables and dot plots, combine rows for events such as \(51\) or fewer matches, and find a missing relative frequency because they all add to \(1\). You will work backwards to a frequency or the number of trials, find expected numbers such as faulty light globes in a batch of \(1500\), explain why \(500\) spins give a better estimate than \(20\), and decide when a die is probably biased. It is part of Year 9 Core Probability.
Every question with a fully worked solution.
- Relative Frequency - Video - How To Make a Relative Frequency Distribution Table Watch
Theory
Relative frequency is how often an outcome actually happened in an experiment, divided by the number of trials. It is the experimental probability. This Year 9 maths page shows how to find it from counts and tables, use it to predict an expected frequency, and compare it with the theoretical probability.
The frequency of an outcome is the number of times it happened. The relative frequency is that frequency divided by the total number of trials. It is also called the experimental or observed probability.
The theoretical probability is the probability expected under fair conditions, worked out from equally likely outcomes. For a fair die, \(P(6) = \dfrac{1}{6}\).
Relative frequency and theoretical probability are rarely exactly equal. With more trials, the relative frequency usually gets closer to the theoretical probability, so more trials give a better estimate.
If, after many trials, the relative frequency is far from the theoretical probability, the object may be biased (not fair).
How to work with relative frequency
- Find the number of trials. In a table, add the whole frequency column.
- Find the frequency of the event. Add every row that fits it.
- Divide frequency by trials. Give a decimal, or simplify the fraction.
- Predict by multiplying a relative frequency by a new number of trials.
- Compare with the theoretical probability. A small gap is normal; a big gap after many trials suggests bias.
Relative frequency is the frequency divided by the number of trials.
| \(\text{relative frequency}\) | \(=\) | \(\dfrac{28}{80}\) |
| \(\) | \(=\) | \(\mathbf{0.35}\) |
| \(\dfrac{28}{80}\) | \(=\) | \(\dfrac{28 \div 4}{80 \div 4} = \mathbf{\dfrac{7}{20}}\) |
The answer is a proportion, so it is between \(0\) and \(1\). The \(28\) on its own is the frequency, not the relative frequency.
| People in car | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 18 | 12 | 6 | 4 |
Add the frequency row to get the number of trials. Then add the rows that fit the event: \(3\) people or \(4\) people.
| \(\text{cars}\) | \(=\) | \(18 + 12 + 6 + 4\) |
| \(\) | \(=\) | \(40\) |
| \(\text{more than } 2\) | \(=\) | \(6 + 4 = 10\) |
| \(\text{relative frequency}\) | \(=\) | \(\dfrac{10}{40} = \mathbf{0.25}\) |
"More than \(2\)" does not include \(2\) itself.
(i) How many faulty parts would you expect in \(2500\) parts?
(ii) A batch had \(45\) faulty parts. About how many parts were in the batch?
(i) Multiply the relative frequency by the number of trials.
| \(\text{expected}\) | \(=\) | \(0.12 \times 2500 = \mathbf{300}\) |
(ii) Work backwards: divide the frequency by the relative frequency.
| \(0.12 \times n\) | \(=\) | \(45\) |
| \(n\) | \(=\) | \(45 \div 0.12 = \mathbf{375}\) |
Both answers are estimates. A real batch will usually be close to these, not exactly on them.
Compare the relative frequency with the theoretical probability for a fair coin.
| \(\text{relative frequency}\) | \(=\) | \(\dfrac{268}{500} = 0.536\) |
| \(P(\text{H}) \text{ for a fair coin}\) | \(=\) | \(0.5\) |
| \(\text{difference}\) | \(=\) | \(0.036\) |
The gap is small. Differences like this happen often with a fair coin over \(500\) tosses, so there is no strong evidence of bias. A relative frequency far from \(0.5\) after many tosses, such as \(0.7\), would suggest bias.
Common pitfalls
Frequently asked questions
What is relative frequency?
Relative frequency is the number of times an outcome happens divided by the total number of trials. It is the experimental probability of that outcome.
How do you calculate relative frequency?
Divide the frequency of the outcome by the number of trials. An event that happened \(28\) times in \(80\) trials has a relative frequency of \(\dfrac{28}{80} = 0.35\).
What is the difference between theoretical and experimental probability?
Theoretical probability is what should happen under fair conditions, worked out from equally likely outcomes. Experimental probability, or relative frequency, is what actually happened in trials.
Why do more trials give a better estimate?
With only a few trials, chance can push the results a long way from the true probability. Over many trials the relative frequency settles close to the true probability.
How do you find the expected frequency?
Multiply the relative frequency, or probability, by the number of trials. A fault rate of \(0.12\) in \(2500\) parts gives about \(300\) faulty parts.