Relative Frequency: KS3 Maths

What is relative frequency?

Relative frequency measures how often an event occurs compared to the total number of trials:

$$\text{relative frequency} = \frac{\text{number of times the event occurs}}{\text{total number of trials}}$$

The result is always a value between 0 and 1 (or equivalently, between 0% and 100%).

Example: A coin is flipped 50 times and lands on heads 23 times. The relative frequency of heads is:

23 ÷ 50 = 0.46 (or 46%).

How do you calculate relative frequency from a table?

Worked example: A spinner with four colours is spun 200 times. The results are recorded below.

Check: All relative frequencies must sum to 1.00 (since one outcome always occurs). ✓

Each relative frequency is that colour's estimated probability for this spinner.

How does relative frequency estimate probability?

When you cannot work out the probability theoretically — for example, because a die might be biased or a spinner's sections are not equal — you use experimental probability instead.

The experimental probability of an event is simply its relative frequency from a set of trials. You treat the relative frequency as your best estimate of the true probability.

In the spinner example above:

• The estimated probability of landing on blue is 0.36.
• If the spinner were fair, each colour would have probability 0.25. Blue's high relative frequency suggests the blue section may be larger than the others.

Why does more trials give a better estimate?

With a small number of trials, chance alone can make relative frequency swing far from the true probability. With many trials, the random variation evens out.

Example: Flipping a fair coin.

• After 10 flips, you might get 7 heads → relative frequency = 0.70 (far from 0.5).
• After 100 flips, you might get 53 heads → relative frequency = 0.53 (closer).
• After 1000 flips, you might get 498 heads → relative frequency = 0.498 (very close).

This settling-down behaviour is known as the law of large numbers: as the number of trials increases, the relative frequency gets closer and closer to the true probability.

A KS3 exam question might give you relative frequencies from two different experiments with different numbers of trials and ask which is the better estimate — always choose the one with more trials.

How does relative frequency differ from theoretical probability?

Theoretical probability is found by reasoning about equally likely outcomes (for example, a fair die has six equally likely faces, so P(3) = 1/6). It does not require any experiment.

Relative frequency (experimental probability) is found by actually running an experiment and counting outcomes. It is used when:

• We cannot assume equally likely outcomes (e.g., a biased die).
• The situation is too complex to reason about theoretically (e.g., weather prediction).
• We want to test whether something is fair.

How do you interpret a relative frequency graph?

Some questions show a graph of relative frequency against number of trials. As the number of trials increases, the graph typically starts with large swings (early variation) and then settles towards a stable value — the estimated true probability.

Reading the graph: The value the line is settling towards (its long-run level) is the best estimate of the true probability. Early fluctuations are expected and do not mean the experiment is faulty.

Frequently asked questions

What is the difference between frequency and relative frequency?

Frequency is the raw count of how many times an event occurs (e.g., "heads appeared 23 times"). Relative frequency is that count expressed as a fraction or proportion of the total trials (e.g., "heads appeared 23 out of 50 times, so the relative frequency is 0.46"). Relative frequency allows you to compare results from experiments with different total numbers of trials.

Do all the relative frequencies in an experiment have to add up to 1?

Yes, if the categories are mutually exclusive and exhaustive (they cover all possible outcomes with no overlap). Every trial produces exactly one outcome, so summing the relative frequencies for all categories always gives 1. If your relative frequencies do not sum to 1, you have made an arithmetic error — check each calculation.

Is relative frequency always a good estimate of probability?

It is the best estimate available from the data, but it is not exact. A biased experiment (for example, repeatedly flipping a coin in the same way) can give a misleading relative frequency. For a reliable estimate, the trials should be independent and carried out under consistent conditions.

How is relative frequency used in real life?

Relative frequency underpins many real-world estimates: the probability of rain in a weather forecast is based on historical frequency data; insurers estimate the likelihood of claims from past records; manufacturers calculate defect rates from production data. In each case, the true probability is unknown, and relative frequency from large numbers of observations provides the best available estimate.

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