Calculating relative frequency
A spinner is spun 50 times and lands on red 18 times. What is the relative frequency of red?
Relative frequency is always a decimal between 0 and 1. It's based on actual experimental data, not theoretical expectations.
Year 9
Relative frequency and theoretical probability
Explain why experimental (relative frequency) results approach theoretical probability with larger sample sizes, and apply this idea in practical decision making.
Lesson overview
Statistics and data — probability
Relative frequency is the ratio of the number of times an event actually occurs to the total number of trials: . Relative frequency is an experimental or empirical probability based on actual data. As the number of trials increases, the relative frequency tends to get closer to the theoretical probability. This is the law of large numbers. For example, flipping a coin 10 times might give 6 heads (relative frequency 0.6), but 1000 flips might give 502 heads (relative frequency 0.502), which is much closer to the theoretical probability of 0.5.
Comparing relative frequency to theoretical probability
A fair coin should have P(heads) = 0.5. Different students flip it different numbers of times. Compare their relative frequencies.
| Number of flips | Heads | Relative frequency | Difference from 0.5 |
|---|---|---|---|
| 10 | 6 | 0.60 | 0.10 |
| 50 | 26 | 0.52 | 0.02 |
| 100 | 48 | 0.48 | 0.02 |
| 200 | 102 | 0.51 | 0.01 |
| 1000 | 501 | 0.501 | 0.001 |
This is the Law of Large Numbers: with more data, experimental results get closer to theory.
Using relative frequency to estimate probability
A die is rolled 600 times and a 6 appears 105 times. Estimate the probability of rolling a 6.
When theoretical probability is unknown, relative frequency from a large sample is a good estimate.
Worked example — testing if a coin is fair using relative frequency
A coin is flipped 400 times and heads appears 225 times. Is this evidence that the coin is unfair?
- Calculate the relative frequency: 225 ÷ 400 = 0.5625 (or 56.25%).
- Compare to theoretical probability: P(heads) = 0.5 (50%) for a fair coin.
- Find the difference: |0.5625 - 0.5| = 0.0625 (or 6.25%).
- Interpret: with 400 trials, small variations (around 5-10%) are normal due to chance.
- Conclusion: 56.25% heads is reasonably close to 50%, so the coin is likely fair.
- Note: if this pattern continued with more trials, it would become stronger evidence of bias.
Try it
Calculate relative frequencies from data, compare them to theoretical probabilities, and explain patterns.
Question 1
A spinner lands on blue 24 times out of 150 spins. What is the relative frequency of blue?
💡 Relative frequency = 24 ÷ 150 = 0.16.
Question 2
Which statement best describes the Law of Large Numbers?
💡 The Law of Large Numbers says relative frequency converges to theory with more trials.
Question 3
A die is rolled 300 times. A 6 appears 48 times. What is the relative frequency of rolling a 6?
💡 Relative frequency = 48 ÷ 300 = 0.16.
Question 4
Theoretical P(heads on a fair coin) = 0.5. After 10 flips, you get 7 heads (relative frequency 0.7). What does this tell you?
💡 With only 10 trials, getting 70% heads by chance is plausible. Need more data to suspect unfairness.
Question 5
An event has theoretical probability 0.4. In 500 trials, it occurs 215 times. Calculate the relative frequency to 2 d.p.
💡 Relative frequency = 215 ÷ 500 = 0.43.
Common mistakes
Watch for these when working through the lesson.
- Confusing relative frequency (experimental) with theoretical probability: relative frequency is based on actual data; probability is theoretical expectation.
- Thinking small samples always match theory perfectly: small samples have natural random variation; large samples converge to theory.
- Forgetting to divide by the total number of trials: relative frequency must be a ratio (frequency ÷ total), not just the frequency.
Related topics
These ideas fit closely with this lesson.
- Theoretical probability and single events
- Estimating populations from samples
- Bias and fairness in data collection