Relative & Expected Frequency
Learn relative frequency and expected frequency for NSW Year 12 Mathematics Standard 2. Relative frequency is how often an outcome happens as a fraction of the total trials — the experimental probability — and it estimates the true probability of an event.
You will calculate relative frequency from a frequency table, compare experimental and theoretical probability, and find the expected frequency as \(n\times p\) — core probability skills for Standard 2, from dice and spinner trials to quality-control samples.
Theory
Relative frequency is the fraction of trials in which an outcome happens, and it estimates probability. This Year 12 Standard 2 (NSW) guide shows how to find relative frequency from a table, compare experimental and theoretical probability, and calculate the expected frequency as \(n\times p\).
The relative frequency of an outcome is the number of times it happens divided by the total number of trials: \(\dfrac{\text{frequency}}{\text{total}}\). It is always a value between \(0\) and \(1\), and is also called the experimental probability.
Because it comes from actually running trials, relative frequency is used as an estimate of the probability. The theoretical probability is worked out from equally likely outcomes (for example \(P(\text{six})=\tfrac{1}{6}\) on a fair die). As the number of trials grows, the relative frequency settles toward this theoretical value.
The expected frequency is how many times an outcome should happen on average: \(n\times p\), the number of trials multiplied by the probability. In this Year 12 Standard 2 (NSW) topic you use the theoretical \(p\) for a fair set-up, or the relative frequency as an estimate of \(p\) from data such as quality-control samples.
With frequency \(f\) out of \(N\) trials, probability \(p\) and \(n\) trials:
The relative frequency estimates the probability, so it is the experimental probability:
How to work with relative and expected frequency
- Find the total number of trials \(N\) (add up every frequency).
- Relative frequency: divide the frequency of the outcome by \(N\); for a group of outcomes, add their frequencies first.
- Estimate the probability: use the relative frequency as the experimental probability \(p\), or use the theoretical \(p\) if the set-up is fair.
- Expected frequency: multiply the number of trials by the probability, \(n\times p\), and interpret it as an average.
| Colour | Frequency |
|---|---|
| Red | 50 |
| Blue | 64 |
| Green | 46 |
| Yellow | 40 |
Divide the frequency of Blue by the total number of spins.
| \(\text{total}\) | \(=\) | \(50+64+46+40 = 200\) |
| \(\text{rel. freq.}\) | \(=\) | \(\dfrac{64}{200}\) |
| \(\) | \(=\) | \(0.32\) |
The relative frequency of Blue is \(0.32\).
| Biscuits per packet | Frequency |
|---|---|
| 22 | 3 |
| 23 | 9 |
| 24 | 16 |
| 25 | 9 |
| 26 | 3 |
Add the frequencies for \(24\), \(25\) and \(26\), then divide by \(40\).
| \(24\text{ or more}\) | \(=\) | \(16+9+3 = 28\) |
| \(\text{rel. freq.}\) | \(=\) | \(\dfrac{28}{40}\) |
| \(\) | \(=\) | \(0.7\) |
The relative frequency is \(0.7\).
Find each theoretical probability, then multiply by \(240\).
| \(E(8)\) | \(=\) | \(240\times\dfrac{1}{8} = 30\) |
| \(P(\text{even})\) | \(=\) | \(\dfrac{4}{8}=\dfrac{1}{2}\) |
| \(E(\text{even})\) | \(=\) | \(240\times\dfrac{1}{2} = 120\) |
About \(30\) eights and \(120\) even numbers.
Use the relative frequency as an estimate of \(p\), then expected \(=n\times p\).
| \(p\) | \(\approx\) | \(\dfrac{15}{250} = 0.06\) |
| \(\text{expected}\) | \(=\) | \(n\times p\) |
| \(\) | \(=\) | \(4000\times0.06 = 240\) |
About \(240\) underweight jars are expected.
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. For example, if Blue comes up 64 times in 200 spins, its relative frequency is 64 divided by 200, which is 0.32. It is always between 0 and 1.
What is the difference between experimental and theoretical probability?
Theoretical probability is worked out from equally likely outcomes, such as one sixth for rolling a six on a fair die. Experimental probability is the relative frequency you actually measure by running trials. They get closer as the number of trials increases.
How do you calculate expected frequency?
Multiply the number of trials by the probability of the outcome: expected frequency equals n times p. For instance, in 240 rolls of an eight-sided die you expect 240 times one eighth, which is 30 rolls showing an eight.
Can you use relative frequency to estimate probability?
Yes. When a situation is not obviously fair, the relative frequency from a sample is used as an estimate of the probability. A quality inspector who finds 15 underweight jars in 250 estimates the probability of underweight as 0.06.
Why doesn't the relative frequency exactly match the theoretical probability?
Because each trial is subject to chance. Over a small number of trials the relative frequency can swing well above or below the theoretical value, but as the number of trials grows it settles toward the theoretical probability.
Does expected frequency have to be a whole number?
No. Expected frequency is an average, so n times p can be a decimal. You report it as about that many, and it tells you what to expect on average rather than guaranteeing an exact count.