Relative Frequency (Edexcel GCSE Maths): Revision Notes

Relative frequency

What is relative frequency?

Relative frequency is a statistical method used to estimate the probability of events occurring based on experimental data. When we conduct experiments or collect data, we can use relative frequency to work out how likely different outcomes are to happen.

The concept is particularly useful when we want to determine whether something is behaving fairly or if there might be some bias affecting the results.

Understanding the concept

When we conduct an experiment multiple times, we record how often each outcome occurs. This count is called the frequency. However, frequency alone doesn't tell us about probability - we need to compare it to the total number of trials we performed.

For example, if we know that rolling a three on a normal dice should theoretically happen 1 out of every 6 times, we can compare our experimental results to this expected outcome. If our results are significantly different from what we'd expect, this might indicate that the dice is biassed rather than fair.

The relative frequency formula

The calculation for relative frequency is straightforward and forms the foundation of this topic:

$\text{Relative frequency} = \frac{\text{Frequency}}{\text{Number of times you tried the experiment}}$

This formula converts raw frequency data into a proportion that can be directly compared to theoretical probabilities or used to estimate the likelihood of future events.

Calculating relative frequencies step by step

To calculate relative frequencies from your data, follow these steps:

1. Count the frequency - Record how many times each outcome occurred
2. Count the total trials - Add up all the frequencies to get your total number of experiments
3. Divide each frequency by the total - This gives you the relative frequency for each outcome
4. Express as a decimal or percentage - Relative frequencies are usually written as decimals between 0 and 1

The example shows a spinner experiment where each outcome's frequency is divided by 100 (the total number of spins) to find the relative frequencies.

Using relative frequency to estimate probability

Once you have calculated relative frequencies, you can use them to estimate probabilities for future events. The more times you repeat an experiment, the more accurate your probability estimates will become.

Score	Frequency
1	140
2	137
3	138
4	259
5	161
6	165

This is particularly important when dealing with real-world situations where theoretical probabilities might not apply due to manufacturing defects, bias, or other factors affecting the outcomes.

Determining if something is fair or biassed

Relative frequency is an excellent tool for detecting bias in experiments. Here's how to approach this:

For fair outcomes: All possible results should have similar relative frequencies. For example, a fair six-sided dice should show each number roughly $\frac{1}{6}$ of the time (approximately 0.17).

For biassed outcomes: Some results will appear much more or less frequently than expected. If one outcome has a much higher relative frequency than others, or if the pattern differs significantly from what you'd expect theoretically, this suggests bias.

When analysing results, look for outcomes that are "far away" from what you'd expect. If the relative frequencies show a clear pattern that doesn't match theoretical expectations, you can conclude that the object or system is probably biassed.

Worked examples and practice

The most effective way to understand relative frequency is through practical examples. When working through problems:

1. Organize your data clearly - Use tables to show frequencies and calculations
2. Show your working - Always demonstrate how you calculated each relative frequency
3. Compare to expected values - Consider what the relative frequencies should be if the situation were fair
4. Draw conclusions - State whether the evidence suggests bias and explain your reasoning

Remember that larger sample sizes generally give more reliable estimates. If you conducted 1000 trials instead of 100, your relative frequencies would likely be closer to the true probabilities.