Relative frequency Revision Notes for Edexcel GCSE Maths

Relative frequency

What is relative frequency?

Relative frequency is a method used to estimate probabilities based on actual data collected from experiments or observations. When we don't know the theoretical probability of an event, we can use relative frequency to make a reasonable estimate.

This approach is particularly useful when dealing with real-world data where the theoretical probability might be unknown or difficult to calculate.

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Relative frequency is especially valuable in real-world scenarios where theoretical calculations aren't practical, such as quality control in manufacturing or medical research studies.

The relative frequency formula

To calculate a probability using relative frequency, we use this formula:

$\text{Probability} = \frac{\text{Frequency of outcome}}{\text{Total frequency}}$

When we calculate probability this way, it's called a relative frequency estimate. This gives us an estimated probability based on observed data rather than theoretical calculations.

How sample size affects accuracy

The accuracy of relative frequency estimates depends heavily on the sample size:

Larger samples produce more accurate probability estimates
Smaller samples may give less reliable estimates
The estimate becomes more trustworthy as the number of trials increases

This is why conducting more trials in an experiment generally leads to better probability estimates.

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The key principle: The most accurate estimate comes from the largest number of trials. Always consider whether your sample size is adequate when interpreting results.

Experimental probability

Experimental probability involves carrying out repeated trials to estimate the likelihood of an outcome. Each time we perform the experiment, we collect data that helps improve our probability estimate.

For example, if we throw a drawing pin multiple times and record how often it lands point up, we can estimate the probability of this outcome occurring.

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Unlike theoretical probability which uses known mathematical principles, experimental probability relies entirely on observed data from actual trials.

Using relative frequency with data tables

When working with frequency tables, follow these steps:

Identify the frequency of your desired outcome
Find the total frequency (sum of all frequencies)
Apply the formula: $\text{Probability} = \frac{\text{Frequency of outcome}}{\text{Total frequency}}$
Express your answer as a fraction, decimal, or percentage

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Worked Example: Calculating Probability from Egg Weight Data

If a sample of 40 eggs shows:

15 eggs weigh 55g or more
10 eggs weigh between 60-65g
Total eggs = 40

To find P(egg weighs 55g or more):

Step 1: Identify the frequency of eggs ≥ 55g = 25 eggs (15 + 10) Step 2: Total frequency = 40 eggs Step 3: Apply the formula

$P(w \geq 55) = \frac{25}{40} = \frac{5}{8} = 0.625$

Comparing experimental and theoretical probability

Understanding the relationship between these two types of probability is crucial:

Experimental probability is calculated from actual trial results
Theoretical probability is calculated using known information about equally likely outcomes
As the number of trials increases, experimental probability gets closer to theoretical probability
For a fair dice, experimental results should approach the theoretical probability of $\frac{1}{6}$ for each face

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This convergence of experimental probability towards theoretical probability is known as the Law of Large Numbers - a fundamental principle in probability theory.

Exam tips

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Essential Exam Guidelines:

Always show your working when calculating relative frequency
Remember to simplify fractions where possible
Comment on accuracy by referring to sample size
Be prepared to compare experimental results with theoretical expectations
Don't round intermediate calculations - keep exact values until the final answer

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Key Points to Remember:

Relative frequency = $\frac{\text{Frequency of outcome}}{\text{Total frequency}}$
Larger samples give more accurate probability estimates
Experimental probability gets closer to theoretical probability with more trials
Always consider whether your sample size is adequate for accuracy
Show all working steps in exam questions for full marks

Relative frequency (Edexcel GCSE Maths): Revision Notes