GCSE Maths: Relative Frequency

Introduction

Relative frequency is a fundamental concept in statistics that bridges the gap between theoretical probability and real-world data collection. While theoretical probability tells us what should happen in perfect conditions, relative frequency shows us what actually happens when we conduct experiments or collect data from real situations.

This topic is closely connected to probability theory, where relative frequency serves as an estimate for theoretical probability. It’s also linked to data handling and statistical analysis, making it an essential building block for more advanced statistical concepts you’ll encounter later in your mathematical journey.

In the real world, relative frequency is everywhere – from market research companies analyzing survey responses to medical researchers studying treatment effectiveness, and from quality control in manufacturing to sports statisticians tracking player performance.

Exam Boards

This topic applies to all exam boards at both Higher and Foundation levels.

What You Need to Know (From the Specification)

• Calculate relative frequency from experimental data
• Use relative frequency as an estimate of probability
• Compare experimental results with theoretical predictions
• Use relative frequency to make predictions about future experiments
• Record and analyse frequency data from probability experiments
• Understand the relationship between relative frequency and expected outcomes

Core Content Sections

5.1 Understanding Relative Frequency

Key Facts

🔑 Key Fact: Relative frequency is the proportion of times a particular outcome occurs in an experiment, calculated by dividing the frequency of that outcome by the total number of trials.

🔑 Key Fact: The formula for relative frequency is: Relative frequency = frequency ÷ total number of trials, or f/n

🔑 Key Fact: Relative frequency is always between 0 and 1 (inclusive), just like probability.

🔑 Key Fact: As the number of trials increases, relative frequency tends to get closer to the theoretical probability – this is known as the Law of Large Numbers.

Explanation

Relative frequency allows us to estimate probability based on experimental results rather than theoretical calculations. When we can’t calculate the exact probability of an event (perhaps because the situation is too complex or we don’t know all the factors involved), we can conduct experiments and use relative frequency as our best estimate.

The key insight is that relative frequency lets us estimate probability from real-life data when theory alone isn’t enough.. If you flip a coin 10 times and get 6 heads, the relative frequency of heads is 6/10 = 0.6. This doesn’t mean the coin is biased – it’s just the result of a small sample. But if you flip it 1000 times and get 600 heads, then you might start to suspect the coin isn’t fair.

Worked Examples

Example 1: A spinner is spun 40 times. The results are shown in the table below:

Outcome	Red	Blue	Green	Yellow
Frequency	12	8	15	5

Calculate the relative frequency for each outcome.

Solution: Step 1: Check the total number of trials. Total trials = 12 + 8 + 15 + 5 = 40

Step 2: Calculate relative frequency for each outcome using the formula f/n.

Red: Relative frequency = 12/40 = 0.3

Blue: Relative frequency = 8/40 = 0.2

Green: Relative frequency = 15/40 = 0.375

Yellow: Relative frequency = 5/40 = 0.125

Step 3: Verify our answers by checking they sum to 1. 0.3 + 0.2 + 0.375 + 0.125 = 1.0 ✓

Example 2: A student throws a six-sided dice 150 times and records getting a six 23 times. What is the relative frequency of throwing a six? How does this compare to the theoretical probability?

Solution: Step 1: Identify the frequency and total number of trials. Frequency of six = 23 Total trials = 150

Step 2: Calculate the relative frequency. Relative frequency = 23/150 = 0.153 (to 3 decimal places)

Step 3: Compare with theoretical probability. Theoretical probability of a six = 1/6 = 0.167 (to 3 decimal places)

The relative frequency (0.153) is close to but not exactly equal to the theoretical probability (0.167). This difference is normal for experimental data.

Practice Questions

When you see a practice question, you’ll learn much more if you attempt it before you look at the solution. Use the solution to check your answer, or if you get stuck.

Practice Question 1: A bag contains coloured balls. A ball is drawn at random, its colour recorded, and then replaced. This process is repeated 80 times with the following results:

Colour	Red	Blue	Green
Frequency	32	28	20

Calculate the relative frequency of drawing each colour.

Solution 1: Step 1: Verify the total number of trials. Total = 32 + 28 + 20 = 80 ✓

Step 2: Calculate relative frequency for each colour. Red: 32/80 = 0.4 Blue: 28/80 = 0.35
Green: 20/80 = 0.25

Step 3: Check our answers sum to 1. 0.4 + 0.35 + 0.25 = 1.0 ✓

Practice Question 2: A biased coin is flipped 200 times, landing on heads 130 times. Calculate the relative frequency of heads and express your answer as a percentage.

Solution 2: Step 1: Identify the values. Frequency of heads = 130 Total trials = 200

Step 2: Calculate relative frequency. Relative frequency = 130/200 = 0.65

Step 3: Convert to percentage. 0.65 × 100% = 65%

5.2 Using Relative Frequency to Estimate Probability

Key Facts

🔑 Key Fact: Relative frequency provides the best available estimate of probability when theoretical calculation is difficult or impossible.

🔑 Key Fact: Larger sample sizes generally give more reliable estimates of probability.

🔑 Key Fact: The estimated probability can be used to predict outcomes in future experiments using the formula: Expected frequency = probability × number of trials.

Explanation

Once we’ve calculated relative frequency from experimental data, we can use it as an estimate of the true probability. This estimated probability can then help us make predictions about what might happen in future experiments.

The reliability of our estimate depends heavily on the sample size. An estimate based on 1000 trials is likely to be much more accurate than one based on 10 trials. However, even with large samples, there will always be some difference between relative frequency and true theoretical probability due to random variation.

Worked Examples

Example 3: In a quality control test, 500 light bulbs are tested and 15 are found to be defective. a) Calculate the relative frequency of defective bulbs. b) Use this to estimate the probability that a randomly chosen bulb is defective. c) In a batch of 2000 bulbs, how many would you expect to be defective?

Solution: Step 1: Calculate relative frequency of defective bulbs. Relative frequency = 15/500 = 0.03

Step 2: Use relative frequency as probability estimate. Estimated probability of defective bulb = 0.03

Step 3: Calculate expected number of defective bulbs in 2000. Expected defective bulbs = 0.03 × 2000 = 60 bulbs

Example 4: A student records the weather for 60 days and finds it rains on 18 days. Based on this data, what is the probability it will rain tomorrow? If this pattern continued for a full year (365 days), on how many days would you expect rain?

Solution: Step 1: Calculate relative frequency of rainy days. Relative frequency = 18/60 = 0.3

Step 2: Use this as probability estimate. Estimated probability of rain tomorrow = 0.3 (or 30%)

Step 3: Calculate expected rainy days in a year. Expected rainy days = 0.3 × 365 = 109.5 ≈ 110 days

Practice Questions

Practice Question 3: A machine produces components, and quality testing shows that in a sample of 800 components, 24 are faulty. a) What is the relative frequency of faulty components? b) In a production run of 5000 components, how many would you expect to be faulty?

Solution 3: Step 1: Calculate relative frequency. Relative frequency = 24/800 = 0.03

Step 2: Calculate expected faulty components. Expected faulty = 0.03 × 5000 = 150 components

Practice Question 4: A football team’s results over 40 matches show 25 wins, 10 draws, and 5 losses. Use relative frequency to estimate the probability of each outcome, then predict the results if they played 80 more matches.

Solution 4: Step 1: Calculate relative frequencies. Wins: 25/40 = 0.625 Draws: 10/40 = 0.25 Losses: 5/40 = 0.125

Step 2: Predict results for 80 matches. Expected wins: 0.625 × 80 = 50 Expected draws: 0.25 × 80 = 20 Expected losses: 0.125 × 80 = 10

5.3 Comparing Experimental and Theoretical Results

Key Facts

🔑 Key Fact: Experimental results (relative frequency) rarely match theoretical probability exactly, even when the theory is correct.

🔑 Key Fact: The difference between experimental and theoretical results usually decreases as the number of trials increases.

🔑 Key Fact: Large persistent differences between experimental and theoretical results might indicate that our theoretical model is incorrect.

Explanation

One of the most important skills in probability is understanding when experimental results support or contradict theoretical predictions. Small differences are normal and expected due to random variation. However, large or persistent differences might suggest that our theoretical model needs revision.

When comparing results, we need to consider the sample size. A difference of 0.1 between relative frequency and theoretical probability is much more significant in 1000 trials than in 10 trials.

Worked Examples

Example 5: A fair six-sided dice should show each number with probability 1/6 ≈ 0.167. A student rolls the dice 120 times and gets the following results:

Number	1	2	3	4	5	6
Frequency	18	22	19	21	20	20

Compare the experimental results with the theoretical predictions.

Solution: Step 1: Calculate relative frequencies for each outcome. 1: 18/120 = 0.150 2: 22/120 = 0.183 3: 19/120 = 0.158 4: 21/120 = 0.175 5: 20/120 = 0.167 6: 20/120 = 0.167

Step 2: Compare with theoretical probability (1/6 = 0.167). All relative frequencies are close to 0.167, with differences ranging from 0.008 to 0.017.

Step 3: Interpret the results. These small differences are consistent with a fair dice. The experimental results support the theoretical model.

Example 6: A coin is flipped 500 times and lands on heads 320 times. Is this evidence that the coin is biased?

Solution: Step 1: Calculate the relative frequency of heads. Relative frequency = 320/500 = 0.64

Step 2: Compare with theoretical probability for a fair coin. Theoretical probability = 0.5

Step 3: Assess the difference. Difference = 0.64 – 0.5 = 0.14

This is a substantial difference (14 percentage points) in a large sample of 500 trials. This provides strong evidence that the coin is biased towards heads.

Practice Questions

Practice Question 5: A spinner should land on red with probability 0.25. In 200 spins, it lands on red 45 times. Calculate the relative frequency and comment on whether this result is consistent with the theoretical probability.

Solution 5: Step 1: Calculate relative frequency. Relative frequency = 45/200 = 0.225

Step 2: Compare with theoretical probability. Theoretical probability = 0.25 Difference = 0.25 – 0.225 = 0.025

Step 3: Interpret the result. The difference of 0.025 (2.5 percentage points) in 200 trials is small and consistent with random variation. The result supports the theoretical probability.

Practice Question 6: A bag should contain equal numbers of red and blue balls. In 300 draws with replacement, 180 red balls and 120 blue balls are drawn. Does this suggest the bag composition is different from expected?

Solution 6: Step 1: Calculate relative frequencies. Red: 180/300 = 0.6 Blue: 120/300 = 0.4

Step 2: Compare with expected probabilities. Expected probability for each colour = 0.5 Difference for red = 0.6 – 0.5 = 0.1 Difference for blue = 0.5 – 0.4 = 0.1

Step 3: Interpret the results. A 10 percentage point difference in 300 trials is quite substantial and suggests the bag may not contain equal numbers of red and blue balls.

Statistical Insight: Why Relative Frequency Matters

Understanding relative frequency is crucial because it bridges the gap between mathematical theory and real-world applications. In many practical situations, we can’t calculate exact probabilities because the systems are too complex or we don’t have complete information. Relative frequency gives us a way to quantify uncertainty and make informed decisions based on observed data.

This concept is fundamental to statistical inference, where we use sample data to make conclusions about entire populations. Whether it’s testing the effectiveness of a new medicine, analyzing customer behavior, or assessing the reliability of manufactured products, relative frequency provides the foundation for evidence-based decision making.

The relationship between relative frequency and theoretical probability also illustrates one of the most important principles in statistics: the Law of Large Numbers. This principle explains why casinos are profitable (they rely on large numbers of customers), why insurance companies can predict their costs, and why quality control systems work in manufacturing.

Common Mistakes and Misconceptions

Mistake 1: Expecting Relative Frequency to Equal Theoretical Probability Exactly

Incorrect thinking: "I flipped a coin 10 times and got 6 heads. The relative frequency is 0.6, but it should be 0.5, so something’s wrong."

Correct approach: Understand that random variation means experimental results will differ from theoretical predictions, especially with small sample sizes. The relative frequency should approach theoretical probability as the number of trials increases, but it won’t be exactly equal.

How to avoid: Remember that probability describes long-term patterns, not short-term results. Small experiments often show significant variation from expected values.

Mistake 2: Confusing Frequency with Relative Frequency

Incorrect thinking: Using the raw frequency count instead of dividing by the total number of trials.

Correct approach: Always calculate relative frequency as frequency ÷ total trials. The relative frequency tells you the proportion, which is what you need for probability estimation.

How to avoid: Always check that your relative frequencies for all outcomes sum to 1 (or 100% if using percentages).

Mistake 3: Assuming Larger Numbers Always Mean More Accurate Estimates

Incorrect thinking: "100 trials is always better than 50 trials for estimating probability."

Correct approach: While larger samples generally give better estimates, the quality also depends on how the data was collected. A biased sample of 1000 can be worse than an unbiased sample of 100.

How to avoid: Consider both sample size and collection method when evaluating the reliability of probability estimates.

Mistake 4: Not Considering Random Variation When Interpreting Results

Incorrect thinking: "The relative frequency is 0.52 and the theoretical probability is 0.5, so the theory must be wrong."

Correct approach: Small differences between relative frequency and theoretical probability are normal due to random variation. Only large or persistent differences suggest problems with the theoretical model.

How to avoid: Learn to distinguish between meaningful differences and normal random variation. Consider the sample size when making this judgment.

Mistake 5: Forgetting That Relative Frequency is an Estimate

Incorrect thinking: Using relative frequency as if it were the exact probability for future predictions.

Correct approach: Remember that relative frequency provides an estimate of probability based on limited data. It’s our best guess, but it’s still an estimate with uncertainty.

How to avoid: Use phrases like "estimated probability" or "based on the data" when making predictions using relative frequency.

End-of-Topic Practice Questions

Question 1: A survey of 250 students found that 175 walk to school, 45 come by bus, and 30 come by car. Calculate the relative frequency for each method of transport.

Question 2: A factory tests a sample of 400 products and finds 12 are defective. a) Calculate the relative frequency of defective products. b) Estimate how many defective products there would be in a batch of 2500.

Question 3: A weather station records the weather for 90 days. The results are: 54 sunny days, 24 cloudy days, and 12 rainy days. a) Calculate the relative frequency for each type of weather. b) Based on this data, what is the probability it will be sunny tomorrow? c) In the next 180 days, how many would you expect to be rainy?

Question 4: A biased spinner has four sections: A, B, C, and D. After 200 spins, the results are: A: 60 times, B: 50 times, C: 70 times, D: 20 times a) Calculate the relative frequency for each section. b) Which section has the highest probability of being selected? c) If the spinner is used 500 more times, how many times would you expect section C to be selected?

Question 5: A standard six-sided dice should show each number with equal probability (1/6 ≈ 0.167). A student rolls the dice 180 times with these results: 1: 25 times, 2: 32 times, 3: 28 times, 4: 35 times, 5: 30 times, 6: 30 times a) Calculate the relative frequency for each number. b) Compare these results with the theoretical probability. c) Do these results suggest the dice is fair? Explain your reasoning.

Question 6: A basketball player practices free throws. In 80 attempts, she scores 68 times. a) What is the relative frequency of successful free throws? b) Based on this data, how many successful free throws would you expect in her next 25 attempts? c) In a game where she attempts 15 free throws, what is the estimated probability she scores at least 12?

Question 7: Two coins are flipped together 120 times. The results are: Two heads: 28 times One head, one tail: 64 times
Two tails: 28 times a) Calculate the relative frequency for each outcome. b) The theoretical probabilities are: two heads = 0.25, one of each = 0.5, two tails = 0.25. Compare your experimental results with these theoretical values. c) Do the experimental results support the theoretical model?

Question 8: A quality control manager wants to estimate the proportion of faulty items produced by a machine. In a sample of 600 items, 18 are found to be faulty. a) Calculate the relative frequency of faulty items. b) The company considers the machine acceptable if less than 4% of items are faulty. Based on this sample, should the machine be accepted? c) If the production rate is 10,000 items per day, approximately how many faulty items would be produced daily?

End-of-Topic Solutions

Solution 1: Total students = 175 + 45 + 30 = 250 Walking: 175/250 = 0.7 Bus: 45/250 = 0.18 Car: 30/250 = 0.12

Solution 2: a) Relative frequency = 12/400 = 0.03 b) Expected defective products = 0.03 × 2500 = 75

Solution 3: Total days = 54 + 24 + 12 = 90 a) Sunny: 54/90 = 0.6, Cloudy: 24/90 = 0.267, Rainy: 12/90 = 0.133 b) Probability of sunny weather = 0.6 c) Expected rainy days = 0.133 × 180 = 24 days

Solution 4: Total spins = 60 + 50 + 70 + 20 = 200 a) A: 60/200 = 0.3, B: 50/200 = 0.25, C: 70/200 = 0.35, D: 20/200 = 0.1 b) Section C has the highest probability (0.35) c) Expected selections of C = 0.35 × 500 = 175 times

Solution 5: Total rolls = 25 + 32 + 28 + 35 + 30 + 30 = 180 a) 1: 25/180 = 0.139, 2: 32/180 = 0.178, 3: 28/180 = 0.156, 4: 35/180 = 0.194, 5: 30/180 = 0.167, 6: 30/180 = 0.167 b) All values are close to 0.167, with differences ranging from 0.006 to 0.028 c) The results suggest the dice is fair, as all relative frequencies are reasonably close to the expected value

Solution 6: a) Relative frequency = 68/80 = 0.85 b) Expected successful attempts = 0.85 × 25 = 21.25 ≈ 21 c) This requires calculating P(at least 12 successes in 15 attempts) which is beyond GCSE scope, but the estimated success rate is 85%

Solution 7: Total flips = 28 + 64 + 28 = 120 a) Two heads: 28/120 = 0.233, One of each: 64/120 = 0.533, Two tails: 28/120 = 0.233 b) Experimental vs theoretical: Two heads: 0.233 vs 0.25 (difference 0.017), One of each: 0.533 vs 0.5 (difference 0.033), Two tails: 0.233 vs 0.25 (difference 0.017) c) Yes, the small differences are consistent with random variation

Solution 8: a) Relative frequency = 18/600 = 0.03 = 3% b) Yes, 3% is less than the 4% threshold, so the machine should be accepted c) Expected daily faulty items = 0.03 × 10,000 = 300

Summary

• Relative frequency bridges theory and practice by providing probability estimates from real data
• The formula is always frequency ÷ total trials, giving values between 0 and 1
• Larger sample sizes generally provide more reliable probability estimates
• Experimental results naturally vary from theoretical predictions due to randomness
• Relative frequency can predict expected outcomes in future experiments
• Understanding when differences are meaningful versus normal variation is crucial for proper interpretation