Relative and Expected Frequency
Relative frequency is an estimate of probability found by dividing how many times an event happens by the total number of trials, becoming more reliable as the number of trials increases. Expected frequency uses a probability to predict how many times an event should happen in a given number of future trials, and both ideas sit in the Probability strand of GCSE Statistics.
Before you start
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Method
- To find relative frequency, count how many times the event happened and divide by the total number of trials carried out.
- Write the relative frequency as a fraction, decimal or percentage, matching what the question asks for.
- To estimate a probability from an experiment, use the relative frequency from the largest available number of trials, since more trials give a more reliable estimate.
- To find expected frequency, multiply the probability, or relative frequency, of the event by the number of future trials.
- Round expected frequency sensibly when it describes a count of people or objects, but keep it exact where the question wants an expected value.
- Compare relative frequencies from different experiments to judge whether a dice, coin or spinner appears biased.
Worked example
A spinner is spun 200 times and lands on red 56 times. Find the relative frequency of red, and use it to estimate how many times red would occur in 500 spins.
- Relative frequency of red = 56 / 200.
- 56 / 200 = 0.28.
- Use 0.28 as an estimate of the probability of landing on red.
- Expected number of reds in 500 spins = 0.28 x 500.
- 0.28 x 500 = 140.
- Final answer: relative frequency = 0.28; expected reds in 500 spins = 140.
Practice questions
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Q1A coin is flipped 50 times and lands on heads 27 times. Find the relative frequency of heads.Show answer
Answer: 0.54 (27/50)
Q2A dice is rolled 120 times and shows a 6 on 15 of those rolls. Find the relative frequency of rolling a 6.Show answer
Answer: 0.125 (15/120 = 1/8)
Q3The probability that a machine produces a faulty item is 0.04. Out of 300 items, how many would you expect to be faulty?Show answer
Answer: 12 items (0.04 x 300)
Q4A spinner is spun 80 times and lands on blue 20 times. Estimate the probability of landing on blue.Show answer
Answer: 0.25 (20/80)
Q5Using the relative frequency from the spinner above, estimate how many times it would land on blue in 300 spins.Show answer
Answer: 75 times (0.25 x 300)
Q6Student A rolls a dice 20 times and gets a 6 four times. Student B rolls the same dice 300 times and gets a 6 forty-eight times. Whose relative frequency is a more reliable estimate of the true probability, and what is it?Show answer
Answer: Student B (more trials); relative frequency = 48/300 = 0.16
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A biased coin is flipped 40 times and lands on tails 28 times. Work out the relative frequency of tails, giving your answer as a decimal.
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A quality control test finds that 9 out of 60 light bulbs tested are faulty. (a) Work out the relative frequency of a faulty bulb. (b) Use this to estimate the number of faulty bulbs in a batch of 500.
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A dice is rolled 150 times. The results are: 1 twenty times, 2 twenty-seven times, 3 twenty-three times, 4 twenty-six times, 5 twenty-nine times, and 6 the remaining number of times. (a) Work out how many times the dice showed a 6. (b) Work out the relative frequency of scoring a 6, as a decimal. (c) A fair dice would give a relative frequency of about 0.167 for each number. Comment on whether this dice appears biased towards 6.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
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This topic is chapter 19 of GCSE Statistics Foundation Workbook 2 and chapter 25 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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