Maths
What is relative frequency, and how do you find it?
Quick answer
Relative frequency is how often something happens compared with the total, frequency ÷ total. If 6 of 20 students walk to school, the relative frequency is 6/20 = 0.3, or 30%.
In one sentence
Relative frequency tells you what fraction of the time something happened: the number of times it happened divided by the total.
Explained
A frequency is a count: how many times something happens. If 14 students in a class of 40 walk to school, the frequency of "walk" is 14. A relative frequency compares that count with the total, so it shows the share instead of the raw number:
relative frequency = frequency ÷ total frequency
For the walkers that is 14 ÷ 40 = 0.35. You can write a relative frequency as a fraction (14/40 = 7/20), a decimal (0.35) or a percentage (35%). All three mean the same thing: 35 out of every 100.
bus
bike
car
Why bother? Because counts are hard to compare when the totals are different. Suppose 12 of the 30 students in class A like football and 15 of the 50 students in class B do. Class B has more football fans, but as a share, class A wins: 12 ÷ 30 = 0.4 (40%) against 15 ÷ 50 = 0.3 (30%). Relative frequencies put groups of different sizes on the same scale.
Two facts are always true:
- Every relative frequency is between 0 and 1 (between 0% and 100%).
- In a complete table, the relative frequencies add up to 1 (100%), because together they cover every result exactly once. This is a quick check on your working.
Relative frequency is also the bridge between statistics and probability. When you cannot work out a probability from equally likely outcomes (will a drawing pin land point up? will a basketball player score?), you do an experiment and use the relative frequency as an estimate. This is called experimental probability. If a pin lands point up 76 times in 200 throws, the estimated probability is 76 ÷ 200 = 0.38.
The more trials you do, the better the estimate. In one experiment, a fair coin gave 7 heads in the first 10 flips (0.7), 46 heads in 100 flips (0.46) and 508 heads in 1,000 flips (0.508). The relative frequency wobbles a lot at first and then settles close to the true probability of 0.5. This is the law of large numbers.
Turn it around and you can predict: expected frequency = relative frequency × number of trials. If the pin lands point up with relative frequency 0.38, in 500 throws you would expect about 0.38 × 500 = 190.
Examples
How 40 students travel to school:
| Travel | Frequency | Relative frequency | Percentage |
|---|---|---|---|
| Walk | 14 | 14 ÷ 40 = 0.35 | 35% |
| Bus | 10 | 10 ÷ 40 = 0.25 | 25% |
| Car | 12 | 12 ÷ 40 = 0.30 | 30% |
| Bike | 4 | 4 ÷ 40 = 0.10 | 10% |
| Total | 40 | 1 | 100% |
This is a relative frequency table (also called a relative frequency distribution). Drawn as a bar chart with relative frequency up the side, it has exactly the same shape as the ordinary bar chart: only the numbers on the vertical axis change. The same goes for a relative frequency histogram of grouped data.
Worked example: a dice is rolled 600 times and lands on 6 a total of 140 times. The relative frequency is 140 ÷ 600 ≈ 0.23. A fair dice should give about 1/6 ≈ 0.17, or about 100 sixes in 600 rolls. 140 is a lot more than that over so many rolls, which suggests the dice may be biased.
Running totals of relative frequencies are cumulative relative frequencies: walk and bus together are 0.35 + 0.25 = 0.60. See cumulative frequency for more.
Common mistakes
- Dividing by the wrong number. Fix: always divide by the total frequency (here 40), not by the number of categories or by another frequency.
- Relative frequencies that don't add up to 1. Fix: check your total. Rounding can give 0.99 or 1.01; anything further off means a mistake.
- Mixing up decimals and percentages. 0.3 is 30%, not 0.3%. Fix: multiply by 100 to get a percentage.
- Treating relative frequency as the exact probability. Fix: it is an estimate from one experiment. Repeat it and you get a slightly different answer; more trials give a better estimate.
Step by step
Step by step
- Count the frequenciesTally how many times each result happened.
- Find the totalAdd up all the frequencies to get the total number of results or trials.
- DivideFor each row, relative frequency = frequency ÷ total.
- Convert if askedMultiply by 100 for a percentage, or simplify the fraction.
- CheckThe relative frequencies should add up to 1 (or 100%).
To go backwards, multiply: frequency = relative frequency × total. That is also how you find an expected frequency from a probability.
Practice questions
Relative frequency
Try each one, then check your answer.
- 8 out of 20 students have a pet. What is the relative frequency?
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8 ÷ 20 = 0.4 - Write a relative frequency of 0.4 as a percentage.
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0.4 × 100 = 40% - A dice is rolled 50 times and lands on 5 nine times. What is the relative frequency of a 5?
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9 ÷ 50 = 0.18 - A coin is flipped 80 times and lands heads 36 times. What is the relative frequency of tails?
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Tails: 80 − 36 = 44, so 44 ÷ 80 = 0.55 - A table has frequencies 3, 5 and 12. Find the three relative frequencies.
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Total 20: 3 ÷ 20, 5 ÷ 20 and 12 ÷ 20 = 0.15, 0.25 and 0.6 - The relative frequency of red is 0.35 out of 60 results. How many results were red?
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0.35 × 60 = 21 - The relative frequencies of A and B are 0.2 and 0.45. There is only one other result, C. What is its relative frequency?
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1 − 0.2 − 0.45 = 0.35 - 25% of the 48 people asked chose blue. How many chose blue?
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0.25 × 48 = 12 - Write 7 out of 25 as a decimal and as a percentage.
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7 ÷ 25 = 0.28 = 28% - In class A, 12 of 30 students like football; in class B, 15 of 50 do. Which class has the greater proportion of football fans?
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12 ÷ 30 = 0.4 and 15 ÷ 50 = 0.3, so class A - A spinner lands on green 46 times in 200 spins. Estimate the probability of green.
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46 ÷ 200 = 0.23 - Using your answer to question 11, how many greens would you expect in 1,000 spins?
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0.23 × 1,000 = 230 - Frequencies are 5, 15, 20 and 10. What are the relative frequencies as percentages?
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Total 50: 10%, 30%, 40% and 20% - A dice lands on 6 a total of 140 times in 600 rolls. Is it likely to be fair?
Show answer
140 ÷ 600 ≈ 0.23, well above 1/6 ≈ 0.17 (we would expect about 100 sixes), so it may be biased
Word problems
Relative frequency in real life
Try each one, then check your answer.
- An ice-cream van sold 250 ice creams, and 90 of them were chocolate. What percentage were chocolate?
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90 ÷ 250 = 0.36 = 36% - A basketball player scored 34 of her last 40 free throws. Estimate how many she will score in her next 60.
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34 ÷ 40 = 0.85, and 0.85 × 60 = 51 - It rained on 9 of the 30 days in April. What is the relative frequency of a rainy day?
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9 ÷ 30 = 0.3 - A factory finds 12 faulty bulbs in a sample of 400. About how many faulty bulbs would you expect in 5,000?
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12 ÷ 400 = 0.03, and 0.03 × 5,000 = 150 bulbs - In town A, 45 of 150 people asked cycle to work; in town B, 60 of 240 do. Where is cycling more common?
Show answer
45 ÷ 150 = 0.3 and 60 ÷ 240 = 0.25, so town A (30% against 25%)
Questions people ask
What is the formula for relative frequency?
Relative frequency = frequency ÷ total frequency, often written f ÷ n. For a percentage, multiply the result by 100.
What is the difference between frequency and relative frequency?
Frequency is the count (14 students walk). Relative frequency is that count as a share of the total (14 out of 40 = 0.35). Frequencies add up to the total; relative frequencies add up to 1.
What is a relative frequency table?
It is a frequency table with an extra column that shows each frequency divided by the total. It may also have a percentage column. The relative frequency column must add up to 1.
Is relative frequency the same as probability?
Not quite. Probability is the true long-run chance; relative frequency is what actually happened in your data. With many trials the relative frequency is a good estimate of the probability, so it is often called experimental probability.
What is a relative frequency histogram or bar graph?
It is a histogram or bar chart with relative frequency (or percentage) on the vertical axis instead of frequency. The shape is the same; it just makes groups of different sizes easier to compare.
How do you calculate relative frequency in Excel or Google Sheets?
Put the frequencies in B2:B5. In C2 type =B2/SUM($B$2:$B$5), then fill the formula down. Format column C as a percentage if you want percentages.
Quick quiz
Relative frequency
Check what you know with 5 quick questions.
Question 1 of 5
15 out of 60 people in a survey chose pizza. What is the relative frequency of pizza?
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Related topics
Probability
Probability measures how likely something is to happen, on a scale from 0 (impossible) to 1 (certain). A fair coin landing heads has a probability of 1/2.
Cumulative frequency
Cumulative frequency is a running total of frequencies. Add each frequency to the total so far; the last value equals the total, and a cumulative frequency graph gives the median and quartiles.
Histograms
A histogram shows grouped numerical data as touching bars on a number scale, one bar per class. When the classes have different widths, the area of each bar shows the frequency.
Percentages
A percentage is a number out of 100, written with the sign %. So 25% means 25 out of every 100, which is the same as one quarter.
The binomial distribution
The binomial distribution B(n, p) gives the probability of k successes in n independent trials, each with success probability p: P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ. Its mean is np.
Conditional probability
Conditional probability is the chance of an event A when you already know event B has happened. It is written P(A|B) and found with P(A|B) = P(A and B) ÷ P(B).