# Relative Frequency Worksheets, Questions and Revision

GCSE 4 - 5KS3AQAEdexcelOCRWJECFoundationAQA 2022OCR 2022

## Relative Frequency

There are two types of probability you will see:

• Theoretical probability – this is the kind of probability that we have prior understanding of. For example, we know that the chance of rolling a $6$ on a fair die is $\dfrac{1}{6}$.
• Relative frequency – this is the kind of probability that we determine from a survey or experiment.

Make sure you are happy with the following topics before continuing.

Level 4-5 GCSE

## Relative Frequency

We calculate relative frequency using the following formula:

$\text{relative frequency }=\dfrac{\text{no. of times an outcome happened}}{\text{total no. of all outcomes}}$

Example: A coin is flipped $100$ times, the coin lands on heads $48$ times.

Calculate the relative frequency for heads.

$\text{relative frequency }=\dfrac{48}{100} =0.48$

for landing on heads.

Level 4-5GCSE

## Expected Frequency

We can use relative frequency to determine expected frequency, which is the number of times we would expect an outcome to happen. It is calculated as follows:

$\text{expected frequency }=\text{ probability }\times\text{number of trials}$

Example: If we roll a dice $60$ times, the expected frequency of sixes is

$\dfrac{1}{6}\times60= 10$

Note: In practice, it probably won’t end up being exactly $10$ times,

i.e. Our relative frequency won’t be $\dfrac{1}{6}$. However, if you do more trials, the relative frequency will get closer to the expected frequency.

Level 4-5GCSE

## Bias

One of the key uses of relative frequency is in testing for bias. We say that an experiment is biased when the probability of a particular outcome is unfairly bigger or smaller than it should be.

Example: A biased coin might be cleverly designed so that it’s lands on tails more than $50\%$ of the time. If we suspect a coin of being biased, then we test it by flipping it a lot of times and recording the results. Then, if the relative frequency of tails is noticeably more than $50\%$, we might suspect the coin of being biased.

Level 4-5 GCSE
Level 4-5 GCSE

## Example 1: Relative Frequency

Prim asks some people in her town about their dietary habits and records the results in the table below.

a) Work out the relative frequency of someone in Prim’s town being vegetarian.

b) There are $20,000$ people in Prim’s town. Using your answer to part a), find an estimate for the number of people in this town who are vegetarian.

[4 marks]

a) Firstly, we need the total number of all outcomes:

$\text{total }=44+23+43=110$

Now, using the formula shown above, we can calculate the relative frequency for vegetarians to be

$\text{relative frequency }=\dfrac{44}{110}=\dfrac{2}{5}$

b) Our answer to part a) suggests that if we picked a random person in Prim’s town, there would be a $\dfrac{2}{5}$ or $0.4$ chance that they are vegetarian. Therefore

$\dfrac{2}{5}\times 20,000=8,000\text{ vegetarians}$

Level 4-5GCSE

## Example 2: Relative Frequency and Bias

Amber has a spinner with the numbers $1$, $2$, $3$ and $4$ on it. She was told that the spinner is fair, i.e. there is an equal chance of landing on any number when you spin it, but she claims that the spinner is biased.

Amber spins the spinner $150$ times and it lands on the number one $54$ times. By calculating the relative frequency of landing on $1$, comment on Amber’s claim.

[3 marks]

Firstly, the relative frequency:

$\text{relative frequency of landing on 1 }=\dfrac{54}{150}=0.36$

So, the relative frequency is $0.36$, or $36\%$.

But we know there are $4$ equally likely outcomes on the spinner, meaning the theoretical probability of landing on $1$ is

$\dfrac{1}{4}=25\%$

$25\%$ is significantly smaller than $36\%$, so it seems that Amber has good to reason to make the claim she does. The results suggests the spinner is biased.

Note: We don’t know this for sure. It’s possible that a fair spinner could land on the number one $54$ times due to chance, but it is unlikely, which leads us to believe the spinner is biased.

We can always do a better experiment and make our conclusion more confident by increasing the number of trials!

Level 4-5 GCSE

## Example Questions

a) $12$ students out of the total of $52$ used public transport so, as a fraction, this can be expressed as:

$\dfrac{12}{52}$

To calculate the relative frequency as a decimal to $3$ decimal places, divide the top by the bottom:

$\dfrac{12}{52}=0.231$ to $3$ decimal places

b) $16$ students out of the total of $52$ walked to school so, as a fraction, this can be expressed as:

$\dfrac{16}{52}$

To calculate the relative frequency as a decimal to $3$ decimal places, divide the top by the bottom:

$\dfrac{16}{52}=0.308$ to $3$ decimal places

c) If $9$ of the $52$ students cycled, then the number who didn’t cycle is:

$52-9=43$

As a fraction, this can be expressed as:

$\dfrac{43}{52}$

To calculate the relative frequency as a decimal to $3$ decimal places, divide the top by the bottom:

$\dfrac{43}{52}=0.827$ to 3 decimal places

In Bev’s experiment, the relative frequency of rolling a $6$ is:

$\dfrac{63}{400}=0.1575$

The probability of rolling a $6$ is:

$\dfrac{1}{6}=0.166666...$

The two results are fairly similar, in fact in Bev’s experiment she has successfully thrown a $6$ slightly less often than would be expected, although not by a significant amount.

So, we can conclude that Bev’s statement is false.  The die does not appear to be biased.

a) The relative frequency of silver cars  using Hannah’s morning data can be calculated as follows:

$\dfrac{32}{32+48}=0.4=40\%$.

So, according to her data, her statement is correct.

b) To calculate the relative frequency based on all the data collected, we will need to combine Hannah’s data.

The total number of cars spotted was $32 +41 + 48 + 111 = 232\text{ cars}$.

The total number of silver cars spotted was $32+41=73\text{ silver cars}$

The relative frequency of a car being silver using all the data can be calculated as follows:

$\dfrac{73}{232}=0.315$ to 3 decimal places

c) To find the expected frequency of silver cars, we will multiply the probability of a car being silver (i.e., the relative frequency calculated in part b) by the total number of cars. The expected frequency of silver cars in the town is therefore:

$0.315\times5,000=1,575$ silver cars

a)  Bob throws the die $100$ times and throws a $7$ on $15$ occasions.  As a fraction this can be written as:

$\dfrac{15}{100}$

The probability of Bob throwing a $7$ is the decimal equivalent of the above fraction, which we can calculate by dividing the numerator by the denominator:

$15\div100 = 0.15$

b)  Susan throws the die $500$ times and throws a $7$ on $60$ occasions.  As a fraction this can be written as:

$\dfrac{60}{500}$

The probability of Susan throwing a $7$ is the decimal equivalent of the above fraction, which we can calculate by dividing the numerator by the denominator:

$60\div500 = 0.12$

c)  Susan’s results are more accurate because the more times you conduct an experiment, the more accurate the estimate will be.

a)  Since the relative frequency for Burger Queen is $0.1$, we should be able to turn this into a fraction without too much difficulty:

$0.1 = \dfrac{1}{10}$

We know that $16$ people in total opted for Burger Queen, therefore:

$\frac{1}{10} = 16\text{ students}$

If $16$ students represents $\frac{1}{10}$ of the total number of students, then the total number of students on the trip can be calculated as follows:

$16\text{ students}\times10 = 160\text{ students}$

b)  The relative frequency of Pizza Cottage is $0.4$ which, as a fraction is $\frac{4}{10}$.  If there are $160$ students in total and $\frac{4}{10}$ chose Pizza Cottage, then the number of students who chose Pizza Cottage can be calculated as follows:

$\dfrac{4}{10}\times160\text{ students} = 64\text{ students}$

c)  The relative frequency of Dazza’s Fish and Chips is $0.5$ which, as a fraction is $\frac{5}{10}$ or $\frac{1}{2}$.  If there are $160$ students in total and $\frac{1}{2}$ chose Dazza’s Fish and Chips, then the number of students who chose Dazza’s Fish and Chips can be calculated as follows:

$\dfrac{1}{2}\times160\text{ students} = 80\text{ students}$

or

$160\text{ students}\div2 = 80\text{ students}$

Level 1-3GCSEKS3

## Worksheet and Example Questions

### (NEW) Relative Frequency Exam Style Questions - MME

Level 4-5 GCSENewOfficial MME

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