# Mean

Here we will learn about the mean, including what the mean is and how to find the mean.

There are also mean worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is the mean?

The mean is a type of average calculated by finding the total of the values and dividing the total by the number of values.

The “number of values” is sometimes referred to as the “number of numbers”.

\text{Mean}=\frac{\text{total}}{\text{number of values}}

E.g. Work out the mean

$\text{Mean}=\frac{\text{total}}{\text{number of values}}=\frac{5+8+10+11+13}{5}=\frac{47}{5}=9.4$

The mean is 9.4

The mean (also known as the arithmetic mean) is a measure of central tendency because it describes a set of numbers by identifying a central position within the data.

### What is the mean? ## How to calculate the mean

In order to calculate the mean:

1. Find the total of the values.
2. Divide the total by the number of values.

### How to calculate the mean ## Related lessons on mean, median, mode

Mean is part of our series of lessons to support revision on mean, median, mode. You may find it helpful to start with the main mean, median, mode lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Mean maths examples

### Example 1: finding the mean

Calculate the mean value of this list of numbers:

2 7 9 10 12

1. Find the total of the values.

Add up all the values in the list

\text{Total}=2+7+9+10+12=40

2Divide the total by the number of values.

There are 5 values in the data set. We need to divide the total by 5 .

\text{Mean}=\frac{\text{total}}{\text{number of values}}=\frac{40}{5}=8

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures.

Here the mean is an integer, so there is no need to round.

The mean is 8

### Example 2: finding the mean

Calculate the mean value of this set of numbers.

13 16 17 17 18 20

Add up all the values in the list

\text{Total}=13+16+17+17+18+20=101

There are 6 values in the data set. We need to divide the total by 6 .

\text{Mean}=\frac{\text{total}}{\text{number of values}}=\frac{101}{6}=16.8333...

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures. Here we have been asked to round to 1 decimal place.

\text{Mean}=16.8333…=16.8 \ \text{(1 dp)}

### Example 3: finding the mean

Calculate the mean value of this set of data. Give your answer to 1 decimal place.

11 13 14 15 19 20 22

Add up all the values in the list

\text{Total}=11+13+14+15+19+20+22=114

There are 7 values in the data set.  We need to divide the total by 7 .

\text{Mean}=\frac{\text{total}}{\text{number of values}}=\frac{114}{7}=16.2857...

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures.

Here we have been asked to round to 1 decimal place.

\text{Mean}=16.2857…=16.3 \ \text{(1 dp)}

### Example 4: finding the mean

Calculate the mean value of this list of numbers.

101 102 105 106 108

Add up all the values in the list

\text{Total}=101+102+105+106+108=522

There are 5 values in the data set. We need to divide the total by 5 .

\text{Mean}=\frac{\text{total}}{\text{number of values}}=\frac{522}{5}=104.4

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures.

We have not been asked to round and the mean is a simple decimal.

The mean is 104.4

## How to solve a problem involving the mean

In order to solve a problem involving the mean:

1. Use the mean and number of values to find the total.
2. Read the question carefully to work the next step.

## Problem solving involving mean examples

### Example 5: problem solving

The mean of 4 values is 10 .

Here are 3 of the values:

6 9 12

Find the 4 th value.

The mean of 4 values is 10 .  We can multiply these together to find the total of the 4 numbers.

\text{Total of 4 values}=\text{mean} \times \text{number of values}=10\times 4=40

We need to find the 4

th value. We can find the total of the first 3 values.

Then we can subtract this total from the total of the 4 values.

\text{Total of 3 values}=6+9+12=27

We need to find the difference of the totals.

40-27=13

The 4 th value is 13

Alternatively, you could use the equation for finding the mean. You could use x as the missing value.

Then rearrange and solve.

$\text{Mean} = \frac{\text{total}}{\text{number of values}}\\ 10=\frac{6+9+12+x}{4}\\ 10 = \frac{27+x}{4}\\ 40=27+x\\ 13=x$

### Example 6: problem solving

The mean of 5 values is 14 .

Here are 4 of the values:

5 11 13 19

Find the 5 th value:

The mean of 5 values is 14 . We can multiply these together to find the total of the 5 numbers.

\text{Total of 5 values}=\text{mean} \times \text{number of values}=14\times 5=70

We need to find the 5 th value. We can find the total of the first 4 values.

Then we can subtract this total from the total of the 5 values.

\text{Total of 4 values}=5+11+13+19=48

We need to find the difference of the totals.

70-48=22

The 5 th value is 22

Alternatively, you could use the equation for finding the mean. You could use x as the missing value.Then rearrange and solve.

$\text{Mean} = \frac{\text{total}}{\text{number of values}}\\ 14=\frac{5+11+13+19+x}{5}\\ 14 = \frac{48+x}{5}\\ 70=48+x\\ 22=x$

### Example 7: problem solving involving 2 means (HIGHER)

Data Set A has 6 values and a mean of 10 .

Data Set B has 4 values and a mean of 8 .

Find the mean of all 10 values.

For Set A – The mean of 6 values is 10 . We can multiply these together to find the total of the Set A numbers.

\text{Total of Set A values}=\text{mean} \times \text{number of values}=10\times 6=60

For Set B – The mean of 4 values is 8 . We can multiply these together to find the total of the Set B numbers.

\text{Total of Set A values}=\text{mean} \times \text{number of values}=8\times 4=32

We need to find the mean of all the values combined. We know there are 10 values. We can find the total of all 10 values by adding the total for Set A to the total of Set B .

\text{Total} =\text{total of Set A values} + \text {total of Set B values} = 60+32=92

We need to find the mean by dividing the total of all 10 values by 10 .

\text{Mean}=\frac{\text{Total}}{\text{number of values}}=\frac{92}{10}=9.2

The mean of all the values is 9.2

### Example 8: problem solving involving 2 means (HIGHER)

Data Set A has 4 values and a mean of 8 .

Data Set B has 7 values and a mean of 6 .

Find the mean of the all 11 values. Give your answer correct to 2 decimal places.

For Set A – The mean of 4 values is 8 . We can multiply these together to find the total of the Set A numbers.

\text{Total of Set A values}=\text{mean} \times \text{number of values}=8\times 4=32

For Set B – The mean of 7 values is 6 . We can multiply these together to find the total of the Set B numbers.

\text{Total of Set A values}=\text{mean} \times \text{number of values}=6\times 7=42

We need to find the mean of all the values combined. We know there are 11 values. We can find the total of all 11 values by adding the total for Set A to the total of Set B .

\text{Total} =\text{total of Set A values} + \text {total of Set B values} = 32+42=74

We need to find the mean by dividing the total of all 11 values by 11 .

\text{Mean}=\frac{\text{Total}}{\text{number of values}}=\frac{74}{11}=6.7272…=6.73 \ \text{(2 dp)}

The mean of all the values is 6.73 \text{ (2 dp)}

### Common misconceptions

• Check which average you are being asked for

Check if you have been asked for the median, mode, or mean average.

• The mean average can be a decimal

The mean does not have to be a whole number. It can be a decimal or a fraction. It may be a decimal which needs rounding.

### Practice mean maths questions

1. Find the mean of this list of values:
5 7 8 8 9

7 7.4 7.5 8 \text{Mean}= \frac{\text{total}}{\text{number of values}}\\ \text{Mean}=\frac{5+7+8+8+9}{5}\\ \text{Mean}= \frac{37}{5}\\ \text{Mean}=7.4

2. Find the mean of this list of values:
3 5 6 9

5.7 5.75 5.8 5.9 \text{Mean}= \frac{\text{total}}{\text{number of values}}\\ \text{Mean}=\frac{3+5+6+9}{4}\\ \text{Mean}= \frac{23}{4}\\ \text{Mean}=5.75

3. Find the mean of this list of values. Give your answer to 3 significant figures:
7 8 9 10 10 11

9.16 9.5 10 9.17 \text{Mean}= \frac{\text{total}}{\text{number of values}}\\ \text{Mean}=\frac{7+8+9+10+10+11}{6}\\ \text{Mean}= \frac{55}{6}\\ \text{Mean}=9.16666…\\ \text{Mean}=9.17 \ \text{(to 3 sf)}

4. Find the mean of this list of values. Give your answer to 3 significant figures:
17 18 18 19 20 21 25

18 19 19.8 19.7 \text{Mean}= \frac{\text{total}}{\text{number of values}}\\ \text{Mean}=\frac{17+18+18+19+20+21+25}{7}\\ \text{Mean}= \frac{55}{7}\\ \text{Mean}=19.7142…\\ \text{Mean}=19.7\ \text{(to 3 sf)}

5. The mean of 4 numbers is 9 .
Here are 3 of the numbers
6 8 15
What is the 4 th number?

7 5 6 8 The total of 4 numbers is:
\text{Total of 4 values}=\text{mean} \times \text{number of values}=9\times 4=36

The total of 3 numbers is: 6+8+9=29

The difference between the totals is: 36-29=7

This is the missing number.

6. The mean of 6 numbers is 12 .
Here are 5 of the numbers
7 9 11 13 18
What is the 6 th number?

12 14 13 15 The total of 4 numbers is:
\text{Total of 4 values}=\text{mean} \times \text{number of values}=12\times 6=72

The total of 5 numbers is: 7+9+11+13+18=58

The difference between the totals is: 72-58=14

This is the missing number.

### Mean maths GCSE questions

1. Here is a list of numbers: Work out the mean of the numbers in this list.

(2 marks)

\frac{1+3+4+6+6}{5}=\frac{20}{5}
For adding the numbers and dividing by 5

(1)

=5

(1)

2. Here are the ages of people in a family Work out the mean of the ages in this family.

(2 marks)

\frac{15+15+19+20+54+59}{6}=\frac{182}{6}
For adding the numbers and dividing by 6

(1)

\frac{182}{6}=30.333…=30.3 \ \text{(to 3 sf)}

(1)

3. Dev has 5 tests and achieves an average of 61% .
Here are 4 of the results. Find the missing result.

(2 marks)

5\times 61-(48+59+65+73)
For subtracting the total of 4 results from the total of 5 results

(1)

305-245=60

(1)

4. HIGHER
5 red bricks have a mean weight of 10 kg .
3 blue bricks have a mean weight of 6 kg .
Find the mean of the 8 bricks.

(3 marks)

5\times 10=50

3\times6=18

For finding the total weights of the red bricks and the blue bricks

(1)

\frac{50+18}{8}=\frac{68}{8}
For dividing the total weight of 8 bricks by 8

(1)

=8.5 \ \text{kg}

(1)

## Learning checklist

You have now learned how to:

• Calculate the mean
• Solve problems involving the mean
• Higher – calculate the combined mean

## Still stuck?

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#### FREE GCSE maths practice papers (Edexcel, AQA & OCR)

8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.

Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.