GCSE Maths · Statistics

Averages GCSE Questions and Worked Answers

Mean shares the total equally; median is the middle of ordered data; mode is the most frequent value. Range measures spread, not an average. Choose the measure the task actually asks for.

Foundation & Higher6 worked examples10 original questions

Ask Ari about averages Start the questions

Free AI tutor · GCSE averages

Practise GCSE averages for free with an AI tutor

Ask Ari for an explanation or work through an original exam-style question together.

Hi, I’m Ari. We can start averages from the beginning, work through an example together, or practise a question. Tell me which step is confusing.

Ari is an AI tutor and can make mistakes. Use the worked answers below to check important results.

Start with the meaning

What you need to know about averages

Three people hold 2, 4 and 9 counters. Together they have 15. If the counters were shared equally, each would have 5. That fair-share value is the mean. It describes the group with one number, even though none of the original amounts was 5.

See the idea first

Different summaries answer different questions

For the ordered data 2, 4, 4, 7, 13, the mean is 30 ÷ 5 = 6. The middle value, 4, is the median. The most frequent value, also 4, is the mode. The range is 13 − 2 = 11 and tells you spread rather than centre.

Four summaries of 2, 4, 4, 7, 13
Measure	Question it answers	Value
Mean	What is the equal share of the total?	6
Median	What is in the middle after ordering?	4
Mode	What occurs most often?	4
Range	How far apart are the extremes?	11

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a named average

What the problem asks: Find the median of 8, 3, 10 and 5.

How to solve it: Order them: 3, 5, 8, 10. With two middle values, take their mean: (5 + 8) ÷ 2 = 6.5.

Recover a missing value

What the problem asks: Four numbers have mean 7. Three are 3, 8 and 9. Find the fourth.

How to solve it: The required total is 4 × 7 = 28. Known values total 20, so the missing value is 8.

Compare distributions

What the problem asks: Team A has mean time 12 s and range 3 s; B has mean 10 s and range 6 s. Compare.

How to solve it: B is faster on average because its mean time is lower. A's times have a smaller spread by the range, so are more consistent by that measure. State both comparisons in context.

A reliable routine

Calculate the summary requested

There is no single operation for every average. Mean preserves the total while sharing it equally; median locates the centre by order; mode identifies the most common outcome. Choose after reading the task, not from a keyword guess.

Check whether values are raw observations or frequencies.
For mean, add all observations and divide by their number.
For median, sort first and find the middle value or mean of the middle pair.
For mode, count occurrences; for range, subtract minimum from maximum.

Check: A very large or small outlier can pull the mean and range strongly. The median is usually less affected by how extreme that one value is.

Fully worked

Averages GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Mean

2 marks

Question

Find the mean of 6, 9, 4, 11 and 5.

\text{Total}=6+9+4+11+5=35

There are five observations.

\text{Mean}=35/5=7

Tip: divide by the number of observations, not their largest value.

Example 2

Odd number of observations

2 marks

Question

Find the median of 9, 2, 7, 4 and 12.

Order the list: 2, 4, 7, 9, 12. The third value has two values on either side, so the median is 7.

Example 3

Even number of observations

2 marks

Question

Find the median of 8, 1, 5, 11, 4 and 7.

Order: 1, 4, 5, 7, 8, 11. The middle values are 5 and 7.

\text{Median}=(5+7)/2=6

It need not be a value in the original list.

Example 4

Mode and range

2 marks

Question

Find the mode and range of 3, 6, 6, 8, 9, 9, 9 and 12.

9 occurs three times, more often than any other value, so mode = 9.

\text{Range}=12-3=9

Mode and range happen to agree here but mean different things.

Example 5

A missing observation

3 marks

Question

Five numbers have mean 8. Four are 6, 7, 9 and 10. Find the fifth.

\text{Required total}=5(8)=40

\text{Known total}=6+7+9+10=32

x=40-32=8

Check: the complete total is 40.

Example 6

Combine groups

Harder4 marks

Question

Six pupils have a mean score of 8 and four other pupils have a mean of 13. Find the mean of all ten.

Recover each group's total.

T_1=6(8)=48

T_2=4(13)=52

\text{Combined mean}=\frac{48+52}{6+4}=\frac{100}{10}=10

The unweighted mean of 8 and 13 would ignore the different group sizes.

10 original questions · total 24 marks

Averages GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 29 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Mean of four

2 marks

Find the mean of 3, 7, 8 and 10.

Show worked answer

T=3+7+8+10=28

\text{Mean}=28/4=7

Median

2 marks

Find the median of 14, 5, 9, 6 and 11.

Show worked answer

Order: 5, 6, 9, 11, 14. The middle value is 9.

Middle pair

2 marks

Find the median of 2, 12, 6 and 8.

Show worked answer

Order: 2, 6, 8, 12.

\text{Median}=(6+8)/2=7

Two modes

2 marks

Find the modes of 1, 2, 2, 4, 4 and 7.

Show worked answer

2 and 4 each occur twice, more than any other value. Both are modes; the data is bimodal.

A range with negatives

2 marks

Find the range of −6, −2, 3 and 8.

Show worked answer

Maximum is 8 and minimum is −6.

8-(-6)=14

Subtract the negative minimum.

Missing value

3 marks

Four numbers have mean 9. Three are 5, 10 and 12. Find the fourth.

Show worked answer

Total needed: $4(9)=36$ . Known total: $5+10+12=27$ . Missing value: $36-27=9$ .

Add one observation

3 marks

Five values have mean 6. A new value of 12 is added. Find the new mean.

Show worked answer

Old total: $5(6)=30$ . New total: $30+12=42$ . There are now six values.

42/6=7

Compare fairly

2 marks

Two delivery firms, A and B, are compared. A has median delivery time 18 min and range 8 min. B has median 21 min and range 4 min. Compare their delivery times.

Show worked answer

A has a shorter typical delivery time by the median. B has a smaller range, so its delivery times are more consistent by that measure.

An outlier

3 marks

For 10, 11, 12, 13, 54, find mean and median. Which better describes a typical value among the first four?

Show worked answer

\text{Mean}=100/5=20

Median is 12. The high outlier 54 pulls the mean above the first four values; median 12 better describes that central group.

Weighted mean

3 marks

A club has three members aged 12 and seven aged 14. Find the mean age.

Show worked answer

Total age: $3(12)+7(14)=36+98=134$ . Ten members:

134/10=13.4\text{ years}

Do not simply average 12 and 14.

Examiner-style feedback

Common averages mistakes

Median before ordering

The middle position of an unsorted list has no special meaning.

Treating range as an average

Range describes spread; mean, median and mode describe centre or a typical outcome.

Mean of group means

Weight each mean by its group size by reconstructing totals.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Name the requested measure.
Order before finding a median.
Use total ÷ count for a mean.
Compare centre and spread in context.

Quick answers

Averages FAQ

Can a mean be a decimal when data is whole numbers?

Yes. An equal share need not be a whole number or one of the original observations.

Can data have no mode?

If every value appears just once, the usual GCSE answer is no mode. If two or more values tie for the greatest frequency above the others, they are all modes.

Build connected skills

What to revise next

Repeated data

Frequency tables

Compute summaries without writing every value out.

Revise frequency tables

Compare distributions visually

Box plots

Use medians and spread on a diagram.

Revise box plots

Content standards

Curriculum and rights review

S4 means, medians, modes, range and comparisons across tiers; grouped estimates are developed in the linked frequency-table guide. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content