Find the mean, median, mode and range from a list of numbers, and choose the right average for the situation. It is a Foundation topic, aimed at grade 2. Free questions with answers are on this page. Worked examples show each line before you try the questions.
There are three averages and they answer different questions. The mode is the value that appears most. The median is the middle value once the list is in order. The mean is the total divided by how many there are. The range is not an average — it is largest minus smallest, a measure of spread.
Always order the list before the median. For 11,4,8,2,6 the ordered list is 2,4,6,8,11, so the median is 6. With an even number of values, the median is halfway between the two middle ones.
If the mean of five numbers is 8, the total is 40. Four of the numbers given, subtract from 40 to find the fifth. That 'mean × how many = total' reverse is a regular 2-mark question.
Objectives
Add the values and divide by how many there are.
Order the list, then take the middle value.
The mode is the value that appears most often.
Key points
Find the mean by adding the values and dividing by how many there are.
Find the median by ordering the values and taking the middle (or the mean of the middle two).
Find the mode (the most common value) and the range (largest − smallest).
Work backwards from a mean to find a total or a missing value.
Worked examples
Example 1
Find the mean of 1, 8, 2, 5.
2 marks
Total: 1 + 8 + 2 + 5 = 16.
There are 4 values, so divide the total by 4: 16 ÷ 4 = 4.
The mean is 4.
Answer: 4
Common slips
Slip: Total: 3 + 8 + 2 + 5 = 16. That changes a number from the question, so it answers a different question.
Slip: There are 7 values, so divide the total by 4: 16 ÷ 4 = 4. That changes a number in the working, so the step no longer follows from the one before.
Slip: The mean is 8. That does not match the working above it.
Example 2
Find the median of 10, 4, 13, 9, 6.
2 marks
Put the values in order: 4, 6, 9, 10, 13.
There are 5 values, so the median is the middle one, the 3rd: 9.
The median is 9.
Answer: 9
Common slips
Slip: Put the values in order: 6, 6, 9, 10, 13. That changes a number from the question, so it answers a different question.
Slip: There are 8 values, so the median is the middle one, the 3rd: 9. That changes a number in the working, so the step no longer follows from the one before.
Slip: The median is 13. That does not match the working above it.
Example 3
Find the range of 8, 13, 5, 4, 10.
2 marks
The greatest value is 13 and the least value is 4.
Range = greatest − least = 13 − 4 = 9.
The range is 9.
Answer: 9
Common slips
Slip: The greatest value is 15 and the least value is 4. That changes a number from the question, so it answers a different question.
Slip: Range = greatest − least = 16 − 4 = 9. That changes a number in the working, so the step no longer follows from the one before.
Slip: The range is 13. That does not match the working above it.
Questions and answers
GCSE Maths averages questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Find the mean of 4, 8 and 12.
Show answer
Answer: 8
Add the three numbers.
Divide by 3.
M1 — Add the three numbers.
A1 — Answer 8
Question 2 Easy2 marks
Find the median of 3, 9, 5 and 7.
Show answer
Answer: 6
Order the numbers first.
Average the two middle ones if there are four.
M1 — Order the numbers first.
A1 — Answer 6
Question 3 Medium2 marks
The mean of four numbers is 8. Three of them are 5, 9 and 6. What is the fourth?
Show answer
Answer: 12
The total is the mean times 4.
Subtract the known numbers.
M1 — The total is the mean times 4.
A1 — Answer 12
Question 4 Medium2 marks
Five scores have a mean of 10. A sixth score of 16 is added. What is the new mean?
Show answer
Answer: 11
The old total is 50.
Add 16 and divide by 6.
M1 — The old total is 50.
A1 — Answer 11
Question 5 Hard2 marks
The values 2, 5 and 8 have frequencies 1, 2 and 1. What is the mean?
Show answer
Answer: 5
Multiply each value by its frequency.
Divide the total by the number of values.
M1 — Multiply each value by its frequency.
A1 — Answer 5
Question 6 Hard2 marks
A group of 3 has mean 6. Another value is 14. What is the mean of all four numbers?
Show answer
Answer: 8
The first total is 18.
Add 14 and divide by 4.
M1 — The first total is 18.
A1 — Answer 8
Question 7 Exam-style3 marks
The values 2, 5 and 8 have frequencies 1, 3 and 1. What is the mean?
Show answer
Answer: 5
Multiply each value by its frequency.
Divide the total by the number of values.
M1 — Multiply each value by its frequency.
M1 — Divide the total by the number of values.
A1 — Answer 5
Common mistakes
Most lost marks on this topic are one slip.
Dividing by the greatest value instead of how many values.
The mean divides the total by the count.
Forgetting to order the list before the median.
The middle of an unordered list is not the median.
Calling the range an average.
The range is the spread.
Spot the slip
The range of a list is
greatest minus least
the middle value
the total divided by how many
Show the slip
greatest minus least Range is a spread, not an average.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Show the total and the division separately for the mean; each is a mark.
Write the ordered list out in full — that line earns the method mark for the median.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.