Find the mean of: 2, 4, 6, 8
Add then divide by 4.
2 + 4 + 6 + 8 = 20 → 20 ÷ 4 = 5.
Averages
Averages summarise data using values such as mean, median and mode. Choosing the correct average is important when interpreting data in statistics.
Overview
An average is a single value used to represent a set of data.
In GCSE Maths, the three main averages are mean, median and mode. You will also often be asked for the range, which shows spread.
Mean = \( \dfrac{\text{total}}{\text{number of values}} \)
What you should understand after this topic
- Calculate mean, median, mode and range
- Order data correctly
- Find the mean from a frequency table
- Choose the most suitable average
- Avoid common exam mistakes
Mean
Uses all the values.
Median
The middle value when the data is in order.
Mode
The most common value.
Range
The biggest value minus the smallest value.
Key Definitions
Average
A value that represents a set of data.
Mean
Add all the values, then divide by how many values there are.
Median
The middle value when the data is written in order.
Mode
The value that appears most often.
Range
The difference between the highest and lowest values.
Frequency
How many times a value appears.
Discrete Data
Data made of separate values, often whole numbers.
Frequency Table
A table showing each value and how often it occurs.
Key Rules
Mean
Add all values and divide by the number of values.
Median
Put the data in order first.
Mode
Look for the value that appears most often.
Range
\( \text{largest} - \text{smallest} \)
Quick Choosing Guide
Best when using all data
Mean
Best when one value is unusually large or small
Median
Best when asking for the most common result
Mode
Best for spread
Range
How to Solve
Step 1: Understand the four main statistics
In GCSE Maths, averages usually refers to the mean, median and mode. The range is often included because it describes how spread out the data is.
Mean, Median, Mode and Range
Exam tip: Always read the question carefully to see which average is required.
Step 2: Finding the mean
The mean uses all the data values. Add them together, then divide by how many values there are.
Mean = \( \dfrac{\text{sum of values}}{\text{number of values}} \)
Why this matters: The mean is affected by every value, including extreme values.
Step 3: Finding the median
The median is the middle value, but the data must be written in order first.
Exam tip: Always arrange the data in ascending order before finding the median.
Step 4: Finding the mode
The mode is the value that appears most often.
Step 5: Finding the range
The range shows the spread of the data by subtracting the smallest value from the largest value.
Range = largest value - smallest value
Step 6: Mean from a frequency table
For a frequency table, multiply each value by its frequency first, then divide by the total frequency.
Mean = \( \dfrac{\sum (x \times f)}{\sum f} \)
This method is often used together with frequency tables in exam questions.
Step 7: Choosing the best average
Exam tip: Be prepared to explain why one average is more suitable than another.
Use the mean
When you want to include every value in the data.
Use the median
When there are extreme values that could affect the mean.
Use the mode
When you want the most common value or category.
Use the range
When you want to describe how spread out the data is.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on calculating mean, median, mode and range.
Edexcel
Find the mean of \( 5,\ 7,\ 8 \).
Edexcel
Find the median of \( 9,\ 2,\ 6,\ 4,\ 7 \).
Edexcel
Find the mode of \( 3,\ 5,\ 5,\ 8,\ 9 \).
Edexcel
Find the range of \( 4,\ 10,\ 6,\ 13,\ 8 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on averages with frequency and multi-step reasoning.
AQA
Find the median of \( 1,\ 3,\ 5,\ 7,\ 9,\ 11 \).
AQA
The values 2, 4 and 6 have frequencies 3, 2 and 1.
Find the mean.
AQA
The table shows the number of goals scored by a team in 6 matches.
| Goals | Frequency |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 2 |
| 3 | 1 |
Find the mean number of goals scored.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning and reverse mean problems.
OCR
The mean of 4 numbers is 9.
The first three numbers are 7, 10 and 8.
Find the fourth number.
OCR
A student says, "The mean is always the best average to use."
Tick one box. True ☐ False ☐
Give a reason for your answer.
OCR
Explain why the median is often used instead of the mean when there are extreme values in a data set.
Exam Checklist
Step 1
Read the question carefully and decide which average is needed.
Step 2
If finding the median, put the data in order first.
Step 3
If using a frequency table, multiply value by frequency.
Step 4
Check whether the answer makes sense in the context.
Most common exam mistakes
Mean mistake
Dividing by the wrong number of values.
Median mistake
Not ordering the data first.
Mode mistake
Choosing the biggest number instead of the most common number.
Frequency mistake
Forgetting to use frequencies when calculating the mean.
Common Mistakes
These are common mistakes students make when calculating averages in GCSE Maths.
Not ordering data for the median
Incorrect
A student finds the middle value without arranging the data first.
Correct
To find the median, the data must be written in order from smallest to largest. The middle value is then selected correctly.
Dividing by the wrong number for the mean
Incorrect
A student adds the values but divides by the wrong total.
Correct
The mean is calculated by dividing the total sum by the number of values. Make sure you count all values correctly.
Confusing mode and median
Incorrect
A student says the mode is the middle value.
Correct
The mode is the most frequent value, while the median is the middle value after ordering the data.
Using incorrect values for the range
Incorrect
A student subtracts the wrong numbers when finding the range.
Correct
The range is found by subtracting the smallest value from the largest value. Always check both extremes carefully.
Forgetting frequency in tables
Incorrect
A student adds values but ignores their frequency.
Correct
In a frequency table, each value must be multiplied by its frequency before adding. This ensures the mean is calculated correctly.
Try It Yourself
Practise calculating the mean, median, mode and range from data.
Foundation
Foundation Practice
Calculate mean, median, mode and range from simple data.
Find the mean of: 3, 5, 7
Add and divide by 3.
3 + 5 + 7 = 15 → 15 ÷ 3 = 5.
Find the median of: 1, 3, 5, 7, 9
Middle number.
The middle value is 5.
Find the median of: 2, 4, 6, 8
Find average of middle two.
(4 + 6) ÷ 2 = 5.
Find the mode of: 2, 3, 3, 4, 5
Most frequent value.
3 appears most often.
Find the range of: 4, 8, 10, 12
Max − min.
12 − 4 = 8.
Which average is the most frequent value?
Think frequency.
Mode = most frequent.
Find the mean of: 10, 20, 30, 40
Add then divide by 4.
100 ÷ 4 = 25.
A student finds the median without ordering the data. What is wrong?
Order matters.
Median requires ordered data.
Find the range of: 15, 20, 35, 40
Max − min.
40 − 15 = 25.
Higher
Higher Practice
Solve problems involving averages, including missing values and reasoning.
The mean of 5 numbers is 10. What is their total?
Mean × number of values.
10 × 5 = 50.
The mean of 4 numbers is 6. What is their total?
Mean × count.
6 × 4 = 24.
Find the median of: 5, 1, 9, 3, 7
Order first.
Ordered: 1, 3, 5, 7, 9 → median = 5.
The mean of 3 numbers is 8. Two numbers are 6 and 10. Find the third number.
Find total first.
Total = 8 × 3 = 24 → 6 + 10 = 16 → third = 8.
Which average is most affected by extreme values?
Uses all values.
Mean is affected by outliers.
Find the range of: 100, 105, 110, 150
Max − min.
150 − 100 = 50.
A set has no repeated values. What is the mode?
Mode needs repetition.
No repeated value → no mode.
The mean of 6 numbers is 12. What is their total?
Mean × count.
12 × 6 = 72.
A student forgets to divide when finding the mean. What is wrong?
Mean requires division.
Mean = total ÷ number of values.
The mean of 5 numbers is 20. Four numbers are 10, 15, 25, 30. Find the fifth number.
Find total first.
Total = 100 → known sum = 80 → missing = 20.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What are the three averages?
Mean, median and mode.
When is median useful?
When there are outliers.
How do I find mean?
Add values and divide by total.