Compare an average and a measure of spread, in context. Range = largest value - smallest value, and it measures how spread out the data is. Free notes.
Updated 24 Mar 2026 · Revise Right Team
Comparing Populations with Statistical Measures is part of Sampling and Statistical Enquiry in GCSE AQA Higher Maths.
4 min
Reading time
11
Lessons in this topic
GCSE AQA Higher
Pathway
Reviewed by Revise Right
Content review
What this lesson covers
Compare an average and a measure of spread, in context. Range = largest value - smallest value, and it measures how spread out the data is. Free notes.
Apply statistical measures — mean, median, mode, and range — to describe and compare populations
Select and justify the most appropriate average for a given data set or context
Interpret statistical measures in context, explaining what they reveal about a population
Compare two populations using averages and measures of spread
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Use this lesson to stay inside Sampling and Statistical Enquiry while connecting the detail back to the wider Statistics topic.
Average: A single value that represents a typical or central value in a data set. The three averages used at GCSE are the mean, median, and mode.
Range: A measure of spread calculated as the largest value minus the smallest value. Used alongside an average to describe how consistent or variable a population is.
Population: The entire group being studied or described. Statistics such as averages and range are used to summarise key features of a population without listing every data value.
Measure of spread: A value that describes how spread out or consistent the data in a population is. The range is the standard measure of spread at GCSE.
Revision notes
Read the core explanations from Comparing Populations with Statistical Measures before testing yourself from memory.
1. Why Statistics Describe Populations
Imagine a school wants to compare how well two Year 10 classes performed in a maths test. Listing all 60 individual scores tells you very little at a glance. What you need are a few carefully chosen numbers that capture the story of each class — that is exactly what statistics does for a population.
Applying statistics to describe a population means choosing the right measures to summarise what is typical (an average) and how consistent the data is (a measure of spread). Together, these two pieces of information let you compare groups, spot patterns, and draw conclusions — all from a handful of numbers.
2. The Three Averages and the Range
There are three averages you need to know: the mean, the median, and the mode. Each one measures the centre of a data set differently, and each has strengths and weaknesses depending on the context.
The median is the middle value when all data is arranged in order. If there is an even number of values, the median is the mean of the two middle values. Because it sits in the middle of ordered data, extreme values do not pull it off-centre the way they affect the mean.
The mode is the value that appears most often. A data set can have no mode, one mode, or more than one mode. It is the only average that can be used for non-numerical (categorical) data, such as favourite colours or favourite subjects. Numerical data such as shoe sizes can also have a mode, but they are not examples of categorical data.
3. Calculating the Measures — Worked Example
Knowing the formulas is one thing; applying them accurately under exam conditions is another. Here is a full worked example that mirrors the style of an exam question, showing every step of the calculation.
Worked example: Calculating Mean, Median, Mode, and Range
The times (in minutes) taken by eight students to complete a puzzle are: 12, 7, 15, 7, 10, 18, 7, 12. Calculate the mean, median, mode, and range. [4 marks]
Values: 12, 7, 15, 7, 10, 18, 7, 12
The value 7 appears three times — more than any other. — Mode = 7 minutes [1 mark]. Scan for the most frequent value before doing any arithmetic.
Largest value = 18, Smallest value = 7
Range = 18 − 7 = 11 — Range = 11 minutes [1 mark]. Always subtract smallest from largest — never the other way round.
Sum = 12 + 7 + 15 + 7 + 10 + 18 + 7 + 12
= 88
Mean = 88 ÷ 8 = 11 — Mean = 11 minutes [1 mark]. Count the values carefully — there are 8 students, so divide by 8.
Ordered: 7, 7, 7, 10, 12, 12, 15, 18
Eight values → two middle values are the 4th and 5th: 10 and 12 — With an even number of values, the median is the mean of the two middle values.
Median = (10 + 12) ÷ 2 = 22 ÷ 2 = 11 — Median = 11 minutes [1 mark]. Here the mean and median happen to be equal, but that is not always the case.
Answer: Mode = 7 min, Range = 11 min, Mean = 11 min, Median = 11 min
4. Choosing the Right Average for the Context
One of the most common exam tasks is not just calculating an average, but justifying which average best describes a population. The choice depends on the nature of the data and whether any extreme values are present.
Average
Use when...
Avoid when...
Mean
Data is fairly symmetrical with no extreme outliers
There are outliers that would distort the result
Median
Data contains outliers or is skewed
You need to use every data value in the calculation
Mode
Data is categorical, or the most common value is the key information
Data has no repeated values, or many values appear equally often
A classic exam scenario involves salaries. If nine employees earn £20,000 and one manager earns £200,000, the mean salary is around £38,000 — far higher than what most employees actually earn. The median of £20,000 is far more representative of a typical worker's pay in that company.
5. Comparing Two Populations
The real power of statistics shows when you use it to compare two groups. To make a valid comparison, you need to comment on both an average (what is typical) and the range (how consistent the data is). Commenting on only one of these will cost you marks in the exam.
Worked example: Comparing Two Populations Using Averages and Range
Class A scores in a test: 45, 50, 55, 60, 65. Class B scores: 30, 50, 55, 60, 80. Compare the two classes using an appropriate average and the range. [4 marks]
Sum = 45 + 50 + 55 + 60 + 65 = 275
Mean = 275 ÷ 5 = 55 — Mean for Class A = 55 [M1]. Five values, so divide by 5.
Sum = 30 + 50 + 55 + 60 + 80 = 275
Mean = 275 ÷ 5 = 55 — Mean for Class B = 55 [M1]. The means are identical — this is where the range becomes essential.
Range = 65 − 45 = 20 — Range for Class A = 20 [M1]. A smaller range means more consistent performance.
Range = 80 − 30 = 50 — Range for Class B = 50 [M1]. A larger range means more variation between students.
Both classes have the same mean score of 55, so on average they performed equally well.
However, Class A has a smaller range (20) compared to Class B (50), so Class A performed more consistently. — A full comparison must address both the average AND the spread. Stating only one earns partial credit. [A1] for the comparison statement.
Answer: Both classes have a mean of 55. Class A has a range of 20; Class B has a range of 50. Class A performed more consistently, even though the typical score was the same.
6. Practice Problems
Now it is your turn to apply these methods. Work through each problem before checking the solution. Pay attention to whether the question asks you to calculate, compare, or justify — these are different exam tasks requiring different responses.
7. Exam Technique and Common Mistakes
The final section pulls together the exam-technique guidance that will protect your marks. These are the errors that appear most frequently in mark scheme reports — knowing them in advance puts you ahead.
Applying statistics to describe a population always comes down to the same core idea: choose a measure of centre to show what is typical, and pair it with the range to show how consistent the population is. Master that two-part structure and you have the foundation for every comparison question the exam can throw at you.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-24.
A data set can have no mode, one mode or more than one mode. The mode is the value that appears most often, so if two values are equally most common, both are modes.
Which average should you use?
Use the mean when the data is fairly symmetrical with no extreme outliers, the median when there are outliers or the data is skewed, and the mode for categorical data. If nine employees earn £20,000 and a manager earns £200,000, the median of £20,000 is more representative than the mean.
How do you compare two sets of data?
Compare an average and a measure of spread, in context. Class A (45, 50, 55, 60, 65) and Class B (30, 50, 55, 60, 80) both have a mean of 55, but Class A has a range of 20 and Class B a range of 50, so Class A performed more consistently.
What does the range tell you?
Range = largest value - smallest value, and it measures how spread out the data is. A small range means the data is consistent, while a large range means there is more variation in the population.
How do you find the mean, median, mode and range?
Mean = total ÷ number of values, median = middle value in order, mode = most common value and range = largest - smallest. For 12, 7, 15, 7, 10, 18, 7, 12: mean = 88 ÷ 8 = 11, median = (10 + 12) ÷ 2 = 11, mode = 7 and range = 18 - 7 = 11.
How do you find the median with an even number of values?
Put the data in order, then find the mean of the two middle values. For 7, 7, 7, 10, 12, 12, 15, 18, the 4th and 5th values are 10 and 12, so the median is (10 + 12) ÷ 2 = 11.
Why is the mean affected by outliers?
The mean is affected by outliers because it uses every data value, so one extreme value pulls it up or down. The median sits in the middle of ordered data, so extreme values do not pull it off-centre in the same way.
Which average can be used for categorical data?
The mode is the only average that can be used for non-numerical, categorical data, such as favourite colours or favourite subjects. The mean and median need numerical data.
Why is the median better for salaries?
The median is better for salaries when a few people earn much more than the rest. If nine employees earn £20,000 and one manager earns £200,000, the mean is about £38,000, but the median of £20,000 better shows a typical worker's pay.
What does it mean if two groups have the same mean but different ranges?
Two groups with the same mean but different ranges performed equally well on average, but the group with the smaller range was more consistent. For classes with mean 55, ranges of 20 and 50 show the first class was more consistent.
When should you use the mean?
The mean is best when the data is fairly symmetrical with no extreme outliers, because it uses every value. When there are outliers that would distort it, or the data is skewed, the median is usually a better choice.