Comparing Distributions

This lesson covers how to compare two (or more) data sets using averages and measures of spread — a skill that appears regularly on the AQA GCSE Mathematics exam. Simply calculating values is not enough; you must be able to write meaningful comparison statements that interpret the numbers in context.

Why Compare Distributions?

In many real-life situations, we need to compare data sets to make decisions or draw conclusions. For example:

Comparing test scores between two classes.
Comparing rainfall in two cities.
Comparing the ages of customers at two shops.
Comparing the performance of two athletes over a season.

To compare distributions, you use two types of measure:

Type	Purpose	Examples
Measure of average (central tendency)	Summarises the typical value	Mean, median, mode
Measure of spread (dispersion)	Shows how spread out the data is	Range, IQR

The Two-Part Comparison Rule

When comparing two distributions, you should always make two statements:

Compare an average (usually the mean or median) — this tells you which data set has higher or lower typical values.
Compare a measure of spread (usually the range or IQR) — this tells you which data set is more or less consistent.

Exam Tip: The AQA mark scheme almost always requires two comparison statements: one about average and one about spread. Writing only one will lose you marks. Each statement must be in context (referring to the actual situation, not just numbers).

Writing Comparison Statements

A good comparison statement has three parts:

State the statistical measure (e.g. "The median for...").
Compare the values (e.g. "is higher than...").
Interpret in context (e.g. "which shows that, on average, Class A scored better").

Template for Comparison Statements

For average: "The [mean/median] for [Group A] is [higher/lower] than for [Group B] ([value A] compared to [value B]), which shows that on average, [Group A] [did better/worse / had more/less / was higher/lower]."

For spread: "The [range/IQR] for [Group A] is [larger/smaller] than for [Group B] ([value A] compared to [value B]), which shows that [Group A's] data is [more/less] consistent (or spread out)."

Worked Example

Two classes took the same test. Here are the results:

	Class A	Class B
Mean	64	58
Median	62	60
Range	45	28
IQR	18	12

Comparison statements:

"The mean score for Class A (64) is higher than for Class B (58), which shows that on average, Class A performed better in the test."
"The range for Class A (45) is larger than for Class B (28), which shows that the scores in Class A were more spread out and less consistent than in Class B."

Or using IQR:

"The IQR for Class A (18) is larger than for Class B (12), which shows that the middle 50% of scores in Class A were more spread out, meaning Class B was more consistent."

graph TD
    A[Comparing Distributions] --> B[Step 1: Compare an Average]
    A --> C[Step 2: Compare a Measure of Spread]
    B --> D[Mean or Median]
    C --> E[Range or IQR]
    D --> F[State which is higher/lower]
    E --> G[State which is larger/smaller]
    F --> H[Interpret in context]
    G --> I[Interpret in context]

Exam Tip: Never just write "Class A is higher" without saying what is higher. Always name the measure: "The median for Class A is higher than for Class B." Then explain what this means in the context of the question. Generic statements without context will not earn full marks.

Choosing the Right Measures to Compare

Situation	Best Average	Best Spread
Data with no outliers	Mean	Range
Data with outliers or skewed data	Median	IQR
Comparing box plots	Median	IQR
Comparing from grouped frequency tables	Estimated mean	Cannot find exact range; comment on class intervals
Data is categorical	Mode	Not applicable

Why IQR Is Often Better Than Range

The range is affected by extreme values (outliers). A single very high or very low value can make the range misleadingly large.

The IQR only considers the middle 50% of the data, so it is not affected by outliers and gives a more reliable measure of spread.

Exam Tip: If you have a choice between using range and IQR, state that the IQR is preferred because it is "not affected by extreme values" and gives a better representation of the typical spread.

Comparing Box Plots

Box plots are particularly useful for comparing distributions because they display the median, quartiles, IQR, and range visually.

Worked Example

Two box plots show the times (in seconds) for students in two sports clubs to complete a sprint.

Measure	Club A	Club B
Minimum	11.2	10.8
Q1	12.5	12.0
Median	13.8	13.2
Q3	15.0	13.9
Maximum	17.5	15.2
IQR	2.5	1.9
Range	6.3	4.4

Comparison:

"The median time for Club A (13.8 seconds) is higher than for Club B (13.2 seconds), which shows that on average, Club B completed the sprint faster than Club A."
"The IQR for Club A (2.5 seconds) is greater than for Club B (1.9 seconds), which shows that Club B's times were more consistent and less spread out than Club A's."