What you'll learn
This topic covers comparing distributions using statistical measures and diagrams. In this guide you will learn how to compare two data sets fairly, which average and which measure of spread to use, how to compare from box plots and other diagrams, and how to write comparisons in context. These interpretation skills are tested across GCSE statistics.
Key terms and definitions
Distribution — the way data values are spread out.
Average — a measure of the centre (mean, median or mode).
Spread — how varied the data is (range or interquartile range).
Interquartile range (IQR) — Q3 − Q1, the spread of the middle half.
In context — referring to what the data actually represents.
Core concepts
Comparing an average and a spread
A fair comparison of two data sets uses two things: a measure of average (centre) and a measure of spread (consistency). Comparing only one is not enough — you need both to describe the difference properly.
Choosing the right average
The median is best when data is skewed or has outliers, because it is not distorted by extreme values. The mean uses all the data and suits symmetrical distributions. The mode is used for the most common value or categorical data.
Choosing the right spread
The range is simple but affected by outliers. The interquartile range (IQR) is more reliable as it ignores the extreme quarter at each end, describing the middle half. A smaller spread means more consistent data.
Comparing from box plots and diagrams
Box plots make comparison easy: compare the medians (averages) and the IQRs or ranges (spreads). The same idea applies to cumulative frequency curves and frequency polygons — compare the centre and the spread.
Writing comparisons in context
Always write comparisons in context, naming the data and what the numbers mean — for example, "Class A had a higher median score, so on average they did better, and a smaller IQR, so their scores were more consistent." Plain numbers without interpretation lose marks.
Worked examples
Example 1: Choosing an average
A data set has a large outlier. Which average is most appropriate?
The median, as it is not distorted by the outlier.
Example 2: Comparing spread
Set A has IQR 8 and set B has IQR 15. Which is more consistent?
Set A — a smaller IQR means less spread.
Example 3: Box plot comparison
Box plot X has a higher median than Y. What does this say?
On average, X's values are higher than Y's.
Common mistakes and how to avoid them
Comparing only the average. Compare a spread as well.
Using the mean with outliers. Prefer the median for skewed data.
Giving bare numbers. Always interpret in context.
Confusing range and IQR. IQR ignores the extreme quarters.
Saying "bigger is better" wrongly. A smaller spread means more consistent, which is often preferable.
Exam technique for Comparing Distributions
Compare an average and a spread — both are needed.
Choose the median for skewed data, the mean for symmetrical.
Use the IQR as a reliable measure of spread.
Read box plots by comparing medians and IQRs.
Write every comparison in context.
Quick revision summary
To compare two distributions fairly, compare both a measure of average (centre) and a measure of spread (consistency). Choose the median when data is skewed or has outliers (it is not distorted), the mean for symmetrical data, and the mode for the most common or categorical value. For spread, the range is simple but sensitive to outliers, while the interquartile range (IQR = Q3 − Q1) is more reliable, describing the middle half — a smaller spread means more consistent data. Box plots (and cumulative frequency curves and polygons) make comparison easy: compare medians and IQRs. Crucially, write every comparison in context, naming the data and meaning. The common errors are comparing only the average, using the mean with outliers, giving bare numbers, and confusing range with IQR. Compare a centre and a spread, pick the right average, use the IQR, and interpret in context.