To compare two distributions fairly, use the same scale for both, compare an average and a spread, and use proportions when the groups are different sizes. Then write each comparison in context, naming both groups.
This skill belongs to data displays and cumulative reasoning. It combines the medians and quartiles from cumulative frequency with the careful reading of scales, which you can also practise in the map scale and contour practice tool.
What makes a comparison fair?
There are three checks. First, scale: are both graphs drawn with the same axis?
Second, size: do the two groups contain the same number of values? Third, measures: are you comparing a typical value and a spread?
A median is usually the safer average when there are extreme values, because it is not pulled by them. The interquartile range (IQR) is the matching spread, since it ignores the most extreme quarter at each end.
How to write a comparison, step by step
- Pick the average (median or mean) for each group and compare them in one sentence, in context.
- Pick the spread (range or IQR) for each group and compare in a second sentence.
- Use proportions when the groups differ in size, and compare percentages, not counts.
- Check scales on any graph before you compare by eye.
- Say what it means: a smaller IQR means the values are more consistent.
Worked example
Two classes sat the same test. Summary statistics:
| Lower quartile | Median | Upper quartile | Range | |
|---|---|---|---|---|
| Class A | 55 | 62 | 69 | 40 |
| Class B | 44 | 58 | 70 | 55 |
Step 1, IQR: Class A: 69 − 55 = 14. Class B: 70 − 44 = 26.
Step 2, compare averages: the median for Class A (62) is higher than for Class B (58), so the typical mark was higher in Class A.
Step 3, compare spreads: the IQR for Class A (14) is smaller than for Class B (26), so Class A’s marks were more consistent.
Step 4, conclusion in one line: Class A had a higher median and more consistent marks than Class B.
Now compare proportions. In Class A, 12 of 30 students scored over 60. In Class B, 28 of 80 students scored over 60.
Step 5: 12 ÷ 30 = 0.40 = 40%. 28 ÷ 80 = 0.35 = 35%.
Class B has more students over 60 (28 against 12) but a smaller proportion (35% against 40%).
The mistake to watch for
The usual slip is comparing counts when the groups are not the same size.
Mistaken answer: “Class B did better on this test because 28 students scored over 60 and only 12 did in Class A.”
The student compared 28 with 12 and forgot that Class B has far more students.
The correction is to turn each count into a proportion of its own group. Another related slip is comparing two graphs drawn with different axis scales by eye, which can make a spread look bigger or smaller than it is.
Check yourself
Try these, then open each answer.
1. Group X has median 45 and IQR 10. Group Y has median 50 and IQR 20. Write two comparisons in context (test marks).
Show answer
Group Y has a higher median (50 against 45), so its typical mark is higher. Group X has a smaller IQR (10 against 20), so its marks are more consistent.
2. In Survey 1, 18 of 45 students walk to school. In Survey 2, 30 of 100 walk. Which survey has the larger proportion walking?
Show answer
18 ÷ 45 = 0.40 = 40%. 30 ÷ 100 = 0.30 = 30%. Survey 1 has the larger proportion, even though Survey 2 has more students walking.
3. One graph has a vertical axis going 0 to 10. A second graph of a bigger group has its axis going 0 to 100. Why is it unfair to compare the bar heights by eye?
Show answer
The same bar height stands for very different frequencies on the two axes, and the groups are different sizes. Redraw both on a common scale, or compare relative frequencies or percentages.
Where this leads next
Once you can compare, the next step is to decide which display to draw in the first place, which is the topic of choosing a display that preserves the data meaning. Practise everything together in the data displays practice set. The non-calculator working trainer can help with fractions and percentages.
Comparison sentences are one of the places where careful wording earns marks. A teacher can read what you wrote and show where a claim needs a number behind it, which is how we work in online one-to-one Mathematics tuition.