Two summaries can be compared fairly only when they use the same scale, the same kind of measure and groups that are fair matches. Check these three things before calculating anything.
This page is part of choosing the right summary of a data set. It follows reconstructing a missing observation from a mean.
What are the three checks?
| Check | Question to ask | Fails when |
|---|---|---|
| Scale | Are both on the same units or maximum? | One is out of 40 and the other out of 100 |
| Measure | Mean with mean, median with median? | A mean is compared with a median |
| Group | Are the groups fair matches and of comparable size? | Five students against five hundred |
Worked example: two tests
A class sits Test 1, out of 100, with a mean of 66 and a standard deviation of 12. The same class sits Test 2, out of 40, with a mean of 28 and a standard deviation of 4.
The scales differ, so rescale Test 2 before comparing. Multiply by 100 ÷ 40 = 2.5.
- Mean: 28 × 2.5 = 70 out of 100.
- Standard deviation: 4 × 2.5 = 10 out of 100.
| Mean (out of 100) | Standard deviation | |
|---|---|---|
| Test 1 | 66 | 12 |
| Test 2 | 70 | 10 |
Conclusion. After rescaling, Test 2 has a higher mean (70 against 66) and a smaller standard deviation (10 against 12). The class scored higher and more consistently on Test 2.
What the raw numbers would have said
Comparing 66 with 28, a student might say the class did much worse on Test 2. Comparing 12 with 4, the same student might say Test 2 was far more consistent.
Both conclusions come from the different scales, not from the marks. Converting first removes the trap.
When a comparison cannot be made
If one set gives a mean and the other gives only a median, say that the summaries are not directly comparable, and state what you would need. For example, you could ask for the median of the first set or the mean of the second.
Writing “cannot be compared because …” with a clear reason is better than forcing a comparison. Try this on invented pairs with the graph evidence comparison lab.
Check yourself
Quiz A is out of 20 with a mean of 14 and a standard deviation of 3. Quiz B is out of 50 with a mean of 36 and a standard deviation of 6. Compare them fairly.
Answer
Convert both to percentages. Quiz A: mean 14 ÷ 20 = 70%, standard deviation 3 ÷ 20 = 15%.
Quiz B: mean 36 ÷ 50 = 72%, standard deviation 6 ÷ 50 = 12%. Quiz B has a slightly higher mean (72% against 70%) and a smaller spread (12% against 15%), so it was slightly higher and more consistent.
What to study next
Test the cluster with the practice set. Return to the cluster overview to revisit any page.
For a teacher to ask you whether a comparison is fair, see online one-to-one Mathematics tuition.