Comparing two data sets properly needs two statements: one about the centre (mean or median) and one about the spread (range, or interquartile range). Exam questions often say “compare the distributions”, and a single statement about averages gets only part of the marks. This is a core skill in statistics and distributions.
Why is the average not enough?
An average is one number that summarises the middle of a set. It cannot show whether the values are packed tightly around it or scattered far from it. The spread answers that.
A useful picture is two archers with the same average distance from the target centre. One clusters arrows close together, the other scatters them widely. The averages match, but you would trust the first archer more in a real contest.
How to compare, step by step
- Choose a measure of centre for each set that suits the data (see choosing an average).
- Choose a measure of spread, usually the range, and check for extreme values.
- Calculate both for each set and write the numbers down.
- Write a centre comparison in context, using the numbers.
- Write a spread comparison in context, then link them: for example, “higher on average but less consistent”.
Worked example
Two classes sit the same test. Class A marks: 50, 55, 60, 65, 70. Class B marks: 30, 45, 60, 75, 90.
Step 1, means: Class A: 50 + 55 + 60 + 65 + 70 = 300, and 300 ÷ 5 = 60. Class B: 30 + 45 + 60 + 75 + 90 = 300, and 300 ÷ 5 = 60.
Step 2, medians: both data sets are in order, and the middle value of each is 60.
Step 3, ranges: Class A: 70 − 50 = 20. Class B: 90 − 30 = 60.
Step 4, compare:
“The two classes have the same mean and median of 60, so they are equal in their centre. Class A has a range of 20 compared with 60 for class B, so class A’s marks are more consistent. Class B has both much higher and much lower marks.”
What the answer does not say: it does not say one class is better. The data supports a statement about consistency, not about which class has the stronger students.
The mistake to watch for
A common slip is to compare only the centres, or to decide from the range alone when one extreme value is involved.
Mistaken answer: “Class A and class B have the same mean, so they performed the same.”
Equal means hide a large difference in spread.
The correction is to always finish with the spread. If the range seems odd, check for an outlier. The data set 4, 5, 5, 6, 30 has a range of 26, but the first four values span only 2, so the range is misleading and the median (5) describes the centre better.
Check yourself
Try these, then open each answer.
1. Two runners’ times in seconds. Amy: 12.0, 12.2, 12.4, 12.6, 12.8. Ben: 11.0, 11.8, 12.4, 13.0, 13.8. Find each mean and range, then say who is more consistent.
Show answer
Amy: total 62.0, mean 62.0 ÷ 5 = 12.4, range 12.8 − 12.0 = 0.8. Ben: total 62.0, mean 12.4, range 13.8 − 11.0 = 2.8.
The means are equal at 12.4 seconds. Amy’s range of 0.8 is smaller than Ben’s 2.8, so Amy is more consistent.
2. Machine A fills bottles with a mean of 500.2 ml and a range of 1.0 ml. Machine B has a mean of 500.0 ml and a range of 6.0 ml. Which machine is more reliable? Give a reason.
Show answer
Machine A. Its range of 1.0 ml is much smaller than 6.0 ml, so the fills are much more consistent. The means are almost the same, so the difference lies in the spread.
3. The data set is 4, 5, 5, 6, 30. Find the range and explain why it may be a poor summary of spread.
Show answer
Range = 30 − 4 = 26. It is poor because one extreme value (30) stretches it. The other four values lie within 2 of each other, so the range suggests more spread than most of the data shows.
Where this leads next
The next lesson, identifying an unsupported conclusion from a sample, tests whether the claim you write can actually be backed by the data. Use the statistics and distribution explorer to add an outlier and watch the range jump while the median barely moves. For more comparison questions, go to the mixed practice set.
Many students write a correct centre comparison and then forget the spread. In online one-to-one Mathematics tuition, a teacher can build a sentence frame with you and then remove it as it becomes habit.