Two classes sit the same test. Two coffee shops time how long customers wait. Two football teams count the goals they score each week. In every case someone wants to know the same thing: which is better?
It is tempting to grab one number — "Class A's average was higher, so Class A is better" — and stop there. But one number tells only half the story. To compare two groups fairly you must always look at two things at once:
A group can win on the average and lose on the spread. That is exactly why you report both — and then say, in a plain sentence, what it all means for the real situation.
A fair comparison has three moving parts. Pick one measure of average and one measure of spread, quote them for both groups, then interpret in context.
The golden rule: use the same pair for both groups. Compare median with median, or mean with mean — never one group's mean against the other's median. And always translate the numbers into words about the real thing: "Class A scored higher on average, and was more consistent because its IQR was smaller."
Two classes take a test marked out of 25. You are handed just their medians and IQRs:
| Class | Median | IQR |
|---|---|---|
| Class A | 14 | 8 |
| Class B | 15 | 4 |
Average: B's median (15) is higher than A's median (14), so Class B scored higher on average. Spread: B's IQR (4) is smaller than A's IQR (8), so Class B was more consistent — its middle-half of marks is packed into a narrower band.
Put together in context: "Class B did better: it had the higher median, so it scored higher on average, and the smaller IQR, so its results were more consistent than Class A's." Here B happens to win on both counts — but that is not always how it goes.
Two coffee shops record how many minutes customers wait. Now the winner is not obvious:
| Shop | Median wait | Range |
|---|---|---|
| Beans & Co. | 3 min | 2 min |
| The Grind | 2 min | 9 min |
Average: The Grind has the lower median wait (2 min vs 3 min), so it is usually faster. Spread: but its range is 9 min against Beans & Co.'s 2 min, so The Grind is far less consistent — some customers are served almost instantly, others wait ages.
In context: "The Grind is quicker on average (lower median), but Beans & Co. is much more reliable (smaller range), so you can predict your wait there." Neither is simply "better" — which you prefer depends on whether you value speed or certainty. That is the whole point of quoting both.
Drawing both box plots on the same scale lets you compare at a glance. Here Class A and Class B sit one above the other. Step through to reveal each one, then read off the two things you need.
Class B's box is shifted to the right of A's — the median line is further along — so B scored higher on average. And B's box and whiskers are bunched tighter (a smaller box = smaller IQR, shorter whiskers = smaller range), so B was more consistent.
In context: "Class B had the higher median, so it scored higher on average; and B's IQR and range were smaller, so its scores were more consistent than A's." Two facts, one sentence — that is a full-marks comparison.
These are the mistakes that lose marks in every exam:
There is an old warning: a person who cannot swim drowned crossing a river that was, on average, only 1 metre deep. The average was true — and useless. The river was ankle-deep at the banks and three metres deep in the middle. The spread was everything, and the average hid it.
This is why comparing distributions properly is what separates real statistics from "lying with statistics". Two data sets can have the exact same mean yet behave completely differently — one calm and predictable, one wild and dangerous. Quote only the average and you can make almost anything look fine. It is precisely because they refuse to look at averages alone that experts in weather, finance, and quality control earn their keep: the spread is where the risk lives.