Comparing Distributions

A single number rarely settles a comparison. To compare two data sets fairly, look at two things at once:

an average — the median or the mean: which set is higher or more typical?
a measure of spread — the range or the IQR: which set is more consistent?

And always say what it means in context. Don't just report "A's median is higher and its IQR is smaller" — translate it: "Class A scored higher on average, and was more consistent, because its IQR was smaller."

Higher on average, and more consistent

Drawing both box plots on the same scale makes the comparison visual. Here Class A and Class B sit one above the other. Class B's box is shifted to the right of A's — so B scored higher on average — and B's box and whiskers are bunched tighter, so B was more consistent. Step through to reveal each one.

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.

Compare one average (median or mean) and one spread (range or IQR).
A higher median or mean means a data set is higher on average.
A smaller IQR or range means the data is more consistent (less spread out).
Two box plots on the same scale make the comparison visual at a glance.
Always interpret in context — say what higher and more consistent mean for the real situation.