You can describe a distribution by its center and spread, and measure spread with the mean absolute deviation (MAD). To compare two groups, you need both.
Key idea
Compare the centers relative to the spread: divide the difference of the means by the MAD (or the difference of the medians by the IQR). A gap of about 2 or more MADs means the groups are clearly different. A gap well under 1 MAD means the groups overlap a lot.
Why it works
A difference of 3 points means a lot if each group's scores barely vary, and very little if scores spread over 30 points. Dividing by the spread measures the gap in "typical distances from the mean." When the gap is many of those, most of one group is above most of the other.
Worked examples
Worked example: Clearly different
Plants with fertilizer A have a mean height of 64 cm; with fertilizer B, 72 cm. Both groups have a MAD of 4 cm. How different are they?
- Difference of the means: $72 - 64 = 8$ cm.
- In MADs: $8 \div 4 = 2$.
- Two MADs apart: the groups overlap only a little, so fertilizer B plants are clearly taller.
Answer: 2 MADs apart: clearly different
Worked example: A lot of overlap
Two teams' jump distances have means of 70 and 72 inches. Both have a MAD of 6 inches. How different are they?
- Difference: $72 - 70 = 2$ inches.
- In MADs: $2 \div 6 = \frac{1}{3}$.
- Only a third of a MAD: the groups overlap a lot, and the difference may be ordinary variation.
Answer: About $\frac{1}{3}$ MAD apart: lots of overlap
Worked example: With medians
Two box plots have medians of 40 and 52, and both IQRs are 12. How many IQRs apart are the medians?
- $52 - 40 = 12$.
- $12 \div 12 = 1$ IQR. That is a moderate gap: the groups still overlap some, so the difference is worth noticing but is not as clear as a gap of 2 or more.
Answer: 1 IQR
Common mistake
Calling two groups different because their means are different. Every two groups have slightly different means. Always compare the gap with the spread: divide by the MAD and look at how much the dot plots or box plots overlap.