The IQR measures spread around the median. To measure spread around the mean, we use distances, which are absolute values.
Key idea
The mean absolute deviation (MAD) is the average distance of the values from the mean. Find the mean, find each value's distance from it, then find the mean of those distances.
Why it works
If you subtract the mean from each value and keep the signs, the results always add to 0: the mean is the balance point, so what is above it exactly cancels what is below it. Averaging those signed differences would say every data set has no spread. Distances are never negative, so they cannot cancel, and their average measures how far a typical value is from the mean.
Worked example: Find the MAD
Find the MAD of 10, 12, 15, 18, 20.
- Mean: $75 \div 5 = 15$.
- Distances from 15: 5, 3, 0, 3, 5.
- Mean distance: $16 \div 5 = 3.2$.
Answer: 3.2
| Value | Value − mean | Distance |
|---|---|---|
| 10 | $-5$ | 5 |
| 12 | $-3$ | 3 |
| 15 | $0$ | 0 |
| 18 | $3$ | 3 |
| 20 | $5$ | 5 |
| Total | $0$ | 16 |
Worked example: Compare spreads
Set A: 4, 5, 6, 7, 8. Set B: 2, 4, 6, 8, 10. Both have mean 6. Which has the greater MAD?
- A: distances 2, 1, 0, 1, 2, so MAD $= 6 \div 5 = 1.2$.
- B: distances 4, 2, 0, 2, 4, so MAD $= 12 \div 5 = 2.4$.
Answer: Set B: its values are farther from the mean.
Common mistake
Averaging the signed differences, which always gives 0. Quick check: the MAD can only be 0 when every value equals the mean. If your data vary and you got 0, take the absolute values.