MISSION · Data and probability
Comparing Spread with Mean Absolute Deviation
How to say that one set is more spread out than another
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The idea in one sentence
Mean absolute deviation is the average distance from the mean — a plain answer to “how spread out is this?”
Why the mean isn’t enough
Two basketball players both average 20 points a game.
The first scores 19, 20, 21, 20, 20. The second scores 5, 35, 12, 28, 20. Same mean, completely different players — one is reliable, the other is unpredictable, and a coach cares enormously about the difference.
The mean describes the middle. Spread describes how much the values wander from it, and you need both before you can say anything useful about a data set.
The simplest measure, and its flaw
The range is highest minus lowest. For the first player that’s 2; for the second it’s 30.
Quick, and easily wrecked. Range uses only two values and ignores everything between them, so one unusual game changes it completely while the other measures barely move.
Mean absolute deviation
MAD uses every value. Four steps:
- Find the mean.
- Find each value’s distance from the mean, ignoring the sign.
- Add those distances.
- Divide by how many there are.
For the first player, mean 20: distances are 1, 0, 1, 0, 0, totalling 2, so \(MAD = \frac{2}{5} = 0.4\) points.
For the second: distances are 15, 15, 8, 8, 0, totalling 46, so \(MAD = \frac{46}{5} = 9.2\) points.
A typical game for the first player lands under half a point from their average. For the second, over nine. That number is the answer to the coach’s question.
Why the signs are dropped
Without it, the deviations always cancel to exactly zero — that’s what being the mean means. Taking the absolute value keeps distances positive so they measure something.
There’s another way to stop the cancelling: square the deviations instead. That leads to variance and standard deviation, which dominate at higher levels because squares are easier to work with algebraically. MAD is easier to understand — it’s literally the average distance — which is why it comes first.
Reading it
MAD carries the same units as the data, so it is directly interpretable. A MAD of 0.4 points means typical games sit 0.4 points from average.
Comparing two sets: bigger MAD means more spread out, less consistent, less predictable. Zero MAD means every value is identical.
Common mix-ups
- Forgetting the absolute value, giving zero every time.
- Dividing by the wrong count. Divide by how many values there are.
- Using the range when there’s an outlier, which it exaggerates.
- Comparing MADs measured in different units.
Check yourself
Find the MAD of 2, 4, 6, 8.
The mean is 5. Distances are 3, 1, 1, 3, totalling 8, so \(MAD = \frac{8}{4} = 2\).
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