Algebra
Data and statistics
Mean absolute deviation
20 practice questions
2 video lessons
Theory + worked examples
Theory
Mean absolute deviation (MAD) measures spread as the average distance from the mean:
- Find the mean.
- Find each absolute deviation \(|x-\bar x|\).
- Average those distances.
A larger MAD means the data are more spread out.
MAD averages the distances from the mean.
Computing MAD.
Mean absolute deviation:
\[\text{MAD}=\dfrac{\sum|x-\bar x|}{n}\]
Use absolute value so distances don't cancel.
How to find MAD
- Compute the mean.
- Find each \(|x-\bar x|\).
- Add them up.
- Divide by the number of values.
Example 1 — Find the mean
Find the mean of \(2,4,5,6,8\).
Solution
Add and divide by \(5\).
| \(\bar x\) | \(=\) | \(\dfrac{25}{5}=5\) |
Example 2 — Distances
Find each absolute deviation from the mean \(5\).
Solution
Take \(|x-5|\).
| \(3,\ 1,\ 0,\ 1,\ 3\) |
Example 3 — The MAD
Find the mean absolute deviation.
Solution
Average the distances.
| \(\text{MAD}\) | \(=\) | \(\dfrac{3+1+0+1+3}{5}=1.6\) |
Example 4 — What it means
What does a larger MAD indicate?
Solution
More spread — data are farther from the mean on average.
Common pitfalls
Use absolute values so positives and negatives don't cancel.
Divide by the count to average.
MAD is a distance, always \(\ge0\).
Frequently asked questions
What is mean absolute deviation?
The average distance of the data from the mean.
Why use absolute values?
So the deviations don't cancel out.
What does a large MAD mean?
The data are more spread out.
Is MAD ever negative?
No — it is an average of distances.
More in Data and statistics