What this calculator does
Mean absolute deviation is the average distance of the data from its centre. For 4, 8, 6, 5, 3, 9, 7 the mean is 6 and the average distance from it is 1.714.
It is often the more intuitive measure of spread, because it is in the original units and means exactly what it says. The standard deviation squares the deviations before averaging and then takes a square root, which gives more weight to large deviations and makes the result harder to describe in plain terms. The two answer slightly different questions.
The formula
The absolute difference between each value and the centre is averaged. Both versions are reported: deviation about the mean, which is the standard definition, and deviation about the median, which is mathematically the smaller of the two for any data set. The absolute value is what makes the deviations add rather than cancel.
| Term | Meaning |
|---|---|
| MAD | Mean absolute deviation, the average distance from the centre. |
| About the mean | The standard form, using the arithmetic mean as the centre. |
| About the median | An alternative, and always less than or equal to the version about the mean. |
| Absolute value | Distance regardless of direction, which stops positive and negative deviations cancelling. |
The inputs explained
| Field | What to enter |
|---|---|
| Data (comma or space separated) | Your data, comma or space separated. |
When to use it
Describing spread in plain language
Average distance from the mean is easier to explain to a non-technical audience than a standard deviation.
Measuring forecast error
Mean absolute error is the same calculation applied to the gap between forecast and actual, and it is a standard accuracy measure.
Working with outlier-prone data
The MAD responds to a large outlier proportionally, where the standard deviation responds to its square.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the two versions differ?
Symmetric and heavily skewed data compared.
| Data set | Mean absolute deviation (about the mean) | About the median | Mean |
|---|---|---|---|
| 4, 8, 6, 5, 3, 9, 7 | 1.714 | 1.714 | 6.000 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 1.500 | 1.500 | 5.000 |
| 1, 2, 3, 4, 100 | 31.200 | 20.200 | 22.000 |
Questions
What is the difference between mean absolute deviation and standard deviation?
The MAD averages the distances directly; the standard deviation averages their squares and then takes a square root. Squaring gives extra weight to large deviations, so the standard deviation is always at least as large and reacts more strongly to outliers.
Why is deviation about the median smaller?
Because the median is the value that minimises the sum of absolute deviations, which is a mathematical property rather than a coincidence. For any data set, no other centre gives a smaller mean absolute deviation, so the median version is always less than or equal to the mean version.
When should I use MAD instead of standard deviation?
When you want a measure in the original units that is easy to explain, or when large outliers should not dominate. Standard deviation remains the default for inferential statistics, because the mathematics of most tests is built around variance rather than absolute deviation.
Is MAD the same as median absolute deviation?
No, and the shared acronym causes genuine confusion. Mean absolute deviation averages the distances. Median absolute deviation takes the median of the distances from the median, which is far more robust again. This calculator computes the mean version about both centres.
For the full set of spread measures, see the descriptive statistics calculator. For the middle value, see the median calculator.