Foundations · Spread
Mean Absolute Deviation Calculator
The mean absolute deviation is the most literal measure of spread: the average distance of your data from the centre. No squaring, no square roots — just how far, on average, a value sits from the mean. This computes it about the mean and about the median (the most robust version), and shows it beside the standard deviation, which squares the deviations and so always comes out larger and more sensitive to a single outlier.
Mean absolute deviation = (1∕n) · Σ|xᵢ − centre|. The “centre” is usually the mean, but the median gives the smallest, most outlier-resistant version.
Result
In plain English
To say how spread out a set of numbers is, take each one’s distance from the centre and average those distances. That is the mean absolute deviation, and it is exactly as simple as it sounds — the typical gap between a value and the middle, in the data’s own units. The standard deviation answers the same question but squares each gap first (then square-roots at the end), which makes far-out points count for much more. The MAD treats every mile of distance equally; the SD charges extra for the long ones.
- mean absolute deviation (MAD)
- (1∕n) · Σ|xᵢ − x̄| — the average of the absolute distances from the mean. A plain, units-preserving measure of spread.
- about the mean vs the median
- You can measure distances from either centre. The median minimises the total absolute deviation, so the MAD about the median is always the smaller of the two — and the most robust to outliers.
- why the SD squares
- Squaring makes the measure differentiable (which calculus and least-squares need), makes variances additive, and ties neatly to the normal distribution. The price is sensitivity: one outlier, squared, can dominate.
- MAD ≤ SD
- For the same data the mean absolute deviation is never larger than the standard deviation; for a normal distribution the MAD is about 0.8 of the SD. A big gap between them hints at heavy tails or an outlier.
- robustness
- Because it does not square, the MAD — especially about the median — is far less rattled by extreme values than the SD, which is why it is favoured when the data are messy.
- not the same as MAD-the-error
- “MAD” also means mean absolute error in forecasting (average |actual − forecast|). Same arithmetic, different target: a forecast instead of the centre of the data.
Frequently asked
How do you calculate the mean absolute deviation?
Find the mean, take the absolute value of each number’s distance from it, and average those distances: MAD = (1∕n) · Σ|xᵢ − x̄|. For 1, 2, 3, 4, 10 the mean is 4, the absolute deviations are 3, 2, 1, 0, 6, and their average is 12 ∕ 5 = 2.4. The absolute value is what keeps the positive and negative gaps from cancelling — without it the deviations from the mean always sum to exactly zero.
What is the difference between mean absolute deviation and standard deviation?
Both measure typical distance from the mean, but the standard deviation squares each deviation, averages, and takes the square root, while the MAD just averages the raw distances. Squaring makes the SD weigh large deviations much more heavily, so it is always at least as big as the MAD and far more sensitive to outliers. The SD wins on mathematical convenience (it underlies least-squares, the normal distribution and additive variances); the MAD wins on plain interpretability and robustness.
Should I measure deviation about the mean or the median?
About the mean is the textbook default and pairs naturally with the standard deviation. About the median gives a smaller, sturdier number, because the median is the point that minimises the total absolute distance — no other centre makes the average distance smaller. If your data have outliers or a skew, the median version (and its cousin the median absolute deviation) is the more honest summary of how spread out the bulk of the data really are.
Why is the standard deviation used more often if the MAD is simpler?
History and mathematics. Squared deviations are smooth and easy to work with analytically — least-squares regression, the variance of a sum, the normal distribution and most of classical statistics are built on them, none of which the absolute value supports cleanly. So the SD became the default even though the MAD is easier to explain. For description, especially with messy or skewed data, the MAD is often the clearer choice; for inference and modelling, the SD’s algebra usually wins.