How this instrument works
Mean absolute deviation (MAD) measures spread by taking the distance of every value from the mean, ignoring whether that distance is above or below (that's the 'absolute' part — negative deviations flip to positive), and then averaging those distances. It answers the question 'on average, how far off from the mean is a typical value?' in the most direct way possible, without the squaring step that variance and standard deviation both use. The result comes out in the same units as your original data, which is part of why it's often the easier of the two to explain to someone who isn't already comfortable with statistics.
The calculation happens in two steps. First, find the mean of the whole data set. Second, for every value, calculate its distance from that mean — |value − mean| — and average all of those distances together. Because the DSL-style single-list aggregate functions behind most of this site's calculators can only reduce one list at a time, computing this elementwise |x − constant| transform before averaging needs a dedicated calculation step rather than a single built-in formula, which is exactly what this instrument performs.
Mean absolute deviation and variance/standard deviation both measure spread, but they weight extreme values differently: because variance squares every deviation before averaging, a single very large outlier inflates variance and standard deviation disproportionately more than it inflates MAD, which treats every unit of distance from the mean equally regardless of size. That makes MAD a somewhat gentler, more outlier-resistant measure of spread than standard deviation, though median absolute deviation (measuring distance from the median instead of the mean) goes further still in that direction.
- Enter your numbers into Data set — separate them with commas, spaces or new lines.
- Read Mean beneath to confirm the average your deviations are being measured from.
- Read Mean absolute deviation (from the mean) for the average distance every value sits from that mean.
- A smaller MAD means the data clusters tightly around its mean; a larger MAD means it's more spread out.
- Need the median-based version instead? This site's median-absolute-deviation instrument measures distance from the median rather than the mean, which is more resistant to outliers.
Worked example — the eight quiz scores
Enter 2, 4, 4, 4, 5, 5, 7, 9 into Data set. Mean reads 5.0 (sum 40 divided by 8 scores). Each score's distance from 5 is: |2−5|=3, |4−5|=1 (three times), |5−5|=0 (twice), |7−5|=2, and |9−5|=4.
Adding those eight distances gives 3+1+1+1+0+0+2+4 = 12. Dividing by the 8 scores, Mean absolute deviation (from the mean) reads 12/8 = 1.5 exactly — on average, a quiz score in this set sits 1.5 points away from the class's mean of 5, a single, easy-to-picture figure describing how tightly (or loosely) these eight scores cluster around their average.
Questions
How is MAD different from standard deviation?
Both measure spread, but standard deviation squares every deviation from the mean before averaging (and then takes a square root at the end), which weights larger deviations much more heavily than smaller ones. MAD skips the squaring step entirely — it just averages the plain (absolute) distances — so it's more straightforward to compute by hand and interpret, and it's less sensitive to a single unusually extreme value pulling the result upward, since that one point's distance is counted at its actual size rather than squared.
How is this different from this site's median-absolute-deviation calculator?
This calculator measures every value's distance from the data's mean and then averages those distances. Median-absolute-deviation instead measures every value's distance from the data's median, and then takes the median (not the mean) of those distances. The median-based version is more resistant to outliers on both ends of the calculation, since neither the reference point nor the final averaging step is pulled by extreme values the way a mean-based calculation can be.
Why take the absolute value instead of just averaging the deviations directly?
Because the deviations from the mean always sum to exactly zero by definition — every value above the mean is balanced out by values below it. Averaging the raw (signed) deviations would always give exactly 0, telling you nothing about spread. Taking the absolute value first (treating a deviation of -3 the same as +3) means every distance contributes positively to the total, so the average genuinely reflects how far, on average, values sit from the mean.
Can MAD ever be negative or zero?
It can never be negative, since it's built entirely from absolute (unsigned) distances, which are always zero or positive. It equals exactly zero only when every single value in the data set is identical to the mean — meaning there's no spread in the data at all, since every distance from the mean is zero.
What units is MAD expressed in?
The same units as your original data — dollars, test-score points, seconds, whatever the values represent — because MAD never squares anything the way variance does. That's one of its main practical advantages: a MAD of 1.5 points on an 8-point quiz is immediately interpretable without needing to take a square root first, unlike variance, which comes out in squared units.