How this instrument works
Median absolute deviation (often abbreviated MAD, distinct from this site's mean absolute deviation, or MAD-from-the-mean) measures spread using medians at every step instead of means. First, find the data's median. Second, calculate every value's absolute distance from that median. Third — and this is the step that gives the statistic its robustness — take the median (not the mean) of those distances, rather than averaging them. The result is a single number describing a 'typical' distance from the center, built entirely without ever computing a mean.
This double-median construction is what makes the statistic so resistant to outliers. A mean, and any calculation built on a mean, can be dragged arbitrarily far by even a single extreme value — one billionaire's income can single-handedly shift an entire town's average income. A median barely budges when one extreme value is added or changed, because it only cares about relative position (which values rank in the middle), not magnitude. Because median absolute deviation uses a median twice over — once to find the center, once to summarize the distances — it inherits that same resistance twice, making it one of the most robust spread measures in common use.
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 − median| transform and then taking a second median of the results needs a dedicated calculation step, distinct from this site's mean-based mad calculator, which performs the analogous two-step process using means throughout instead of medians.
- Enter your numbers into Data set — separate them with commas, spaces or new lines.
- Read Median beneath to confirm the center point your deviations are measured from.
- Read Median absolute deviation (from the median) for the typical distance a value sits from that median.
- A smaller value means the data clusters tightly around its median; a larger value means it's more spread out.
- Working with data that might contain outliers? This median-based statistic will generally hold up better than a mean-based spread measure like standard deviation or mad.
Worked example — eight days of small-business orders
A small shop logs its daily order counts over 8 days: 10, 12, 23, 23, 16, 23, 21, 16. Enter those eight values into Data set. Sorted, the data reads 10, 12, 16, 16, 21, 23, 23, 23; with an even count, Median is the average of the 4th and 5th sorted values, 16 and 21, giving Median = 18.5.
Each day's distance from 18.5 is: |10−18.5|=8.5, |12−18.5|=6.5, |23−18.5|=4.5 (for each of the three 23s), |16−18.5|=2.5 (for each of the two 16s), |21−18.5|=2.5. Sorting those eight distances gives 2.5, 2.5, 2.5, 4.5, 4.5, 4.5, 6.5, 8.5; the median of those (average of the 4th and 5th) is (4.5+4.5)/2 = 4.5. Median absolute deviation (from the median) reads 4.5 exactly — a typical day's order count sits about 4.5 orders away from the 8-day median of 18.5.
Questions
How is this different from this site's mean absolute deviation (mad) calculator?
Both measure spread using absolute distances, but this one anchors every step to the median instead of the mean: it measures distance from the data's median, then takes the median of those distances, rather than the mean of distances from the data's mean. The median-based version is markedly more resistant to outliers, since neither of its two steps is a mean that a single extreme value could drag off course — the mean-based mad, by contrast, is pulled (gently) by any extreme value in the data.
Why is median absolute deviation considered so robust to outliers?
Because it's built entirely from medians, and a median only depends on which values rank in the middle of a sorted list, not on how large or small any individual value actually is. Adding one wildly extreme value to a data set barely shifts its median at all, and it barely shifts the median-of-distances calculation either — compare that to a mean or a standard deviation, both of which a single sufficiently extreme value can distort dramatically.
Is a larger median absolute deviation always a bad sign?
Not inherently — a larger value simply means the data is more spread out around its median, which may be entirely expected and normal for the kind of data you're measuring (naturally variable measurements, say, versus a tightly controlled manufacturing process). It's most useful as a comparison figure: comparing the median absolute deviation of two data sets, or of the same process measured at two different times, tells you which one is more (or less) tightly clustered.
Can median absolute deviation be used to flag outliers in a data set?
Yes — this is one of its most common practical uses. A common rule of thumb flags any value whose distance from the median exceeds some multiple (often around 3) of the median absolute deviation, scaled by a constant factor, as a likely outlier. Because the underlying calculation is itself resistant to outliers, this approach tends to catch genuine outliers more reliably than a rule based on standard deviation, which the outliers themselves can inflate.
How much data do I need for a meaningful result?
At least one value is technically all this calculator requires to run, but a meaningful measure of spread needs more than one distinct value — with only a single number, or with every value identical, there's no actual variation to measure, and the median absolute deviation comes out to exactly zero. A handful of values or more will generally give a more stable, more interpretable result.