Mean Absolute Deviation Calculator
Enter a list of numbers and get the mean absolute deviation, a per-value deviation table, and a number-line diagram showing every point's distance from the mean.
What is Mean Absolute Deviation?
Mean absolute deviation (MAD) is the average distance between each value in a dataset and the mean of that dataset. It answers the question, "on average, how far are my values from the middle?" — in the same units as the original data.
MAD vs standard deviation. Both measure spread, but standard deviation squares each deviation before averaging, which makes it more sensitive to outliers. MAD averages the raw absolute distances, so it stays in the original units and reacts to a far-away point in proportion to its actual distance — not the square of it.
MAD explained, step by step
Each concept below has a plain-English definition, its formula (with every symbol spelled out), a small diagram, and a worked example — all in one card so you never have to jump between sections to piece it together.
1. Start with the mean (x̄)
MAD measures how far values sit from a chosen center. That center is the arithmetic mean — add every value, divide by how many there are. The mean is the balance point of the data.
2. Absolute deviation of each value
For every value, measure its distance from the mean and drop the sign. Signed deviations always sum to zero (that's a property of the mean), so absolute values are what let you meaningfully average them.
3. Average the absolute deviations
MAD itself is just the mean of those absolute deviations — total distance divided by the count. The result is in the same units as the original data, which makes it easy to interpret.
4. Why MAD ≠ standard deviation
Standard deviation squares each deviation before averaging, then takes a square root. Squaring makes far-away points count much more, so SD is more sensitive to outliers. MAD grows linearly with distance — a point 4 units away contributes 4 to MAD but 16 to the SD sum-of-squares.
MAD around the median
A common variant computes the mean absolute deviation around the median instead of the mean. Because the median is itself an outlier-resistant center, MAD around the median is even more robust: extreme values change it much less.
Quick example. Dataset 1, 2, 3, 4, 100. Mean = 22, median = 3. MAD around the mean = (21 + 20 + 19 + 18 + 78) / 5 = 31.2. MAD around the median = (2 + 1 + 0 + 1 + 97) / 5 = 20.2 — a smaller, more representative spread for the typical value. This calculator uses the mean by default.
Common mistakes
- Forgetting the absolute value. Signed deviations always sum to zero — that's a property of the mean, not a spread measurement.
- Confusing MAD with variance or standard deviation. Variance is in squared units; SD is in the same units as MAD but is a different value.
- Dividing by n − 1. Unlike sample SD, MAD has no Bessel correction — divide by n whether the data is a sample or the whole population.
Features of this calculator
- Transparent per-value table — see the deviation and absolute deviation for every data point, not just the final MAD
- Number-line diagram plots each point relative to the mean with dashed lines showing the deviation distance visually
- Show/hide step-by-step working: mean, sum of |xᵢ − x̄|, final divide
- Cleans messy input — commas, tabs, currency symbols, thousand separators, brackets
- Explains MAD vs standard deviation and MAD around the median in plain language
Frequently asked questions
+When should I report MAD instead of standard deviation?
When your audience is non-technical, when a few outliers would exaggerate standard deviation, or when you need a spread number in the original units without the "square then square-root" round trip.
+Can MAD ever be zero?
Yes — only when every value in the dataset is identical, so every deviation is 0.
+Is MAD always smaller than standard deviation?
For most real datasets, yes. There's a rough rule of thumb: for a normally-distributed dataset, MAD ≈ 0.7979 × σ.
+Does the order of the numbers matter?
No. MAD depends only on the set of values, not their order.