What Is Mean Absolute Deviation?
Mean absolute deviation (MAD) is the average of the absolute differences between each data value and the mean. Because it skips the squaring used by variance and standard deviation, it reads directly in the units of your data: a MAD of 3 on exam scores means a typical score sits about 3 points from the average.
People who search for a "MAD calculator" or ask "how do I find the MAD" sometimes mean different statistics. This tool gives you the mean absolute deviation about the mean (the version in most school courses), the mean absolute deviation about the median, and the median absolute deviation, so you can use the one your course or project asks for. The standard deviation is shown next to it for comparison.
Below the calculator you will find the MAD formula, a table explaining every variable, a worked example, common mistakes and answers to questions such as MAD versus standard deviation.
How to Use the Mean Absolute Deviation Calculator
- Enter your numbers separated by commas, spaces or new lines.
- Read the MAD at the top of the results. The other versions are listed underneath.
- Open the step-by-step solution to see each distance from the mean, or use the table and bar chart to see which values contribute most.
Mean Absolute Deviation Formula
Mean absolute deviation (about the mean)
Median absolute deviation
| Measure | Centre used | Averaged with |
|---|---|---|
| Mean absolute deviation | Mean | Mean of the distances |
| Mean absolute deviation about the median | Median | Mean of the distances |
| Median absolute deviation | Median | Median of the distances |
Mean Absolute Deviation Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| MAD | Mean absolute deviation | Same units as your data |
| xᵢ | Each individual data value | i runs from 1 to n |
| x̄ | Mean of the data | Σxᵢ ÷ n |
| n | Number of values | MAD divides by n, not n − 1 |
| |xᵢ − x̄| | Absolute deviation of one value | Distance from the mean, never negative |
Mean Absolute Deviation Worked Example
Using the example values (press “Load example” above to try them):
- Find the mean: x̄ = 40 ÷ 8 = 5.
- Find how far each value is from the mean, ignoring the sign: |x − x̄|. These are 3, 1, 1, 1, 0, 0, 2, 4.
- Add the distances: Σ|x − x̄| = 12.
- Divide by n: MAD = 12 ÷ 8 = 1.5.
Result: Mean absolute deviation (MAD) = 1.5; Mean (x̄) = 5.
What the MAD tells you
The mean absolute deviation answers a simple question: on average, how far is a value from the mean? A MAD of 3 on test scores means a typical student scored about 3 points above or below the class average. That directness is why MAD is taught in school statistics courses before standard deviation.
The comparison row in the results shows the standard deviation alongside the MAD. For normally distributed data the SD is about 1.25 times the MAD (precisely √(π/2) ≈ 1.2533); a ratio far above that suggests heavy tails or outliers.
MAD versus standard deviation versus range
| Measure | Uses all values | Sensitive to outliers | Easy to explain |
|---|---|---|---|
| Range | No (two values) | Extremely | Very |
| Mean absolute deviation | Yes | Moderately | Very |
| Standard deviation | Yes | Strongly (squares) | Moderately |
| Median absolute deviation | Yes | Barely | Moderately |
| Interquartile range | Middle 50 % | No | Moderately |
Mean absolute deviation in school math
Grade 6 and 7 curricula ask students to find the MAD of small data sets to compare variability, for example the daily temperatures of two cities with the same mean. The method is always the same: find the mean, find each distance from it, then average the distances. The table above is laid out exactly like the one in textbooks.
Where Mean Absolute Deviation Is Used
Classroom statistics
Comparing the variability of two small data sets, such as daily temperatures in two cities that share the same mean.
Repeated measurements
Summarising how consistent a set of readings is in the original units, without squaring.
Data with a few extreme values
Giving a spread figure that is less dominated by outliers than the standard deviation.
Explaining spread to non-specialists
"On average, values are this far from the mean" is easy to say out loud and to put in a report.
Common Mean Absolute Deviation Mistakes to Avoid
- Forgetting absolute values, which makes the deviations sum to zero.
- Mixing up the mean absolute deviation with the median absolute deviation.
- Using the median instead of the mean as the centre when the question says “about the mean”.
- Dividing by n − 1; the standard MAD divides by n.
- A larger MAD means the values are more spread out. A MAD of 0 means every value is the same.
- For roughly normal data, the standard deviation is about 1.25 times the mean absolute deviation.
- The median absolute deviation barely moves when you add an outlier, which makes it a good robust measure.
Mean Absolute Deviation Terms and Definitions
- Absolute value
- The distance of a number from zero, always positive: |−3| = 3.
- Deviation
- How far a value is from the centre.
- Robust statistic
- A measure that is not much affected by outliers.
- Mean absolute deviation
- The average distance of values from the mean.
- Median absolute deviation
- The median distance of values from the median.
Mean Absolute Deviation FAQs
Why take the absolute value?
The signed deviations from the mean always add to zero. Absolute values make every distance positive so they can be averaged.
How is MAD different from standard deviation?
SD squares the deviations, so large ones count disproportionately. MAD treats all distances equally, which is easier to interpret but harder to use in further formulas.
What is a good MAD?
There is no universal value. It depends on the units. Compare it with the mean (shown as a percentage in the results) or with the MAD of another data set.
Is MAD affected by outliers?
Less than the standard deviation, but more than the median absolute deviation or the interquartile range.
What is the MAD used for in forecasting?
Forecasters use the mean absolute deviation of forecast errors (often called MAE) to measure accuracy. See the mean squared error and percentage error calculators for related measures.
How do you calculate the mean absolute deviation?
Find the mean of the data, subtract it from each value and drop the signs to get the absolute distances, then average those distances.
What is the difference between MAD and standard deviation?
Both measure spread. Standard deviation squares the distances before averaging, which gives extra weight to values far from the mean. MAD uses the plain distances, so it is simpler and less sensitive to outliers.
Is MAD the same as the median absolute deviation?
Not always. The abbreviation is used for both. This calculator shows the mean absolute deviation and the median absolute deviation separately, so you can pick the right one.
Can the mean absolute deviation be negative?
No. Every distance is zero or positive, so the MAD is always zero or greater.