📐 Mean Absolute Deviation (MAD) Calculator
By ToolNimba Editorial Team · Updated 2026-06-19
| Value (x) | x minus mean | |x minus mean| |
|---|
MAD is the average distance of each value from the mean.
The mean absolute deviation (MAD) measures how spread out a set of numbers is by averaging how far each value sits from the mean. Paste or type your data, and this calculator returns the mean, the absolute deviation of every value, and the MAD itself, with each step shown so you can follow the working. It is a simple, intuitive measure of variability that is easy to teach and easy to check by hand.
What is the Mean Absolute Deviation Calculator?
Mean absolute deviation answers a plain question: on average, how far is a typical value from the center of the data? You first find the mean of all the values, then measure the distance of each value from that mean, ignoring whether it is above or below (that is the absolute part), and finally average those distances. The result is reported in the same units as your original data, so a MAD of 3 on a set of test scores literally means values sit about 3 points from the mean on average.
The absolute value step is what keeps the measure meaningful. If you simply averaged the raw deviations (some positive, some negative), they would always cancel out to exactly zero, because the mean is the balance point of the data. Taking the absolute value of each deviation strips away the sign so the distances add up instead of cancelling. This is the same problem that variance and standard deviation solve, except they square the deviations rather than taking absolute values.
That difference is the main reason to know both. Standard deviation squares each deviation, which gives extra weight to values far from the mean, so a single outlier can pull it up sharply. MAD treats every distance in proportion, so it is more resistant to outliers and arguably easier to interpret. Standard deviation is more common in formal statistics because it has convenient mathematical properties (it links neatly to the normal distribution and to least-squares methods), but MAD is often a clearer first description of spread, especially in the classroom.
When to use it
- Teaching or learning variability in a statistics class, where MAD is introduced before variance and standard deviation.
- Describing the spread of a small data set in plain units without the squaring used by standard deviation.
- Comparing the consistency of two data sets, for example two machines or two students, where a smaller MAD means more consistent values.
- Measuring forecast accuracy, where the mean absolute deviation of forecast errors is a standard error metric.
How to use the Mean Absolute Deviation Calculator
- Enter your numbers in the box, separated by commas, spaces, or new lines.
- Read the mean, which the calculator finds first as the center of your data.
- Check the step table to see the deviation and absolute deviation for each value.
- Read off the MAD, the average of those absolute deviations.
Formula & method
Worked examples
Find the MAD of the data set 2, 4, 6, 8, 10.
- Mean = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6
- Absolute deviations: |2-6|=4, |4-6|=2, |6-6|=0, |8-6|=2, |10-6|=4
- Sum of absolute deviations = 4 + 2 + 0 + 2 + 4 = 12
- MAD = 12 / 5 = 2.4
Result: Mean = 6, MAD = 2.4
Find the MAD of the data set 3, 6, 6, 7, 8, 11, 15, 16.
- Mean = (3 + 6 + 6 + 7 + 8 + 11 + 15 + 16) / 8 = 72 / 8 = 9
- Absolute deviations: 6, 3, 3, 2, 1, 2, 6, 7
- Sum of absolute deviations = 6 + 3 + 3 + 2 + 1 + 2 + 6 + 7 = 30
- MAD = 30 / 8 = 3.75
Result: Mean = 9, MAD = 3.75
Worked breakdown for the data set 2, 4, 6, 8, 10 (mean = 6)
| Value (x) | x minus mean | |x minus mean| |
|---|---|---|
| 2 | -4 | 4 |
| 4 | -2 | 2 |
| 6 | 0 | 0 |
| 8 | 2 | 2 |
| 10 | 4 | 4 |
| Sum | 0 | 12 |
MAD compared with related measures of spread
| Measure | How it is found | Sensitive to outliers? |
|---|---|---|
| Range | Maximum minus minimum | Very (uses only extremes) |
| MAD | Average of |x - mean| | Moderately |
| Variance | Average of (x - mean) squared | Strongly |
| Standard deviation | Square root of the variance | Strongly |
Common mistakes to avoid
- Forgetting the absolute value. If you average the raw deviations (x minus mean) without taking absolute values, the positives and negatives always cancel to zero, because the mean is the balance point. You must use the absolute value of each deviation.
- Dividing by n minus 1 instead of n. Sample standard deviation divides by n minus 1, but the ordinary mean absolute deviation divides by the full count n. Using n minus 1 here gives a different, non-standard figure.
- Confusing MAD with standard deviation. They both measure spread but are not equal. Standard deviation squares the deviations and then takes a square root, so for most data sets it is larger than the MAD. Reporting one as the other is a common error.
- Measuring deviations from the median by mistake. The mean absolute deviation uses distances from the mean. There is a separate measure, the median absolute deviation, that uses the median. Make sure you are deviating from the center you actually intend.
Glossary
- Mean
- The arithmetic average: the sum of all values divided by how many values there are.
- Deviation
- The signed distance of a value from the mean, found as x minus the mean.
- Absolute deviation
- The size of a deviation ignoring its sign, written |x - mean|, so it is always zero or positive.
- Mean absolute deviation (MAD)
- The average of all the absolute deviations, a measure of how spread out the data is around the mean.
- Standard deviation
- A related spread measure that squares the deviations, averages them, then takes the square root, giving more weight to large deviations.
Frequently asked questions
What is the mean absolute deviation?
The mean absolute deviation (MAD) is the average distance between each value in a data set and the mean of that data set. You find the mean, measure how far each value is from it ignoring sign, then average those distances. It tells you how spread out the data is in the original units.
How do you calculate MAD step by step?
First add up all the values and divide by the count to get the mean. Next, for every value, subtract the mean and take the absolute value to get its absolute deviation. Add all the absolute deviations together, then divide that sum by the number of values. The result is the MAD.
What is the difference between MAD and standard deviation?
Both measure spread around the mean. MAD averages the absolute deviations, while standard deviation averages the squared deviations and then takes a square root. Squaring makes standard deviation react more strongly to outliers, so it is usually a bit larger than MAD, and MAD is generally easier to interpret.
Can the mean absolute deviation be zero?
Yes. The MAD is zero only when every value in the data set is identical, because then each value equals the mean and every deviation is zero. Any spread at all in the values gives a MAD greater than zero.
Do you divide by n or n minus 1 for MAD?
For the ordinary mean absolute deviation you divide the sum of absolute deviations by n, the full count of values. The n minus 1 adjustment belongs to the sample standard deviation, not to MAD, so this calculator divides by n.
What units is the MAD in?
The MAD is expressed in the same units as your original data. If your values are in dollars, the MAD is in dollars; if they are test scores, the MAD is in points. This is one reason it is considered easy to interpret.