Mean Absolute Deviation (MAD) Calculator

Understanding the mean absolute deviation — concept and principle

The mean absolute deviation (MAD) is the average distance of the data values from their mean. It answers a simple question: on average, how far is a typical value from the centre? Because it uses plain distances rather than squares, it is easy to explain and is widely taught as a first measure of spread. The calculator also gives the median absolute deviation, a robust alternative that is hardly affected by outliers.

How it is calculated

• Mean absolute deviation: MAD = Σ|xᵢ − x̄| ÷ n, where x̄ is the arithmetic mean and n the number of values.
• Median absolute deviation: find the median m, take every distance |xᵢ − m|, and report the median of those distances.

Both results are in the same unit as the data (points, cm, minutes), which makes them easy to interpret.

The principle

Deviations from the mean always add up to zero, because values above and below the mean cancel. To measure spread you must remove the sign. The variance and standard deviation do this by squaring; the MAD takes absolute values. Squaring gives large deviations extra weight, so the standard deviation is always at least as large as the MAD and reacts more strongly to outliers. The median absolute deviation goes further: it is based on medians at both steps, so up to half the data can be extreme before it breaks down.

Worked example

1. Data: 12, 15, 11, 19, 22, 15, 18, 25, 14, 16; n = 10, sum 167, mean x̄ = 16.7.
2. Distances from 16.7: 4.7, 1.7, 5.7, 2.3, 5.3, 1.7, 1.3, 8.3, 2.7, 0.7; total 34.4.
3. MAD = 34.4 ÷ 10 = 3.44.
4. Sorted data: 11, 12, 14, 15, 15, 16, 18, 19, 22, 25, so the median is (15 + 16) ÷ 2 = 15.5.
5. Distances from 15.5, sorted: 0.5, 0.5, 0.5, 1.5, 2.5, 3.5, 3.5, 4.5, 6.5, 9.5.
6. Median absolute deviation = (2.5 + 3.5) ÷ 2 = 3.

How to read the result

A MAD of 3.44 means values lie on average about 3.4 units from the mean of 16.7. For comparison, the population standard deviation of the same data is about 4.15, larger as expected. For normally distributed data the MAD is about 0.80 times the standard deviation, and the median absolute deviation is about 0.674 times it. Multiplying the median absolute deviation by 1.4826 therefore gives a robust estimate of the standard deviation (here about 4.45).

Common mistakes

• Forgetting the absolute value, which always gives zero.
• Mixing up the two quantities both abbreviated MAD; always state which one you mean.
• Comparing MAD from one data set with a standard deviation from another.

Frequently asked questions

Should I divide by n or n − 1?

The MAD is normally defined with n, as here; the n − 1 correction belongs to the sample variance.

What does MAD mean in forecasting?

In forecasting, MAD or MAE usually means the mean absolute forecast error, the same formula applied to errors.

Sources: Hampel, F. R. (1974). Journal of the American Statistical Association, 69(346), 383–393.