Last updated: 2026-07-01

The mean absolute deviation

The mean absolute deviation (MAD) of a data set is the average distance of each value from the mean. For values x₁, x₂, …, xₙ with mean x̄, the MAD is:

MAD = (1/n) Σ |xᵢ − x̄|

Each term |xᵢ − x̄| is the absolute deviation of a single value from the mean — how far it sits from the center, ignoring direction. Averaging those distances gives a single number, in the same units as the data, that answers a direct question: on average, how far is a typical value from the middle?

Because the deviations are taken in absolute value, positive and negative differences do not cancel. A MAD of zero means every value equals the mean; a larger MAD means the data are more spread out.

How to calculate it

The calculation follows three steps:

1. Find the mean. Add up all the values and divide by the count.
2. Find each absolute deviation. Subtract the mean from each value and drop the sign.
3. Average the deviations. Add the absolute deviations and divide by the count.

That last average is the mean absolute deviation.

Worked example

Take the data set 4, 8, 6, 5, 3, 7.

• Count: n = 6
• Mean: (4 + 8 + 6 + 5 + 3 + 7) / 6 = 33 / 6 = 5.5

Next, the absolute deviation of each value from 5.5:

• |4 − 5.5| = 1.5
• |8 − 5.5| = 2.5
• |6 − 5.5| = 0.5
• |5 − 5.5| = 0.5
• |3 − 5.5| = 2.5
• |7 − 5.5| = 1.5

The sum of the absolute deviations is 1.5 + 2.5 + 0.5 + 0.5 + 2.5 + 1.5 = 9. Dividing by the count gives:

MAD = 9 / 6 = 1.5

So on average each value lies 1.5 units from the mean.

MAD versus standard deviation

The MAD and the standard deviation both measure spread, but they treat deviations differently. The MAD averages the absolute deviations, while the standard deviation averages the squared deviations and then takes a square root. Squaring emphasizes large deviations, so the standard deviation responds more sharply to outliers than the MAD does.

For a normal distribution the MAD is approximately 0.8 times the standard deviation. The MAD is easier to explain and more resistant to extreme values; the standard deviation has algebraic properties that make it the standard tool in inferential statistics. When the goal is a plain, interpretable description of scatter, the MAD is often the clearer choice.

MAD about the median

The mean absolute deviation is usually measured from the mean, but it can also be measured from the median:

MAD(median) = (1/n) Σ |xᵢ − x̃|

where x̃ is the median. This version is more robust to outliers, because the median itself is not pulled by extreme values. A useful fact is that the median minimizes the sum of absolute deviations, so the MAD about the median is always less than or equal to the MAD about the mean. When the data are symmetric, the mean and median coincide and the two versions give the same answer.

For the example above the median of 3, 4, 5, 6, 7, 8 is (5 + 6) / 2 = 5.5, which equals the mean, so the MAD about the median is also 1.5.

When to use the mean absolute deviation

The MAD is a good spread measure when interpretability matters. Stated in the original units, it reads directly as an average distance from the center, which makes it common in introductory statistics, quality control, and forecasting — the mean absolute error of a set of forecasts is precisely a MAD of the residuals.

When the result feeds into further statistical machinery such as regression, confidence intervals, or hypothesis testing, the standard deviation and variance are generally preferred for their mathematical convenience. For a straightforward summary of how scattered a data set is, the MAD is frequently the more transparent measure.

Related calculators

• Variance and Standard Deviation Calculator — spread based on squared deviations
• Mean, Median, Mode Calculator — measures of center for a data set
• Descriptive Statistics Calculator — full summary statistics for a dataset