Mean Absolute Deviation: Meaning, Formula, and Examples

What mean absolute deviation tells you

Mean absolute deviation is a measure of spread: it summarizes how dispersed a set of numbers is around its arithmetic mean. A small MAD means the values tend to sit close to the mean; a large MAD means they tend to be farther away. It provides a single, interpretable number for variability, complementing the mean, which describes the data’s center.

For example, imagine two groups of daily temperatures with the same mean. If one group’s readings cluster tightly around that mean and the other’s swing widely, their MADs will differ even though their averages match. The MAD describes typical distance, not the direction of the difference: a value above the mean and one equally far below it both contribute the same positive distance.

The term can be used with different centers, such as the median, but in many introductory statistics settings, “mean absolute deviation” specifically means the average absolute distance from the arithmetic mean. Check how a textbook, calculator, or software package defines it before comparing results.

How to calculate mean absolute deviation

For data values x₁, x₂, …, xₙ and their arithmetic mean x̄, the formula is:

MAD = (|x₁ − x̄| + |x₂ − x̄| + … + |xₙ − x̄|) ÷ n

The vertical bars mean “absolute value”: take the size of a difference without its positive or negative sign. The denominator is n, the number of observations in the data set. In a basic calculation, each observation receives equal weight.

Find the mean. Add all values and divide by how many values there are.
Find each deviation. Subtract the mean from each value.
Make deviations absolute. Remove negative signs so every distance is nonnegative.
Average the distances. Add the absolute deviations and divide by the number of values.

Consider the data set 2, 4, 6, 8. Its mean is 5. The deviations from 5 are −3, −1, 1, and 3; their absolute values are 3, 1, 1, and 3. Their sum is 8, and dividing by 4 gives a MAD of 2. The values are, on average, two units from the mean.

Why use absolute values?

Ordinary signed deviations from the mean always add to zero. In the example, −3 and −1 cancel out 1 and 3. That cancellation makes the sum unsuitable as a measure of spread. Taking absolute values prevents values on opposite sides of the mean from erasing one another.

Another common measure, variance, squares deviations instead. Squaring also prevents cancellation, but it gives greater influence to large deviations and uses squared units. The standard deviation is the square root of the variance, so it returns to the original measurement units. MAD instead uses absolute distances directly, which can make its meaning easier to explain: it is an average distance, not an average squared distance.

Interpreting and comparing MAD values

A MAD of 2 does not mean every value is exactly two units from the mean. It means the average of all the absolute distances is two. Individual distances can be smaller or larger, and the MAD alone does not reveal the shape of the distribution, the direction of values, or whether unusual observations are present.

Because MAD is expressed in the same units as the data, it is usually meaningful to compare data sets measured on the same scale and in comparable contexts. A two-degree spread is not directly comparable with a two-mile spread. If all values are shifted by the same amount, the MAD stays the same; if every value is multiplied by a constant, the MAD is multiplied by that constant’s absolute value.

For a quick sense of scale, consider MAD alongside the mean and the actual data. A value such as 2 may be substantial for measurements that vary by only a few tenths, but small for measurements that span hundreds of units. The mean and MAD do not tell the whole story, so a range, plot, or description of the distribution may be useful as well.

Mean absolute deviation and outliers

Since the mean itself is affected by extreme values, a very large or very small observation can move both the center and the MAD. Absolute deviations do not square the distances, so MAD is less sensitive to extreme values than variance or standard deviation in an important sense, but it is not immune to them.

There is also a related measure called the median absolute deviation, often abbreviated median MAD: take absolute distances from the median and find their median. It is a different statistic, generally more resistant to outliers. The two measures should not be treated as interchangeable. If a source says only “MAD,” look for its definition, especially in robust statistics or data-analysis software.

Common mistakes and useful checks

Keeping the signs: Negative and positive deviations would cancel. Use absolute values before averaging.
Using the wrong center: The standard mean-based MAD uses the arithmetic mean; a median-centered version is different.
Dividing by the wrong number: For the usual descriptive MAD, divide by the number of observations, not by n − 1. The latter appears in some sample variance calculations, not this basic formula.
Confusing MAD with standard deviation: Both describe spread, but their formulas and numerical values differ.
Forgetting units: MAD has the original data’s units, whereas variance has squared units.

A useful check is that MAD cannot be negative. It is zero only when every value equals the mean, which for an ordinary finite data set means all values are identical. If a calculation gives a negative result, a sign or formula error has occurred.

Frequently Asked Questions

What is the mean absolute deviation formula?

Add the absolute differences between every value and the arithmetic mean, then divide by the number of values: MAD = Σ|xᵢ − x̄| ÷ n.

Can mean absolute deviation be zero?

Yes. It is zero when all observations are identical, because every observation is then at the mean and each distance is zero.

Is mean absolute deviation the same as standard deviation?

No. MAD averages absolute distances, while standard deviation is based on squared deviations and then takes a square root. They both describe spread but are calculated differently.

Is MAD resistant to outliers?

Mean-based MAD can be affected by outliers because both the mean and the distances can shift. Median absolute deviation is a separate, more outlier-resistant measure.

Why is mean absolute deviation useful?

It gives a straightforward estimate of the average distance from the mean in the data’s original units. This can make variability easier to describe than variance, which uses squared units.

Further reading: American Statistical Association. Details can change, so check the official page for anything that matters to you. Spotted a mistake? Tell us. How we write an answer.