What is mean absolute deviation?
Mean absolute deviation (MAD) measures how far values in a dataset typically sit from the arithmetic mean. For each observation, you find the distance to the mean, convert that distance to an absolute value so negative and positive deviations cannot cancel each other, then average those absolute distances.
The result uses the same units as the original data. A MAD of 2.4 centimeters means values are, on average, 2.4 centimeters away from the mean—not 2.4 “percentage points” or a unitless score. Whether a MAD looks small or large depends on the scale and context of the dataset; the same number can be meaningful for one measurement and unimportant for another.
This page discusses mean absolute deviation about the arithmetic mean. It is not median absolute deviation (which centers on the median) and not mean absolute percentage error (which scales deviations by the observed values).
Mean absolute deviation formula
CalcLume uses the following definition:
x̄ = Σxᵢ / n
MAD = Σ|xᵢ − x̄| / n
- xᵢ — each observation in the dataset
- n — the number of observations
- x̄ — the arithmetic mean of the observations
- |xᵢ − x̄| — the absolute deviation of each value from the mean
The denominator is n, the count of observations. This calculator does not offer an n − 1 sample option. Intermediate arithmetic keeps full floating-point precision; only the displayed values follow the decimal-places setting you choose.
How to calculate MAD
- Enter the numeric values that form your dataset.
- Compute the arithmetic mean: add the values and divide by n.
- For each value, subtract the mean and take the absolute value to get |xᵢ − x̄|.
- Add the absolute deviations.
- Divide that sum by n. The quotient is the mean absolute deviation.
On this page, press Calculate MAD after entering or loading a dataset. The calculator shows the result summary, formula working, a deviation table, and a neutral interpretation of the average distance from the mean.
Complete worked example
Dataset: 12, 15, 14, 10, 19
Count: n = 5
Mean: x̄ = (12 + 15 + 14 + 10 + 19) / 5 = 70 / 5 = 14
Absolute deviations from 14:
- |12 − 14| = 2
- |15 − 14| = 1
- |14 − 14| = 0
- |10 − 14| = 4
- |19 − 14| = 5
Sum of absolute deviations = 2 + 1 + 0 + 4 + 5 = 12
MAD = 12 / 5 = 2.4
Minimum = 10, maximum = 19, and range = 19 − 10 = 9. Range describes the span of the raw values; MAD describes average distance from the center.
Interpretation: values in this dataset are, on average, 2.4 units away from the arithmetic mean of 14.
Mean absolute deviation versus standard deviation
Both MAD and standard deviation describe spread around a center, but they treat deviations differently.
- MAD averages absolute distances from the mean. Extreme values influence the result linearly.
- Standard deviation averages squared distances (then takes a square root). Extreme values influence the result more strongly because squaring amplifies large deviations.
MAD stays in the original data units and is often easier to explain as “average distance from the mean.” Standard deviation is the more common choice in inference, probability models, and many software defaults. Neither measure is automatically “better”; the right choice depends on the question and the audience.
Naming note: this CalcLume calculator computes mean absolute deviation about the arithmetic mean. NIST material discusses that idea as average absolute deviation (a scale measure). In some statistical references—including NIST contexts—“MAD” means median absolute deviation. Those are different statistics; do not treat them as interchangeable.
When MAD is useful
- Teaching dispersion with a direct “average distance from the mean” interpretation
- Summarizing spread when absolute (rather than squared) deviations match the decision context
- Comparing datasets that share similar units and scale, when a simple average distance is enough
- Checking how far typical values sit from a reported mean without introducing squared units
Limitations and interpretation
- MAD depends on the arithmetic mean. Outliers still affect the center and therefore the deviations, even though they are not squared.
- MAD is not median absolute deviation and is not a percentage error measure.
- Absolute MAD values are not given qualitative dispersion ratings on this site without context. Compare MAD to the dataset’s scale, units, and purpose.
- This calculator always divides by n. There is no sample n − 1 mode.
- Display precision affects rounding in the interface only; it does not change the underlying calculation.
For how CalcLume selects formulas and reviews calculator pages, see the methodology and sources pages.