Mean Absolute Deviation Calculator

Background.

How spread out are these numbers? The most literal answer statistics offers is the mean absolute deviation: find the mean, measure each value's distance from it — ignoring direction — and average those distances. A MAD of 2 means the data sit, on average, 2 units from their centre. No squaring, no square roots, no reinterpretation: it is spread in the same units as the data, computed the way a person would naturally do it by hand.

That literalness is why MAD anchors the school statistics curriculum as the first dispersion measure taught, and why it remains useful long after: forecasters report demand-prediction error as MAD because ‘we miss by 30 units on average’ needs no translation, and quality engineers quote average absolute error for the same reason.

Its famous sibling, the standard deviation, squares the deviations before averaging and then square-roots the result. Squaring weights large deviations disproportionately — one value 10 away contributes as much as twenty-five values 2 away — so standard deviation reacts more sharply to outliers, while MAD treats a distance of 10 as exactly five times a distance of 2. Neither is ‘right’: standard deviation's algebra makes it the native currency of variance-based theory (normal distributions, regression, ANOVA), while MAD is the more honest descriptive summary when you simply want typical distance from centre. For any dataset, MAD ≤ standard deviation, with equality only in degenerate cases.

This page computes the population-form MAD about the arithmetic mean, along with the mean and count it used. One naming caution, flagged in the scope note: robust statistics uses ‘MAD’ for a different quantity — the median absolute deviation about the median — so check which MAD a source means before comparing numbers.

What is mean absolute deviation calculator?

The mean absolute deviation (MAD) of a dataset is the average of the absolute differences between each value and the arithmetic mean: MAD = Σ|xᵢ − x̄| / n. It answers ‘how far from the mean is a typical value?’ directly, in the data's own units. Compared with standard deviation — which averages squared deviations and back-transforms with a square root — MAD weights all deviations linearly, making it less sensitive to extreme values and more literally interpretable, at the cost of the algebraic conveniences that make variance central to statistical theory. It is always at most the standard deviation (about 0.8 of it for normally distributed data).

How to use this calculator.

1. 01Enter the values as a comma-separated list — order is irrelevant, duplicates count each time they appear.
2. 02Read the three outputs together: the arithmetic mean (the centre the deviations are measured from), the count, and the MAD itself.
3. 03Interpret MAD in the data's units: test scores with mean 75 and MAD 6 typically land within about 6 points of 75.
4. 04Comparing spread across datasets is only meaningful on comparable scales — for different scales, divide each MAD by its mean for a unitless relative spread.
5. 05If your source or curriculum says ‘MAD’ meaning the robust median absolute deviation (median of |xᵢ − median|), this is not that quantity — the two differ substantially on skewed or outlier-laden data.

The formula.

MAD = Σ|xᵢ − x̄|/n formalises ‘average distance from the centre’. The absolute value is load-bearing: raw deviations from the mean always sum to zero by the mean's definition (the positives and negatives cancel exactly), so without it every dataset would score 0. Taking magnitudes keeps each value's contribution proportional to its distance — the linearity that distinguishes MAD from variance, where squaring makes one 10-unit deviation outweigh twenty-five 2-unit ones. Two structural facts are worth knowing. First, deviations about the mean minimise the sum of squares, but the sum of absolute deviations is minimised by the median — so mean-anchored MAD is not the smallest possible average distance when data are skewed, a hint toward the robust median-based variant. Second, for normal data MAD ≈ 0.7979·σ, so the two spread measures translate when normality is defensible. The engine computes the mean, the n absolute differences, and their average in Decimal arithmetic, rounding once at the output.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST/SEMATECH e-Handbook, Measures of Scale.
2. [2]OpenStax, Introductory Statistics 2e.
3. [3]NIST/SEMATECH e-Handbook of Statistical Methods.