Mean absolute deviation, MAD, is the most direct measure of spread in statistics: how far, on average, are the values from their own mean? Compute the mean, find each value distance from it, ignore the signs, and average those distances. The result is in the original units, dollars, points, or degrees, which makes it immediately readable in a way that variance, in squared units, is not. This guide works through the formula, three worked examples, the comparison with standard deviation, and the spreadsheet function that computes it in one step.

The formula, step by step

Mean absolute deviation is written as the sum of the absolute values of (x minus mean), divided by n, the number of values. Absolute value means distance without direction: a value 4 below the mean and a value 4 above it both contribute 4. Without the absolute value, the deviations would cancel to zero, because the mean is defined as the balancing point of the data.

The steps are: one, compute the mean by adding all values and dividing by n. Two, subtract the mean from each value. Three, take the absolute value of each difference. Four, add those distances and divide by n. The calculator shows every step, so the arithmetic can be checked by hand on small sets, which is the right way to learn it.

Population versus sample is a detail that matters less for MAD than for variance. The calculator divides by n, the population form, which is standard for describing a complete set. Some texts divide by n minus 1 for a sample estimate, but for a classroom data set or a complete list, divide by n.

Worked examples: three data sets

Example one: 3, 5, 7, 9, 11. The mean is 35 divided by 5, or 7. The deviations are negative 4, negative 2, 0, 2, and 4. The absolute distances are 4, 2, 0, 2, and 4, summing to 12. Divide by 5 and the MAD is 2.4. On average, each value sits 2.4 units from the mean of 7.

Example two: 7, 7, 7, 7, 7. The mean is 7, every deviation is 0, and the MAD is 0. Zero spread means zero average distance, which is the only way MAD reaches zero: every value equal. Example three: 1, 2, 9, 10. The mean is 5.5, the distances are 4.5, 3.5, 3.5, and 4.5, summing to 16, and the MAD is 4. The two clusters sit far from the centre, and the MAD reports that honestly.

Example four, with an outlier: 5, 6, 7, 8, 54. The mean is 16, pulled up by the outlier, and the distances are 11, 10, 9, 8, and 38, summing to 76, giving a MAD of 15.2. The outlier inflates MAD, but only linearly: it contributes its distance, not its squared distance, which is the key difference from standard deviation below.

MAD versus standard deviation, and when to reach for each

Standard deviation squares each deviation before averaging, then takes the square root. Squaring punishes large deviations disproportionately: a deviation of 4 contributes 16, while a deviation of 2 contributes 4, four times as much for twice the distance. MAD treats a deviation of 4 as exactly twice a deviation of 2. When large deviations should count more, for example in quality control where a big miss matters, standard deviation is the right tool.

When every unit of error should count equally, MAD is the right tool, and it is the same formula as mean absolute error in forecasting: the average size of the forecast miss, in original units. A forecasting model with a mean absolute error of 2.4 units is wrong by 2.4 units on average, a sentence anyone can understand, while a root-mean-square error of 2.8 units on the same data needs translation.

MAD is also more robust. A single wild value moves MAD by its distance divided by n, while it moves variance by its squared distance divided by n. For messy real data with occasional outliers, MAD gives a steadier picture of typical spread, and the median absolute deviation, the median of the distances rather than their mean, is steadier still.

Three data sets, side by side

The table shows why the same mean can hide different spreads, and why MAD and standard deviation rank sets differently when outliers appear.

MAD in a spreadsheet

In Excel, MAD is =AVEDEV(range), which returns the average of the absolute deviations from the mean, dividing by n. In Google Sheets the same function exists. For a manual check, compute the mean with =AVERAGE(range), the distances with =ABS(cell-mean), and average those.

The common spreadsheet error is forgetting the absolute value: averaging signed deviations always gives zero, because the mean balances the data. If a MAD computation returns zero on varied data, the absolute step was skipped. The calculator shows each deviation with its sign and its absolute value side by side, so the step is visible.

Five mistakes with MAD

First, confusing MAD with median absolute deviation: MAD averages distances from the mean, while median absolute deviation takes their median. Second, dividing by n minus 1 when the set is complete: the population form divides by n. Third, reporting MAD without the mean: a MAD of 2.4 means little without knowing the mean is 7 and the units are points.

Fourth, comparing MADs across different units: a MAD of 2.4 dollars and a MAD of 2.4 points are not comparable, because the units differ. Fifth, using MAD where squared errors matter: in quality control and in fitting models by least squares, the squared measure is the right loss, and MAD will underweight the big misses that matter most.

Frequently Asked Questions

What is mean absolute deviation?

The average distance of each value from the mean. Compute the mean, find each value absolute distance from it, and average those distances. For 3, 5, 7, 9, and 11, the MAD is 2.4.

How is MAD different from standard deviation?

MAD averages absolute distances, so every unit of deviation counts equally. Standard deviation squares deviations first, so large deviations count disproportionately. MAD is in original units; variance is in squared units.

What is the MAD of 3, 5, 7, 9, 11?

2.4. The mean is 7, the absolute deviations are 4, 2, 0, 2, and 4, summing to 12, and 12 divided by 5 is 2.4.

Can MAD be zero?

Only when every value is equal. If all values are 7, every deviation is 0 and the MAD is 0. Any variation gives a positive MAD.

How do I calculate MAD in Excel?

Use =AVEDEV(range), which returns the average of the absolute deviations from the mean. In Google Sheets the same function exists.

Is MAD the same as mean absolute error?

The formula is the same: average absolute difference. Mean absolute deviation measures distances from the mean of a data set, while mean absolute error measures forecast errors against actual values.