Mean absolute deviation
Mean absolute deviation is a measure of dispersion that summarizes how far observations typically lie from a central value by averaging absolute distances. Because absolute values are used, positive and negative deviations do not cancel, and the result is expressed in the same units as the data.
Mean absolute deviation about the mean equals the average distance from the arithmetic mean. Mean absolute deviation about the median uses the median as the center and is more resistant to outliers.
Definitions (sample and population)
For ungrouped sample data \(x_1, x_2, \dots, x_n\) with sample mean \(\bar{x}\), the mean absolute deviation about the mean is
\[ \operatorname{MAD}_{\bar{x}}=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-\bar{x}\right| \]
For a finite population of size \(N\) with mean \(\mu\), the analogous population quantity is
\[ \operatorname{MAD}_{\mu}=\frac{1}{N}\sum_{i=1}^{N}\left|x_i-\mu\right| \]
For a median-centered version (useful when extreme values are present), letting \(\tilde{x}\) denote the sample median gives
\[ \operatorname{MAD}_{\tilde{x}}=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-\tilde{x}\right| \]
Interpretation
Mean absolute deviation describes a typical absolute error from the chosen center. A value of \(\operatorname{MAD}=3.4\) indicates that observations lie about \(3.4\) units away from the center on average, measured by absolute distance rather than squared distance.
Worked example
Consider the data set 2, 4, 7, 10, 13. The sample mean is \[ \bar{x}=\frac{2+4+7+10+13}{5}=\frac{36}{5}=7.2 \]
| \(x_i\) | \(x_i-\bar{x}\) | \(\left|x_i-\bar{x}\right|\) |
|---|---|---|
| 2 | \(2-7.2=-5.2\) | 5.2 |
| 4 | \(4-7.2=-3.2\) | 3.2 |
| 7 | \(7-7.2=-0.2\) | 0.2 |
| 10 | \(10-7.2=2.8\) | 2.8 |
| 13 | \(13-7.2=5.8\) | 5.8 |
\[ \operatorname{MAD}_{\bar{x}}=\frac{1}{5}(5.2+3.2+0.2+2.8+5.8)=\frac{17.2}{5}=3.44 \]
Visualization of absolute deviations
Relationship to standard deviation
Mean absolute deviation uses linear distance, while standard deviation uses squared distance. Squaring emphasizes large deviations, making standard deviation more sensitive to extreme values. Mean absolute deviation typically changes less under a single outlier, although it still increases when an outlier is added.
Under a normal model \(X\sim N(\mu,\sigma^2)\), the expected absolute deviation from the mean satisfies \[ E\left[\lvert X-\mu\rvert\right]=\sigma\sqrt{\frac{2}{\pi}} \] so a rough conversion is \(\sigma\approx \operatorname{MAD}\sqrt{\pi/2}\) when normality is a reasonable approximation.
Common pitfalls
Mixing centers changes the meaning. Mean absolute deviation about the mean and mean absolute deviation about the median are different dispersion summaries. A second frequent error is using absolute deviations from the mean but dividing by \(n-1\); the canonical mean absolute deviation uses division by \(n\) because it is an average of distances rather than an unbiased variance estimator.