Mean absolute deviation (MAD)

The mean absolute deviation answers one plain question: on average, how far does a value sit from the mean? Drop the sign on every deviation and average what is left, giving $\text{MAD} = \frac{1}{n}\sum |x_i - \bar{x}|$, where $\bar{x}$ (x-bar) is the mean and $n$ is the count. The divisor is $n$, not $n - 1$. There is no bias correction here, which is a real difference from the sample standard deviation.

For 2, 4, 6, 8 the mean is 5, the distances are 3, 1, 1, 3, and the MAD is $8/4 = 2$. Read that as: a typical value sits about 2 units from the mean. The sample standard deviation of the same four numbers is 2.58, larger, because squaring hands the two distances of 3 more say than the two distances of 1.

"MAD skips the squaring, so it is the outlier-resistant version of the standard deviation." It is not resistant at all. Change 1, 2, 3 into 1, 2, 1000 and the MAD goes from 0.67 to 443.78 while $s$ goes from 1 to 576.48. Both chase the outlier, and which one chases harder depends on how you ask. Here $s$ gains more in absolute terms, 575.48 against 443.11, while MAD gains more as a multiple, 665.7 times against 576.5 times. There is no general rule that MAD is the less inflated of the two. The resistant summaries of spread are the median and the IQR.

One inequality does hold for every data set: $\text{MAD} \le \sigma \le s$, where $\sigma$ (sigma) is the population standard deviation. MAD ties $\sigma$ only when every value sits the same distance from the mean, as with 10 and 20, where both come to 5. Otherwise MAD is strictly smaller.

Watch the abbreviation. Some software writes MAD for the median absolute deviation, the median of the distances from the median, which is a different number and a genuinely resistant one. In an introductory course MAD means the mean version about the mean.