MAD vs Standard Deviation

The mean absolute deviation answers one plain question: on average, how far does a value sit from the mean? You take each deviation, drop its sign, and average the results, which gives $\text{MAD} = \frac{1}{n}\sum |x_i - \bar{x}|$, where $\bar{x}$ (x-bar) is the mean and $n$ is the number of values. For the data 2, 4, 6, 8 the mean is 5, the absolute deviations are 3, 1, 1, and 3, and the MAD is $8/4 = 2$. It reports spread in the original units the way the standard deviation does, but without squaring and then taking a square root.

The standard deviation summarizes spread as the typical gap between a value and the mean. The sample standard deviation is $s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}$, where $x_i$ are the values, $\bar{x}$ (x-bar) is the mean, and $n$ is the number of values. For example, a small $s$ means the data cluster tightly around the mean, while a large $s$ means they spread out widely. It is the square root of the variance, which returns the measure to the original units.