# Why is the standard deviation preferred over the mean deviation?

To me, the mean deviation, which is the average distance that a data point in a sample lies from the sample's mean, seems a more natural measure of dispersion than the standard deviation; Yet the standard deviation seems to dominate in the field of statistics. I rarely see the mean deviation reported in studies; generally only the sample mean or median and the standard deviation are provided.

Question: Why is the standard deviation preferred over the mean deviation as a measure of dispersion?

Standard deviation is the square root of variance. Variance is exceptionally well-behaved algebraically; by linearity of expectation we have

\begin{align} \operatorname{Var} X &:= \mathbb{E}[(X - \mathbb{E}X)^2] \\ &= \mathbb{E}[X^2 - 2 X (\mathbb{E}X) + (\mathbb{E}X)^2] \\ &= \mathbb{E}X^2 - 2(\mathbb{E}X)^2 + (\mathbb{E}X)^2 \\ &= \mathbb{E}X^2 - (\mathbb{E}X)^2 \end{align}

It follows, for instance, that if we have a random variable which is a linear combination of other random variables that we can express its variance in terms of the variances and covariances of its constituent pieces:

\begin{align} \operatorname{Var} \left[\sum_i c_i Y_i\right] &= \mathbb{E}\left[\left(\sum_i c_i Y_i\right)^2\right] - \left(\mathbb{E}\left[\sum_i c_i Y_i\right] \right)^2 \\ &= \sum_{i, j} c_i c_j \mathbb{E}\left[Y_i Y_j\right] - \left(\sum_i c_i \mathbb{E} Y_i\right)^2 \\ &= \sum_{i, j} c_i c_j \mathbb{E}\left[Y_i Y_j\right] - \sum_{i, j} c_i c_j (\mathbb{E}Y_i)(\mathbb{E}Y_j) \\ &= \sum_{i, j} c_i c_j \left(\mathbb{E}\left[Y_i Y_j\right] - (\mathbb{E}Y_i)(\mathbb{E}Y_j)\right) \\ &= \sum_i c_i^2 \operatorname{Var} Y_i - \sum_{i \neq j} c_i c_j \operatorname{Cov}[Y_i, Y_j] \\ &= \sum_i c_i^2 \operatorname{Var} Y_i - 2 \sum_{i < j} c_i c_j \operatorname{Cov}[Y_i, Y_j] \end{align}

If we work with mean absolute deviation, on the other hand, the best we can typically get in situations like this is some kind of inequality.

So we like using variance because it lets us perform a long sequence of calculations and get an exact answer. Being able to string together long sequences of simple operations without losing something at each step is often a very big deal.

There are some studies suggesting that, unsurprisingly, the mean absolute deviation is a better number to present to people. But typically you'd still want to use variance in your calculations, then use your knowledge about the distribution to calculate or estimate the mean absolute deviation from the variance.