Nope. The alternative (mean average distribution) also has the same units. The problem with the MAD was that it used an absolute value and so introduced a discontinuity. If you are a mathematician working with a system of equations, each discontinuity doubles the size of the system when taking integrals/derivatives. The SD was a clever hack to hide the discontinuity. It makes the mathematician's job of juggling equations much easier by changing the workload from O(n^2) to O(n). If you are not doing calculus on the system then the SD has much less utility. Negative utility even, because the square/square-root hack used by the SD is identical to the MAD only for perfectly normal distributions. For real-world distributions it tends to give wrong results.