In general, the mean absolute deviation and standard deviation are different quantities. For normally distributed data, their expected ratio is sqrt(2/pi).
The standard deviation is the best estimate of dispersion for normally distributed data, but if the data has heavier tails than the normal, the mean absolute deviation may be preferable since it does not rely on the existence of the second moment.
It may be interesting to compare the probability densities of the
Normal distribution: p(x) = c*exp(-0.5*x^2)
Laplace distribution: p(x) = c'*exp(-|x|)
The Normal distribution has an x^2 term in the exponential, vs. an |x| term for the Laplace. That is why the mean and standard deviation are used to estimate the parameters of the normal distribution, whereas the median and median absolute deviation are used to fit the Laplace distribution. The median and mean absolute deviation are more "robust" measures of location and scale.
A good place to ask statistics questions is
Cross Validated.
