Is there a theoretically-sound way to perform propagation of errors with robust statistics? I am trying to characterize the errors inherent in a measurement and propagate the uncertainty through calculations involving that estimate.

In the past, I have assumed that my errors are normally distributed and then used uncertainty propagation. However, this is clearly a bad assumption (my data is symmetric but guaranteed to have significant outliers), and I am intrigued by using a measure like Median Absolute Deviation.

Is it valid to treat MAD like a standard deviation? Is there a different measure I should be using for this application?