# Propagation of errors with median absolute deviation from the median?

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?

• @Glen_b has answered the question I asked, but I've realized that I didn't frame the question in a way that would get at what I'm really trying to figure out. I will try again in a separate question, unless somebody tells me that goes against site norms. – user670416 Mar 26 '13 at 2:10

## 1 Answer

Beware your acronyms; MAD has been used for more than one thing.

I assume you mean 'median absolute deviation from the median'; speak up if you meant one of the other MADs.

Variances (and hence s.d.'s) have some nice properties. e.g. sums of r.v.s have nice simple variance results.

MAD doesn't have a result like that.

Consider $X_1, X_2$ both take the value $-1$ with probability $1/3$ and $1$ with probabiity $2/3$, while $Y_1$ and $Y_2$ take the values of $-1$, $0$ and $1$ with probability $1/3$, and $Z_1$ and $Z_2$ take the values of $-1$ and $1$ with probability 0.1 each and $0$ with the remaining probability (0.8). All variables are independent.

The $X$ and $Z$ variables have a MAD of 0. The $Y$'s have a MAD of 1.

$X_1 + X_2$ has a MAD of 2. (0,0) -> 2

$Z_1 + Z_2$ has a MAD of 0. (0,0) -> 0

$Y_1 + Y_2$ has a MAD of 1. (1,1) -> 1

There's clearly no such neat relationship.

MAD's of linear rescalings will scale - MAD(2X+3) = 2 MAD(X), but most other manipulations don't behave in neat ways.

• Yes - I had been referring to "median absolute deviation from the median". – user670416 Mar 26 '13 at 0:03
• Great, no problem then. – Glen_b Mar 26 '13 at 0:04
• (sorry - accidentally posted incomplete comment) You've shown that it will not work for my case - is there another statistic that I should use that will allow me to produce error estimates on a linear combination of my variables? – user670416 Mar 26 '13 at 0:06
• 'Robust' implies 'nonlinear' ... so I wouldn't generally expect linear combinations to have robust-scale-measures that are neat functions of the scale-measures of the components. Quantiles don't work that way for example, so doing complicated things with functions of quantiles certainly wouldn't be expected to. – Glen_b Mar 26 '13 at 0:09
• OK, I think in my ignorance of statistics I'm asking the wrong question. Maybe a better phrasing would be: "Given that I need to provide error estimates of some form for a linear combination of these variables, and I know that they are not normally distributed, what is my best option?". I can give more information about the application if that would be helpful. The standard approach in my field's literature seems to be the head-in-sand, pretend that everything's gaussian and just report standard deviation. – user670416 Mar 26 '13 at 0:15