Q:

Are the measurement outcomes of an observable gaussian distributed?

Suppose in an experiment we perform $n$ independent measurements to find the true value of an observable $X$. Let the outcomes of $n$ measurement are denoted by $x_1,x_2,...x_n$. If $n$ is sufficiently large, will these measured outcomes $\{x_i\}$ be Gaussian distributed? 
Please note that I am not asking whether the means of the measurements are Gaussian distributed. I know they are.

A:

It's certainly possible to identify an observable with a non-Gaussian distribution.  
An example from my professional life was a detector which collected Cherenkov photons from relativistic electrons. We needed the same detector to be sensitive to a macroscopic electron current (nanoamps) but also, with a gain change, to be able to trigger on single electrons. In the counting mode we determined that each electron sent about ten photons to the photomultiplier --- a number small enough that we had to account for the asymmetry in the Poisson distribution ($10\pm\sqrt{10}$) rather than using the Gaussian approximation.

