I know how to make a minimum-chi square fit when I have two variables both affected by statistical error. But how do I perform the fit when one of them has no statistical error, just the uncertainty of the instrument (for example say I am using a ruler which carries a constant uncertainty of 1mm) and according to my model this uncertainty cannot be neglected?
I use Root by Cern to make fits. I assume it follows this procedure to minimize $\chi^2$ of the fit when both the $x$ and $y$ variables are affected by statistical error (picture taken from L. Lyons, Statistics for nuclear and particle physicists, page 139)
This hinges on the condition that both values of $x$ and $y$ are gaussian distributed, so that you can evaluate the $\chi^2$ value corresponding to a given choice of the fit parameters. But what happens when you have no info on the statistics of $x$ and you just know that each measurement is sistematically affected by some fixed uncertainty of the measuring instrument?