When the uncertainties are Gaussian and their variances $\sigma_{yi}^2$

setting all $\sigma_{yi}^2=S$, that is, ignoring the uncertainties and replacing them all with the same value $S$

\begin{problem}\label{prob:chi2} 
 Re-do the fit of \problemname~\ref{prob:easy} but setting all 
 $\sigma_{yi}=S$, that is, ignoring the uncertainties and replacing 
 them all with the same value $\sqrt(S)$.  What uncertainty variance $S$ would 
 make $\chi^2 = N-2$?  Relevant plots are shown in 
 \figurename~\ref{fig:chi2}.  How does it compare to the mean and 
 median variance of the uncertainty $\allsigmay$? 
 \end{problem}