Reduced chi-squared value for noiseless spectra I am fitting synthetic spectral data. This is data in the form of a gaussian lineshape. However, although the fit looks appropriate, the value of the reduced chi-squared parameter: 
$$\chi^2_{\rm red} = \frac{1}{\nu}\sum\frac{(O-E)^2}{\sigma^2} \,, $$
is a lot larger than 1.0, the value for a desirable fit. Since I do not have any noise, the value of $\sigma$ in the above equation is unknown. How can I lower my reduced chi-squared value?
 A: I don't really understand how you have calculated a reduced chi-squared using the formula in your question if you don't have a value for $\sigma$?
Obviously, the value of the reduced chi-squared is smaller if you provide a better model that more closely fits your data.
Finally, if you do not know the value of $\sigma$ for your data, then you can still use a Pearson's chi-squared to do the fit (see https://en.m.wikipedia.org/wiki/Pearson%27s_chi-squared_test ), replacing $\sigma^2$ with $E$. You can then examine the residuals to see if they show any trend or asymmetry that might indicate a problem with the model, but without an independent estimate of $\sigma$ you cannot evaluate the quality of the fit.
If you know that the binned data obey Poissonian statistics, then $\sigma^2$ is the number of counts in the bin (so long as that is more than about 10 in all bins). I imagine though, that this is what your fitting software has assumed.
If on the other hand you are saying that you believe your model is correct, but that you need plausible chi-squared values to estimate parameter uncertainties, then you could apply a uniform $\sigma$ to each point such that the reduced chi-squared becomes unity.
A: Each observation should have some error (sigma). This is a physical fact. You would have to use the estimated sigma if you do not know. Most probably your estimated sigma are low. Also you should use the correct(parent) mean and sigma for the Gaussian in the numerator.
