# Uncertanty in Chi-squared measurements, spectra-like

Let's say I have some observational data. For example: in a certain wavelength interval I measure the flux, 10 000 points, and I have the uncertainty for each one of them. For example:

Wavelength Flux Sigma_flux
5000.00    0.988  0.00012
5001.00    0.986  0.00013
...


And I have a grid of parameters that, for four different inputs give me a simulated data in the same wavelengths. For example:

Wv-simul Flux-simul
5000.00  0.991
5001.00  0.989
...


For the 10000 points.

Now I can calculate the chi-squared and see which model better fits the data. First basic question, should I calculate using:

$$\chi ^{2}= \sum \frac{(O_i - E_i)^2}{dO_i^2}$$

or

$$\chi ^{2}= \sum \frac{(O_i - E_i)^2}{E_i}"$$

And after I calculate my chi-squared, let's say I fix all the parameters except one. In that way I can see how the chi squared varies for that parameter, and I have some kind of parabola where I can identify where the minimum is. I'm not really sure of the method to calculate the confidence interval. From what I read I could use a relation that says that

$$\sigma_j^2 = 2\left (\frac{\partial \chi^2}{\partial a_j^2}\right )^{-1}$$

Or I could add +1 to the minimum chi-squared and see at which values the intersection with the "parabola" would happen (+1 because I'm working with 1 parameter), or even if I should add all the value equivalent to all the degrees of freedom, that I'm guessing in includes the $$N$$ data points.

• It should approach a Gaussian as the number of degrees of freedom increase. Given the number of degrees of freedom required by the model to fit the data, then you should expect the degrees of freedom to one less - since you have already used 1 parameter, namely the mean. – Cinaed Simson Apr 11 at 23:36