Your covariance matrices are very interesting! I’ll try to focus on your final question:
As a beginner with data analysis, I would also like to ask you:
Assuming the value doesn’t vary more than that, which one should I take and with what error?
So if I understand correctly, you want to measure your “mean” and quote an uncertainty, and are now unsure what to take (keep in mind what you do should depend on your goal: do you want to measure the
mean parameter precisely, or do you care about a good fit for the whole range of x). Here is how I’d think about the problem of measuring the mean, maybe it’s interesting for you.
The full fit with the 9 parameters shows that the power-law tail on the right side of the crystal ball is difficult to estimate because the parameters are strongly correlated (0.83 in the cov matrix). That’s why they have these large errors. Another problem with the right tail is that its parameters are strongly correlated with the background parameters (Correlation between c0 and n is 0.866).
In this situation I would try a fit with the right tail excluded. The transition to the power law is at
mean - sigma * alpha, which is rounded down 495. So I would fit only the region up to x = 495. Two less parameters to worry about, and systematic uncertainties from the shape you choose for the right tail are not relevant anymore. This is particularly good because alpha was correlated with the mean, so getting ignoring the right tail will hopefully reduce your uncertainty on the mean!
Besides noting that the right tail is problematic, I’m afraid I can’t help you more without knowing the physics behind the plot and your measurement goals. Nobody can tell you which results to take without knowing which components of the fit to trust, and that depends also on the physics. Some things to think about:
- The crystal ball fit told you the power law tail is on the right. Is that what you expect from the physics? In a standard crystal ball, the tail is on the left because it’s the power law from the final state radiation (assuming x is energy or mass or something like that). Would there be a reason for it being on the right? If not, one more reason to be careful with the right tail.
- Is there a motivation for the two sigmas? Are there actually two different resolution effects you can think about, or is it just a ad-hoc solution?
- How did you decide on the Chebychev polynomials for the background? Is it maybe possible to fit far-away sidebands of x to constrain the background?
I hope this rambling helped a bit and gave you some ideas! By the way, I’m still thinking these are luxury problems because your mean is already so precise. I would probably just take the difference between the largest and smallest mean values obtained, divide it by two and write it off as the systematic uncertainty