Uncertainty for maximum (peak)

Question

I have to find $theta$s for which we get maximum values for this plot:

Let's take the first peak. I've found $theta$ to be $26.1^o$. Now I have to determine its uncetrainty. The thing is that my professor told us to "think of something" for finding uncertainties. He's very fussy about anything in reports so I want to use a method that leaves nothing to complain about.
In my previous post I asked about normal distrubution for finding maxima - he didn't like it so I'm searching for something different.
So how do I find the uncertainties of $theta$ for which we get these peaks?
EDIT: This is the first peak zoomed in.

Semoi · Accepted Answer

These resonances are described by a Lorentzian distribution. So you can just fit this distribution. However, note that the result won't differ significantly from the result obtained by fitting a normal distribution -- which is available in most programs.
Your key problem is that you have only few datapoints per peak. Therefore, if you do not have a higher resolution in the dataset, and if you do not have a model, which describes the hole dataset simultaneously, you will obtain approx. the same result. A possible work-around would be to measure the same data several times, and obtaining $N=10$ "similar" dataset. Now, you could calculate the average peak location and its standard deviations (use the scaling factor $1/sqrt{N-1}$, due to the central limit theorem).
An different method is called leave-one-out-cross-validation. It is often used for model selection. The key idea is similar to what I describe before you edited your question:

Choose a region around the peak, which contains $n$ datapoints,
Leave the $i^{th}$ datapoint out and fit the model,
Repeat step 2, and leave each datapoint out once.
Calculate the standard deviation.

Uncertainty for maximum (peak)

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