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I am trying to do some error analysis for an experiment involving laser diagnostics and I have a question about computing the standard error for a spatial average.

The experiment involves taking images of the laser with a camera. The laser is focused into a 'line' using a cylindrical lens, which effectively results in a 1D profile of the laser intensity on the camera sensor. Several hundred images/frames are taken and these are averaged to give a mean beam intensity profile and to eliminate noise.

So, essentially, there are M pixels along that 1D line, which each have their own measure of intensity, and there are N frames, which are averaged to give the mean intensity of each pixel.

The question I have is that we also want to take a spatial average of the intensity profile along a subset of those M pixels (let's say M'); however, I am unsure how to calculate that standard error for that. The calculations I have tried seem to be giving a vanishingly small error, which seems implausible.

So, essentially we have an average over the N frames for each pixel intensity:

$$\mu_{Nj} = \frac{1}{N} \sum_{i=1}^{N} x_i$$

and then the spatial average is as follows:

$$\mu_{M'} = \frac{1}{M'} \sum_{j=1}^{M'} \mu_{Nj}$$

So, it is an 'average of averages' and I am not sure what the proper way is to find the standard error of that. For the $\mu_{Nj}$, I believe we can use the typical standard error formula:

$$S_{Nj} = \frac{\sigma_{Nj}}{\sqrt{N}}$$

but for the $\mu_{M'}$ I am not so sure. It seems that this must be quite a common situation and I suspect there is a probably standard way of doing it.

Please let me know if the question seems unclear and I will do my best to clarify.

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The most straightforward thing to do here is to simply repeat the entire experiment several times, extract the same derived value from your data in each run, and then apply the statistical formulas directly to the set of derived values. This has the benefit of capturing not only the error associated with a single run (say from detector noise) but also error from variations between runs (say from alignment drift).

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  • $\begingroup$ I think your suggestion is a good one and I agree it should give a way to estimate the error in the overall spatial-averaged mean, which we are looking for. However, in our case, it wouldn't be easy to do, since we had to travel to a lab in a different institution to do the experiment, who have the laser diagnostic equipment we need. So, I do really need some way to estimate the error in the overall spatial mean from the data set we have. $\endgroup$
    – Time4Tea
    Commented Sep 29, 2021 at 17:09
  • $\begingroup$ Perhaps a similar idea could be to split up the data we have into subsets and look at the variation of the mean between those subsets? Let's say split the data up into 10 groups of 20 frames each and look at that variance? $\endgroup$
    – Time4Tea
    Commented Sep 29, 2021 at 17:14
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    $\begingroup$ @Time4Tea yes, given the limitations, I think that would be a good idea. As long as you have no reason to believe that there would be systematic differences between your subsets, this would provide a decent estimate. The key here is to clearly report exactly how you calculated your error bar. Then, people will be able to properly interpret your data. $\endgroup$
    – Gilbert
    Commented Sep 29, 2021 at 17:35
  • $\begingroup$ Thanks again for your reply. No, I don't think there should be any systematic variations between the subsets, as they were all taken in very quick succession. $\endgroup$
    – Time4Tea
    Commented Sep 30, 2021 at 13:11

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