In my research, I analyse pixel data from a 2D microscope image. Each pixel represents a unique spatial position. In my image, each pixel $i$ has a corresponding brightness value $x_i$. I assume the image is representative of the entire potential imaging area.

I want to take the mean and provide an appropriate measure for the error of the mean (presumably the standard error + confidence interval) in order to provide the reader with a measure of how reliable value is if I repeated the measurement in a different imaging area. So for a $10\times10$ pixel image I would average over $100$ pixels to receive a sample mean and a standard error.

Experimentally, I am able to increase my pixel size arbitrarily. That means I can image the same area, but with a higher pixel count, e.g. $100\times100$. However, at some point my pixel size becomes smaller than the theoretical spatial limit that my microscope can resolve. This is what my field refers to as oversampling.

My question stems from the following observation. As I increase my pixel size, I increase my sample size because I treat each pixel as an individual data point. Therefore my standard error drops because it is defined as:

$SE = \frac{s}{\sqrt{n}}$

with the sample standard deviation $s$ and the sample size/pixel count $n$. But because I cannot resolve below a certain threshold I am actually not obtaining additional data. I may be effectively measuring the same point multiple times. So it seems like I am artificially reducing my standard error.

If I had to come up with an explanation it would be: My method for estimating the mean and its error is based on the assumption that each element within my sample is independent because they are spatially apart. As I increase my pixel size, I am reducing my spatial separation and therefore the elements within my sample become increasingly dependant. My method is not made to deal with dependent data.

So my question is, how do I deal with this redundancy of data within my sample? How can I include and acknowledge oversampling in my error estimate? Or is this the wrong approach?



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