In the most simple mathematical sense, say I take a number of measurements and average them, the noise reduces by the square root of the repetitions. This is elementary. Now I take the maximum of the repetitions. By doing the exercise for a measurement setup I am using I found out that the uncertainty goes down a lot faster than the square root of the repetitions.
Now, I can measure the uncertainty of the input parameter, and it would be useful if I can estimate the uncertainty of the output parameter without having to estimate it from my dataset. So, I am searching for a way I can transfer an uncertainty statistically through the maximum function. In general one way to transfer uncertainties is by taking the derivative, but what is the derivative of the maximum of some points?
Specifically, the reason why I want to transfer the uncertainty is that I want to test values for being out of bounds. This is also the reason why approximating it by averaging is not desirable.