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Say I have 400 measurements of some variable. I take a sliding window of 50 events and take the standard deviation of each set of 50 events. That would be 350 measurements. Now I want to take the first and second 175 events, take the average of each, and subtract them.

Normally to propagate the error on this final measurement you would, for each side, find the error of each standard deviation std/sqrt(2*(50-1)) then take use sqrt(sum of the squares)/175 to find the error on the average std.dev. for each side. Then use sqrt(sum of the squares) of these two errors for the final error on the subtraction of the averages.

The problem with this is that each measurement of the std.dev shares 49 events with the previous so this method would overestimate the final error. What to do?