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One general comment. You say:
"We usually assume that the distribution of <x> is Gaussian and that the margin of error, on each side is equal to the standard deviation, s, obtained from measurements. "
Do you mean the distribution of the individual values of x is Gaussian, or do you mean that the distribution the actual averages is Gaussian (i.e. you measured <x> several times and the distributions of those values of <x> follow a Gaussian distribution)? The "margin of error" for the distribution of x is s, but the "margin of error" of <x> is generally taken as s/(N^0.5) wher N is the number of values averaged to find <x>.
I think you mean the distribution of x(i) itself, but this distinction needs to be clear throughout the discussion.