Mathematics of uncertainty

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I used this method for uncertainties back in umep physics and I think it made sense at the time, but is it really OK, or do we have to use other techniques? (they might be more rigorous in 2nd year? I don't know)

Say you have a set of data points , and say you have a function on these data points:



My method was to choose the sign of each that would maximize f. The opposite signs would then minimize f. Once and had been obtained then the midpoint of this range would be the mean, so we would have:



Is this reasoning valid?



Also, a while back on the interwebs I saw that if you had and , then the sum would be



Could I get a brief explanation as to why we must take the quadratic mean?

Thx