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Advice is what we ask for when we already know the answer but wish we didn’t. ( Erica Jong ) Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages). Disclaimer : The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues.
The most direct way of computing sample variance or standard deviation can have severe numerical problems. Mathematically, sample variance can be computed as follows. The most obvious way to compute variance then would be to have two sums: one to accumulate the sum of the x's and another to accumulate the sums of the squares of the x's. If the x's are large and the differences between them small, direct evaluation of the equation above would require computing a small number as the difference of two large numbers, a red flag for numerical computing.