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selylindi comments on Stupid Questions February 2015 - Less Wrong Discussion
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Background: Statistics. Something about the Welch–Satterthwaite equation is so counterintuitive that I must have a mental block, but the equation comes up often in my work, and it drives me batty. For example, the degrees of freedom decrease as the sample size increases beyond a certain point. All the online documentation I can find for it gives the same information as Wikipedia, in which k = 1/n. I looked up the original derivation and, in it, the k are scaling factors of a linear combination of random variables. So at some point in the literature after the original derivation, it was decided that k = 1/n was superior in some regard; I lack the commitment needed to search the literature to find out why.
The stupid questions:
1) Does anyone know why the statistics field settled on k = 1/n?
2) Can someone give a relatively concrete mental image or other intuitive suggestion as to why the W-S equation really ought to behave in the odd ways it does?
I am guessing that this is the default assumption of equal weighting or equal scaling of the variances that you are pooling. If you want to assign non-equal weights you should have some specific reason to do so.
I don't think it's "superior", it's just the simplest default in the absence of any additional information.