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?