What I have seen is that the classic SSer has a lower SD per hand seen, but a MUCH higher SD per hand played.

I am no mathematician, but I suspect that this is the crux of the argument here. It's comparing apples to oranges, because the huge difference is not in pre-flop VPIP, it's in how things go down once VPIP is assumed. I speculate that nits have a lower SD/hr than LAGs but a much higher SD/flop-seen. This could lead to the empirics noted that nits tend to have bigger and more violent swings.

That's what I'm saying. Measuring N by hands or hours is the culprit for why theory does not match application in this context. The LAGs sample size is inflated by playing more streets/bets/decisions. So, let's just concede that sigma/std. dev. is higher for LAGs, and thus, their numerator in the SEM calculation is higher, all else equal. Still, if we see a significantly higher N value in the denominator across an equal amount of play, we can end up with a smaller SEM value.

Now, we don't end up with a higher N when we measure by hands/hours. But we obviously would have a higher N, all else equal, if we measured by bets/streets/decisions. And SEM may well be smaller in that case using the traditional formula.

p4f, probability is the furthest field in the Milky Way from being black and white. Everything is gray.

Yes, we will never reach an infinite sample and no one will ever reach their exact EV. But we do indeed converge closer to EV by a meaningful amount if you always run it twice. That is worth pursuing even though it's no guarantee of anything. Also, amount of convergence is much greater the earlier you are in the sample. So even a bit of it is more impactful than one may think.

+1 to this

Wanted to get back to this because it's important.

Let's pretend we build a regression model and regress winrate on talent, stakes, opponent skill, age, amount of sleep last night, etc, etc, etc. The assumption of homoskedasticity will be violated because our dependent variable will vary at different amounts over different values of X. Mike Caro is correct about this. But it doesn't invalidate all analysis of poke statistics.

The consequences of this violation are that the standard errors for our coefficient estimates will be biased, and so our inferences will be suspect. We won't be able to say why winrate varies with X reliably.

However, the coefficient estimates themselves remain unbiased. And, thus, can still be used for prediction. We would still be able to effectively predict winrate, but we couldn't reliably test hypothoses.