Sorry, maybe I'm too dumb, but I don't understand how this explains away the phenomenon. This seems to describe the thing that's happening, but doesn't explain why it happens. Saying X happens is not an explanation for why X happens.

Why is it that for many cognitive tasks, the best in the world are so far better than the median? Stating that they are is not an explanation of the phenomenon.

My point is that while intelligence is well approximated by a normal distribution (Not perfectly, and there may even be mild log-normal distributions), the others aren't well approximated by a normal distribution at all, which means that the controlling variable of intelligence has very small variance, but the variables that are controlled have very large deviations ala power laws are very heavy tailed log-normals, thus the distribution has very high variance, often multiple orders of magnitude variance large.

The point about using 0 elo as a stand in for dumb human doesn't seem to be germane to our actual disagreements, so I wouldn't address it.

Fine with me, but I just want to note again that I am not using 0 Elo as a "stand-in" nor saying you were doing that, I'm estimating roughly where a dumb human happens to end up on the scale, and for the case of FIDE chess ratings I think it turns out to be near zero. To be clear, I'm not trying to reopen any discussion about whether that's right (since I agree it isn't particularly important for our actual disagreement), it's just that it seems like you're being quite insistent about describing something I did in a way that doesn't match what I think I was actually doing, and I would prefer the last thing about it in the discussion not to misrepresent my intentions.

(I notice that that sounds more annoyed than I actually am. Not annoyed, just wanting to avoid future misunderstandings.)

I'm not sure there exactly is an "underlying quantity" here. Differences in rating, hence odds ratios in results, are fairly well defined (though, note, it's not like there's any sort of necessary principle along the lines of "if when A plays lots of games against B their odds are a:b, and if when B plays lots of games against C their odds are b:c, then when A plays lots of games against C their odds are a:c", which the Elo scale and the usual ways of updating Elo ratings in the light of results are effectively assuming IIUC). But I don't think there's an absolute thing that the differences are sensibly regarded as differences in or ratios of.

I guess you could try to pick some "canonical" single player -- a truly random player, or a literally perfect player -- and look at odds ratios there. But I think the assumption I mentioned in the previous paragraph really does break down in that case.

I'm not sure I understand this point well.

I expanded on it a couple of paragraphs below. If that still didn't clarify, can you say a bit about what doesn't make sense?

That ~600 point difference is much smaller than the 1800+ point difference between Carlsen and the median chess player [...] the argument that I made [] that Carlsen was closer to optimal play than to median, not that optimal play was not much better than Carlsen.

Hmm, maybe I misunderstood something? You wrote

It seems that for many cognitive tasks, the median practitioner is often much closer to beginner/completely unskilled/random noise than they are to the best in the world [... footnote:] My intuitive sense is that [...] the gap between Magnus Carlsen and the median human in chess ability is 10x - 1000x the gap between the median human and a dumb human.

It's the "10x-1000x"  I'm disputing, not any version of "much closer" that's compatible with the "right" numbers being on the order of 0 / 1000 / 3000.

As for any relationship to optimal play, I was getting that from assuming that what you said about chess was intended as support for drawing a scale with "idiot" on the left and "Einstein" on the right rather than one with "mouse" on the left and "vast superhuman intelligence" on the right. (My own feeling is that what we actually want is more likely a scale with all four of those points on it, and no pair super-close together. Idiots really are much smarter than mice; Einstein really is much smarter than an idiot; God really is much smarter than Einstein; and for many purposes none of those gaps is so large or so small as to make the others negligible. For difficult enough tasks, idiots and mice may be indistinguishable, or even idiots and mice and average people, but those are also the tasks for which I expect hypothetical superintelligences to have big advantages over the smartest humans.)

So I took you to be suggesting that the median-Carlsen gap is large enough that at least one of these pictures is not misleading: ||---------| (vertical bars are idiot, median, Carlsen) or |------------|| (vertical bars are median, Carlsen, superhuman machine). And I don't agree with either; I think the idiot-median-Carlsen-machine picture is more like |-----|----------|-----| except that I don't know how much room there is for that last gap to grow as the machines continue to improve.

And my reason for thinking this is that (1) if you use something like log odds (roughly equivalent to Elo ratings) to measure the gap sizes, then that's what happens, and (2) if you use the odds themselves[1] then indeed the larger gaps become larger by huge factors, but in that case the |-----|--------| pictures where you put gaps next to one another and compare their sizes are completely misleading, because the correct way to combine two gaps is not to put them side by side (which amounts to adding up their sizes), and (3) these large-factor gaps don't seem to me to be reason to prefer your "just idiot...Einstein" scale to the sort that Yudkowsky likes to draw, because the point Yudkowsky is trying to make by drawing them is about what happens to the right of Einstein, and there is good reason to think that there's plenty -- in particular, there's a large-odds-ratio gap -- to the right of Carlsen.

[1] Insert Schiller quote here :-).