> Why is that important? Unless the runners are conscripted into the race, it's not telling you anything about women or men
It's not saying something about _all_ men or women. If you did try to work with a random sample wouldn't you mainly find that almost all people of both sexes can't run an ultra marathon?
But in making comparative statements even about people who choose to participate in such races, I think a critical distinction made in that article is that there's a difference between "E(Pace_W) < E(Pace_M)" vs "Min(TotalTime_W) <? Min(TotalTime_M)".
The earlier anecdote was making a statement about who won a canoe race and using it as evidence of a group level difference ... But race winners are the extreme end of the distribution and are poor information about the overall behavior.
If you know the extreme values, the shape of the distribution ["normal"], and the number of samples [for a competitive task, generally taken to be the entire population; not so much for e.g. quilting bees], you know the overall behavior. The extreme end of the distribution is evidence of a group-level difference, and very high-quality evidence.
>> How, for example, do we determine a distribution of running ability within an entire population? Can we find a representative sample of tribesmen, provide each with motivation and training, and finally measure their times for some event? Not very likely. There is, however, a way out. In Aggressiveness, Criminality and Sex Drive by Race, Gender and Ethnicity, we introduced the method of thresholds. It applies nicely to this problem. The proportion of each tribe meeting or exceeding some threshold of performance is the only input it requires. When all is said and done, the precise definition of "ability" will still be fuzzy, a characteristic of the method of thresholds. That aside, we will have established running ability distributions in tribes relative to one another.
>> Some of the data we need are available from chroniclers of track and field. All-time-best lists are particularly useful. For a given event, such a list might contain 100, 500, 1500 or any number of the best times ever run. The slowest time on a list serves as the threshold of performance required by the method of thresholds.
> If you did try to work with a random sample wouldn't you mainly find that almost all people of both sexes can't run an ultra marathon?
No, you'd find that people managed to go different distances before failing. You would have to be intentionally avoiding the result you expected to find to binarize your outcome data like that. The data you're appealing to right now isn't binarized.
> But in making comparative statements even about people who choose to participate in such races, I think a critical distinction made in that article is that there's a difference between "E(Pace_W) < E(Pace_M)" vs "Min(TotalTime_W) <? Min(TotalTime_M)".
The article itself provides the explanation: there are very, very few women running. What lesson do you feel we should draw? To me it looks like the lesson is "men are a lot more interested in distance running than women are".
The methodology at the page you link to seems really sketchy and inappropriate to the kinds of groups under discussion.
> Suppose PA(x) and PB(x) differ only by a translation in x, such that fB(x) = fA(x - Δ), where Δ is the mean difference in x between the groups.
... but you can't really just assume that the variance within two groups is the same. Especially when comparing a small group like the Nandis (a "subtribe of a half million") vs a large and diverse group like "Europeans", as they do in the "Augmentation of Small Differences" section, it seems pretty cavalier to just assume that the variance is the same.
But for comparing stuff between sexes, the "variability hypothesis" about men having greater variability across a range of traits dates back to Darwin, and has a pile of research results. It would seem especially irresponsible to rely on an assumption that the variance was equal between the sexes. One might have prior reason to expect otherwise, and differences in variance may materially contribute to different fractions above or below an extreme threshold.
Further, given the kind of selection effects you were alluding to before, it may not even be safe to assume these distributions are normal. While the endurance of the broader population ought to be normally distributed, if there are various hurdles on the path to participating (e.g. the organizers say you should probably have completed a 50 mile race at a given pace before signing up for their 100 mile race?) one might well see a different overall shape.
> But for comparing stuff between sexes, the "variability hypothesis" about men having greater variability across a range of traits dates back to Darwin, and has a pile of research results. It would seem especially irresponsible to rely on an assumption that the variance was equal between the sexes.
That would imply a much larger advantage for men. If you were seeking to show that women outperform men, you'd gloss over that point as much as you thought you could get away with.
(In a little more detail: if you determine that it takes an athleticism factor of 10.8 to run 200 miles, greater male variability immediately implies that among all people who have that much athleticism, the average male athleticism will be quite a bit higher than the average female athleticism. The average of a thresholded normal distribution is quite close to the threshold, but it gets farther away as the standard deviation increases.)
> Further, given the kind of selection effects you were alluding to before, it may not even be safe to assume these distributions are normal. While the endurance of the broader population ought to be normally distributed, if there are various hurdles on the path to participating (e.g. the organizers say you should probably have completed a 50 mile race at a given pace before signing up for their 100 mile race?) one might well see a different overall shape.
...you don't seem to have understood the method. The assumption is that the broader population is normally distributed - you know, what you already said it was safe to assume - and that the selected population consists of that part of the broader population's normal distribution that exceeds some threshold.
Or in pictures, we assume that the population looks like this:
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and the running-200-miles population looks like this:
It's not saying something about _all_ men or women. If you did try to work with a random sample wouldn't you mainly find that almost all people of both sexes can't run an ultra marathon?
But in making comparative statements even about people who choose to participate in such races, I think a critical distinction made in that article is that there's a difference between "E(Pace_W) < E(Pace_M)" vs "Min(TotalTime_W) <? Min(TotalTime_M)".
The earlier anecdote was making a statement about who won a canoe race and using it as evidence of a group level difference ... But race winners are the extreme end of the distribution and are poor information about the overall behavior.