Human Height is not Bimodal

[atoo]

This is a bit of a niche topic I know, but I thought it might be interesting to at least some people.

If there's a scalar characteristic which has a different mean in men and in women, then the distribution for the whole adult population will be Bimodal if the difference in the means is large compared to the variation within each sex.

This is the case for example in gamete size or testosterone levels. But it is not the case for height - the difference between the average man and average woman is less that the difference within each sex. So human height is not in fact a bimodal distribution. It's just a bit flatter than a bell curve (image attached).

Much more detail in this paper: https://faculty.washington.edu/tamre/IsHumanHeightBimodal.pdf,.including why people often find bimodality in practice - e.g. small sample sizes, and men lying about their height.

From a statistical perspective, I’m not sure that the fact the peaks overlap means it is not a bimodal distribution. The curve is still better fitted by two populations with different means, each with a normal bell curve distribution. Isn’t that still bimodal?

I think according to PP there is a specific statistical definition for bimodal. Something something standard deviations and means

If the combined distribution only has one peak then it's unimodal.

Statistically speaking bimodal is when the combined frequency density graph has two local peaks. Two points for which that value is more likely than one just above or below it.

The thing about standard deviations is just a rule of thumb for when it's likely to happen. Probably based on if the individual distributions were normal.

If you combine two different but overlapping single-peaked distributions, like height, the result may be a combined distribution with two peaks or one with one broader peak.

It's obvious they're two different populations just by looking at the separate distributions before adding up. You don't need to be able to deduce it from the combined graph, you deduce that from the differences in the individual graphs. If there was any doubt you could do a significance test. Spoiler: if you do this with height, the difference is going to be significant as long as you have any decent sample size.

As there is very close to equal numbers of M and F to get the combined distribution shape you pretty much add up the densities of the two separate distributions (the heights of those bell curves people keep showing us) and divide by 2. So it's about whether one graph is falling away between the two individual peaks faster than the other one is growing, or not.

Well, yes, except that a significance test doesn't infallibly tell you whether they are different distributions: it merely provides a decision rule on whether to accept the null hypothesis that the distributions are the same or not.

Nothing is absolute, but significance levels is how people convince themselves of stuff in medical research. it could technically all be a huge coincidence that every time we measure the heights of a group of men and an equivalent group of women the men are taller on average, but it's incredibly vanishingly water might fall upwards-ly unlikely that would happen.

If you have a p value a tiny bit below your threshold you would reject the null hypothesis and if you have a p value a tiny bit above your threshold you would not reject it. There's nothing magic about the threshold, it's just the arbitrary level that you decided ahead of time to use as your cutoff.

I'm not sure what point you are trying to make. If you want to convince medical people that two populations have a different measurement (say that a treatment has worked better than the placebo, or that two populations have different mean heights), what you do is carry out a significance test.

Technically what the test result will say is that if men and women were from the same height distribution, then the chance of your data showing a difference at least as big as it has in mean heights is less than say 0.1% (if that's the level I chose for my test). That is very convincing evidence that they are not from the same distribution. A 1-in-1000 unlikely thing would have had to have happened. And the more times the test is repeated with the different samples and the same result, the more unlikely it becomes that they'd all say men were taller if they weren't.

I've explained my point to you twice and genuinely don't know what you're struggling with. You seemed to think that a significance test actually tells you if there are two different populations. It doesn't: it is merely a rule that you specify beforehand to decide whether you will reject the hypothesis that the two groups are the same.

This is exactly why ideas like “Men are from Mars, Women are from Venus” are nonsense when they get turned into stereotypes such as “boys are better at Physics”.

Just because there is an observable or measurable difference between male and female populations, it tells you nothing whatsoever about individual males and females within the population. I am not weird because I’m female, good at Maths and hate pink.

OMG I’m either a man or a basketball player - why did no one tell me in the last 50 years? Maybe I’ve been passing so well as a woman?

If you're passing well, you're probably the basketball player!

I've explained my point to you twice and genuinely don't know what you're struggling with. You seemed to think that a significance test actually tells you if there are two different populations. It doesn't: it is merely a rule that you specify beforehand to decide whether you will reject the hypothesis that the two groups are the same.

I think you may be confusing "evidence" and "proof". Something happening that would be highly improbable under the null hypothesis is a piece of evidence pointing to the null hypothesis being wrong. That's the whole point of doing significance tests. It's how medical science works. We can't 100% prove that treatment A works better than a placebo, but if we can repeatedly in multiple independent studies reject a null hypothesis that it is not better, that gives us a high degree of confidence that it /is/ better.

(To be completely pedantic, we already know there are two populations or we couldn't run the test at all. The question is whether the two populations have different height distributions. They do. It has been very long established. Nobody seriously doubts it. But if you wanted additional evidence you could find some data and do a hypothesis test.

I am always bemused by mumsnet's obsession with height.

No, you were the one confusing evidence and proof which is why I corrected you in the first place.

No, you were the one confusing evidence and proof which is why I corrected you in the first place.

You must have misunderstood me, I was not confusing anything. I confirmed in my first reply to you that I wasn't claiming anything was absolute.

All of which is rather beside the point I was trying to make, which is we don't need to look at the combined distribution of M+F height to demonstrate that there are two different distributions involved, it's much better to look at the distributions of M and F separately.

Though it is true in the other direction that if something does have a bimodal shape, and you didn't already know there were two distinct populations involved, you would go looking for them.