I’m not sure that’s a good example. For most species, the number of limbs is not a genetically variable trait (excluding gross abnormality). For something like a centipede, it goes with body length and number of segments, so I wouldn’t be surprised if it’s approximately normally distributed.
You’re evidently thinking of traits that are continuously variable. Of course there are many traits, like eye color in humans, that are not continuously variable. But even continuously variable traits can show bimodal or even fancier distributions.
One obvious example is height of humans, which also incidentally illustrates one of the dangers in the assumptions behind normality. Human heights are influenced by a wide variety of genes and environmental factors, which is usually what’s described as needed for the Central Limit Theorem. But one of those genetic influences is almost purely discrete, with two values, and that causes the total distribution to be strongly bimodal, as well.
You had me going there until I figured out what you were talking about.
So now they’re armoring themselves. Great.
Surprisingly, that’s not the case. Human adult height distribution is not strongly bimodal, despite being a mixture of two normal distributions. The means are too close and the variance of each (especially males) too wide to yield twin peaks.
There’s a cite here (scroll down to read the abstract).
Edit: you can read the paper in full here.
{sarcasm on}
Are you saying that more than 50% of koalas have below average intelligence?
{sarcasm off}
Probably giving his Winnie the Pooh Halloween costume a dry run…
Somehow, yes. The few that evolved into drop bears bring up the mean.
Well, that’s using a particularly strong definition of “bimodal” that requires the means be different from each other by at least the sum of their standard deviations. That’s certainly not the case for humans with respect to gender bias if you take the entire population of humans as the sample, as the variance of various groups of humans dwarfs the difference in average height between men and women. But if you were to looking at more restricted population than the American population as a whole, a population much more homogeneous, you might see a clearer double peak. Is it enough to satisfy the definition of bimodal used? Maybe not, but it’s clear that women are on average shorter than men by a non-negligible amount and this is consistent across all human populations, so to describe the height of humans as bimodal isn’t entirely wrong even if the definition technically excludes it.
ISTM that it might bump the mean up slightly, due to loss of all the dumb bears, but it would leave a non-normal, one-tailed distribution behind. The variance would go down due to the loss of one tail, but most of the surviving bears would be of less than median intelligence.
Which leads me to wonder about just how much survival advantage there is to greater intelligence - beyond some arbitrary threshhold - in solitary animals.
No, loss of one tail of the distribution could not happen when you consider the underlying mechanism that creates the normal distribution in a quantitative trait like intelligence. The normal distribution arises because the trait is due to small contributions from a large number of genetic loci. A simplified model would be: 100 genetic loci, each gene having two alleles, a “dumb” and “smart” allele. Higher than average intelligence is attributable to inheriting the “smart” allele are more of these loci than the average bear. Natural selection against being dumb would reduce the frequency of dumb alleles at each locus in the next generation. At some loci, the frequency of dumb alleles might go to zero, i.e. the smart allele would go to fixation.
So the initial effect of the selection pressure would be to increase the mean of the distribution; under strong selection pressure, if the smart allele went to fixation at a significant number of loci, the variance of the distribution would also narrow. New mutation at these or other loci that affect intelligence could increase the variance again over time.
The shape of the distribution would tend to become non-normal only if the trait comes under the influence of a small number of genes with large effect, rather than a large number of genes with small effect. For intelligence, this would mean a de novo mutation that arose (randomly, of course) in a single individual, making that individual much smarter, and then increased (non-randomly, through natural selection) in the population.
You do know that Koalas are not bears?
If there’s some threshold of dumbness, below which a bear is incapable of surviving to adulthood, and if you looked at the population of adult bears, you would indeed see one tail cut off of the distribution.
By definition, exactly half of the surviving bears would be of less than median intelligence. You probably meant to say that most of them would be of less than the mean intelligence. Alternately, you could say that most of them would be of more than the modal intelligence.
I think you are reading it backwards. It’s not that they start with a narrow definition of “bimodal” that results in human height distribution not being bimodal. It’s that a mixture of two distributions with different means can have a single mode (human adult height distribution being an example of that) and then they state the conditions under which this occurs.
Short version: the fact that women are on average shorter than men does not imply that the distribution of heights of all adults is bimodal.
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You do realize that was a joke?
(In any case, he didn’t actually say that koalas were bears.)
Drop bears are also not true bears.
Actually, I can think of another common human population which does have a bimodal height distribution: The heights of students (or especially, of boys) in a middle school. Some have hit their growth spurts, some haven’t, and the spurts are quick enough that at any given time, very few are in the midst of one.
No, this is wrong. For a quantitative trait influenced by a large number of loci all with small effect, at equilibrium you will never see a cut-off normal distribution except at the extreme tail where the cutoff is so far below the mean that very few individuals are born below the cutoff, and it has no significant effect on allele frequency.
Suppose that some new environmental challenge arises imposing extremely strong selection against intelligence below x that’s (say) 1SD below the mean of the prior normal distribution. Now, of course, in that generation bears below 1SD are all dead, so you temporarily see a normal distribution cut off at -1 SD. But if this is a trait under the influence of a large number of loci with small effect, the effect on the next generation will be a lower frequency of “dumb” alleles at the large number of loci that influence intelligence. So in the next generation, you will see a new approximately normal distribution with a higher mean and smaller variance, chopped off at intelligence level x that is now more than 1SD below the mean of the new distribution. If the strong selection pressure is maintained for intelligence below x, in successive generations the frequency of dumb alleles is reduced further, so the mean of the distribution is pushed higher and its variance is reduced, until the cutoff point x is so far below the mean that so few individuals are below the cutoff that it’s not having any further significant affect on allele frequency. So the new equilibrium is still approximately normal, cut off at x which is now at the extreme tail of the distribution, with very few individuals born with intelligence below x.
Are sea stars not made of plasma!?!?
But in all seriousness, koalas are very stupid by animal standards. Their brains are tiny and smooth (=less surface area per volume). They are obligate on eucalyptus but fail to eat the leaves if they are picked. Infested with chlamydia. Yeah they’re sort of cute, but missed buying tickets to the genetic lottery.
The lowest brain to body ratio is the bony-eared assfish. They already have small brains, no need to kick them while they’re down.
As far as what is a bear and what isn’t, I’m still recovering from being constantly told as a child that giant pandas are not true bears. Turns out they are.