# Being abnormal

Two populations may have a large overlap and differ only slightly in their means. Still, the most outstanding individuals will tend to come from the population with the higher mean.

This is a trivial observation. It is biologically relevant because heritable quantitative traits are to a great extent the raw material for evolution, and, they generally follow an approximate normal distribution. The reasoning is simple, many loci of small independent additive effects are a good approximation of the genetic architecture of many phenotypes, and this structure simulates, roughly, the independent random variables which result in a normal distribution because of the central limit theorem. Obviously two of the most important parameters in the normal distribution are the mean (which is also the mode & median in a perfectly ideal distribution) and variance around that mean.

Unfortunately, human minds are not unbiased statistical inference devices. Otherwise, cognitive psychologists would be shorted many interesting questions. It seems that the implications of the normal distribution and its most famous parameters (the mean and the variance) should be obvious to all college educated individuals. But my experience is that this isn't true, unfortunately. Experience indicates that principles are often more profitably imparted visually, so I took 10 minutes and cranked out a pretty picture via Excel that you can view below the fold.

For the record, what I'm trying to show here is a comparison between two populations. In one, the mean height is 70 inches, 5'10 (about the American mean for males, 1.78 meters for the rest of the world). I assume the standard deviation (square root of the variance) is 4 inches, so in a perfect normal distribution 68% of the population will be within 1 standard deviation, 95% within 2, and 99% within 3. Obviously you chop the remainder in half for the tails of the distribution, so that for 2 standard deviations about 2.5% is at the top and 2.5% at the bottom. Surely uninteresting to the moment warriors out there, but I wanted to state it plainly and clearly.

Now imagine a second population, the same sample size, also normally distributed. Let's keep the standard deviation the same, 4 inches, but let's move one parameter, the mean. When comparing the populations let us fix one at 70 as the other moves, so one population is of "average" height, while the second is "tall"(er). I simply incremented the second population by 0.5 inches until its mean was 6 inches above the first (i.e., 76 inches - 70 inches is an difference between means of 6 inches). So, at the end of the process, the mean of the "tall" population was 76 inches, so the population average in this group would be in the 6.5th percentile of the "average" population (mean 70 inch population).

The graph below is self-explanatory, on the x axis you have the difference between the heights of the two populations. On the y axis you have the ratio of the numbers of individuals in the "tall" population to the "average" that are above the respective standard deviations of the average population. In other words, I'm displaying (assuming the populations are the same size) the difference in the number above an absolute threshold between the two populations. It is clear that as the mean of the "tall" population increases, its numerical advantage deviated above the norm of the "average" population increases at an exponential rate. Not only that, but as you increase the deviations about the norm the extent of the exponent of growth also increases as the "tall" population gains further advantange. At 8 inches above the "average" population norm you are only 6'6. This would be a medium sized individual in the National Basketball League in the United States, someone who could play either of the two guard slots. To be a center you probably have to be at least 6'10 inches, which is a little above 3 standard deviations above the mean of the "average" population. The graph below shows that when the "tall" population has a mean 6 inches above the "average" population, 3 standard deviations above the mean of the "average" population there will be more 40 times as many individuals from the "tall" population.

All trivial of course.

Now, some of you may ask about biological relevance. As I said, to a great extent this sort of variation is the raw material for microevolutionary processes. Selection upon a population occurs at a rate proportional to the regression coefficient of the value of the trait in the parents on the offspring, the narrow sense heritability (see the breeder's equation). As Jim Crow noted, "Nature seems to follow least-squares principles." In more plain language, if you plotted the values of height of parents on the x axis, and offspring on the y axis, the slope of the fitted trendline would equal the extent of additive genetic variance. A slope of 0 would imply that there was no heritable relationship, a slope of 1 implies 100% (perfect) heritability. Here is Francis Galton's regression of heights which illustrates the principle:

Of course, biological variation doesn't follow the normal distribution necessarily to a close approximation. Until recently, the famed population biologist Luigi Luca Cavalli-Sforza could state that beyond 4 loci a polygenic trait exhibits a normal distribution experimentally (see The Genetics of Human Populations). In other words, 4 random variables, often slightly interactive and of non-equal phenotypic contributions might be able to pass themselves off as gaussian for biological work! Of course, the reality is that there might be non-genetic variables generating the distribution, but you get the picture. Gene-gene interactions and a whole host of other factors probably lead to the fact that the "tails" of biological distributions tend to be "fat." Selection results in the increase of rare gene frequencies whose combinations may result in non-independent (epistatic) effects, just as one example.

The overall point is that when I say heritable, I mean something mathematically simple and trivial (the regression noted above on a normally distributed trait), but biologically rather subtle and the object of great disputation.

I end with David Hume:

In every system of morality, which I have hitherto met with, I have always remark'd, that the author proceeds for some time in the ordinary ways of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surpriz'd to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, 'tis necessary that it shou'd be observ'd and explain'd; and at the same time that a reason should be given; for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.

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You know, in every stats course I've ever taught, the hardest thing for the students to grasp has been the concept of variance. They tend to get mean and mode (median can be more difficult, for some reason), but it takes forever for them to grasp variance, and some never do. Until they do, they look at me like I am speaking Swahili. Once they grasp it, everything else comes easy... until we get to ANOVA. Trying to convince students that you really can detect differences in means by looking at variances is damn near impossible.

well...the whole squaring of units is probably part of the issue. standard deviation is more intuitive, i believe, because it has a rough relationship to what you'd expect based on just inspecting the distribution.

You know, I think that there was an appendix in Herrnstein's The Bell Curve called "Statistics for people who are sure that they cannot learn statistics"(something like that) that dealt with this sort of stuff(graphs, trend lines, regression). It strange because I *think* I get the whole gist of what is being discussed, but my understanding of it is such that I'll never be able to discuss it in normal discourse. I'm just not clear on the normal, or Guassian, distribution yet, as dumb as that seems, altthough I can grasp the logic of a "bell curve". I can, sort of, I think, grasp what "variance" means, but when things start getting algebraic, I get lost. It seems like there's so much fun in math that I'm just missing out on.

By Rietzsche Boknekht (not verified) on 30 Jun 2006 #permalink

I have a genuine question, rather than a comment. It concerns regression towards the mean. Simply stated, my question is: towards WHAT mean, and how do you tell?

As I understand it, it is pretty well established that the progeny of two parents from the same population group will tend to have children with IQs which fall somewhere between the average of the parents' IQ's and that of the population group from which they come. So kids of parents of European extraction will tend to regress towards an IQ of 100, Afro Americans presumably towards their mean IQ of abt 85, NE Asians towards about a 105, and presumably Askenazi Jews towards their population IQ of what is it, abt 110?

But what if you come from an especially bright family? Take the Galton /Darwin/Wedgewood clan. How far out in a family do you look to take a family mean? If you had a family, or group of families which married endogamously within their clan groups, would this become a new population group for regression to the mean purposes? Well, I suppose obviously it world at some point but how many generations of endogamy would it take?

Also how much regression towards the mean does there tend to be on average? Are kids on average halfway towards the mean. Or only 1/3 in that direction, or less? Or does this depend on the wider family?

Has anyone studied any of this?

dougjnn, regression to the mean refers to the mean of the relevant population. So, if a clan/group whatever comprises a statistically distinct population on a certain variable (like intelligence), regression to the mean would be to the mean of that population.

How much regression towards the mean does there tend to be on average? re kids on average halfway towards the mean. Or only 1/3 in that direction, or less? Or does this depend on the wider family?

Well, the CLT says that in a normal distribution, the population will be equally distributed above and below the mean. The proportion of the population that will be within a certain distance of the mean, in a normal distribution, is also defined within CLT.

So, on average, a higher percentage of the population will be close to the mean (68.3% within one standard distribution in either direction from the mean in a normal distribution). That means that every time you encounter an individual from the population, that individual is more likely than not to be within one standard deviation (the very best bet is the mean, of course). Regression to the mean really just says that if you get an individual far away from the mean, the chances are that the next individual will be closer, because that's where most individuals fall anyway.

Chris:

"Regression to the mean really just says that if you get an individual far away from the mean, the chances are that the next individual will be closer, because that's where most individuals fall anyway."

I was talking about children of parents quite removed (upwards) from the mean.

I believe you are wrong (because it is inconsistent with much other discussion of this issue I have seen as well as yes with common sense) if you are saying that the children of parents whose average IQ is say 3 SD's up from average are as likely to fall anywhere on the IQ continuum as anyone else's children.

Wrong.

Next?

dougjnn,

i'm in a hurry, so think of the regression as a outcome of measurement error/noise. the noise resolved, more or less, by adding more information into the system. e.g., two parents from a human population would expected to regress to 90 IQ (the world wide mean). if i told you that the parents were both ashkenazi jews, you would expect a regression back to a mean of around 110. if i told you that we have a lot of pedigree information about 1st and second seconds, you might be able to narrow down the expected point of regression even more....

Also how much regression towards the mean does there tend to be on average? Are kids on average halfway towards the mean. Or only 1/3 in that direction, or less? Or does this depend on the wider family?

in quant. genetics regression is proportional to the heritability. if you assume .5, 1/2 regression. if you assume 1.0, no regression. if you assume .1 heritability, almost total regression back. in other words, the non-heritable proportion of variance is simply the error. if you can account/eliminate this, then the regression would reduce. e.g., high SES parents should have less regression since IQ heritability is higher because they are less impacted by environmental stresses.

hope that clears up a little....

Razib--

Yeah, that clears up quite a bit, especially on the last question, about degree of regression.

So if high SES parents are looking at total broad sense heritability of maybe 70-80% the amount of regression is likely to be the reciprocal of that then. But the mean to which regression actually tends will be somelike the mean of family IQ;s on each side, weighted by fraction of genes inherited.

So if high SES parents are looking at total broad sense heritability of maybe 70-80% the amount of regression is likely to be the reciprocal of that then.

minor note, regression is the reciprocal of the narrow sense heritability. this is what is illustrated by the plot of parent to offspring regression, broad sense includes dominance effects and that is different.

I was just explaining regression to the mean in general. I don't know enough about the heritability of intelligence to make predictions about the intelligence of childre based on their parents' IQs. I don't think anything I did say was wrong, though. Feel free to point out the errors if you like.