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John Cook, in his blog https://www.johndcook.com/blog/2015/03/09/why-isnt-everything-normally-distributed/, writes that many aren't:

Adult heights follow a Gaussian, a.k.a. normal, distribution [1]. The usual explanation is that many factors go into determining one’s height, and the net effect of many separate causes is approximately normal because of the central limit theorem.

If that’s the case, why aren’t more phenomena normally distributed? Someone asked me this morning specifically about phenotypes with many genetic inputs.

The central limit theorem says that the sum of many independent, additive effects is approximately normally distributed [2]. Genes are more digital than analog, and do not produce independent, additive effects. For example, the effects of dominant and recessive genes act more like max and min than addition. Genes do not appear independently—if you have some genes, you’re more likely to have certain other genes—nor do they act independently—some genes determine how other genes are expressed.

Height is influenced by environmental effects as well as genetic effects, such as nutrition, and these environmental effects may be more additive or independent than genetic effects.

I have two questions regarding this. First, I can't come up with any obvious example of phenotypes that don't follow a normal distribution. Could anyone help me? And second, his claims seem to contradict this other paper: https://www.uvm.edu/~dstratto/bcor102/readings/4_Evol_of_Phenotypes.pdf

To understand the genetic basis of quantitative traits, it is important to think about the effect of a particular allele, not simply its presence or absence. A single locus can produce three discrete phenotypes, but as more and more loci contribute to a trait the phenotypic distribution comes closer and closer to a normal (bell shaped) distribution

Who is right here? Does the fact that genes are digital rather than analog really make a difference? And what about the second argument that they aren't independent. Is that really necessary (second paper seems to indicate otherwise if I am understanding properly).

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    $\begingroup$ This may be more suited to the biology site. $\endgroup$ – GeoMatt22 Jun 9 '17 at 20:02
  • $\begingroup$ I realized this, but I posted here because I wanted opinion from someone with a statistics/math background. Because to me, it seems like even the mathematical claims are not necessarily true (binomial can be approximated by normal and it seems like CLT can be generalized to not require independence). $\endgroup$ – samlaf Jun 9 '17 at 20:34
  • $\begingroup$ Well, due to things like allometry, obviously all possible measurable "phenotypic traits" cannot be normally distributed (this does not depend on genetics). The empirical question of "what types of quantitative-trait distributions are normal vs. not" could possibly be answered at the biology site. And that community could also unpack the simplifications in your quotes (e.g. epigenetics). $\endgroup$ – GeoMatt22 Jun 9 '17 at 20:58
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    $\begingroup$ In practice I expect that none of the variables you deal with are really normally distributed. Physical measurements of non-negative quantities (like lengths) will be necessarily non-normal -- and in many cases obviously so -- often somewhat right skew, in some cases multimodal and so on. In some particular cases the normal distribution will be a reasonable model, but don't imagine for a moment that such a simple model is actually the truth. It's a convenient approximation, and sometimes a very useful one. $\endgroup$ – Glen_b Jun 10 '17 at 4:50
  • $\begingroup$ This really belongs in biology. Mendel's peas had seven characteristics, each with two values, eg, smooth and wrinkled. There are a vast number of examples for fruit flies. Several diseases depend on a single locus. $\endgroup$ – David Smith Jun 10 '17 at 18:18

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