# Why the length of the beak of birds of a certain specie follows a normal distribution. Also, IQ

A friend of mine is studying biology, and in a project of hers, she had to measure the beak (and other parts of the body) of some dozens of birds of a same specie, and analyze the data. In particular, she found out (as it was supposed) that the distribution of the lengths of the beaks is close to a normal distribution. I wonder why that happens.

The central limit theorem states that

If $X_1,...,X_n$ are independent and identically distributed random variables and have the same mean $\mu$ and variance $\sigma^2$, then the distribution of $$\frac{X_1+...+X_n}{n}$$ converges to $$N(\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ as $n\rightarrow\infty$.

We also know that $X_1+...+X_n$ itself also has a normal distribution. So, by the "why" above, I simply mean: "what are these $X_1,...,X_n$?".

My thoughts:

Let us say that all the birds of the given specie are born with a beak of the same length $a$. This seems to me to be approximately true. Also, let us say that the beaks of these birds stop growing when they are $b$ years hold, and that all measured birds are older than that (so all of them have full-grown beaks). I would say that this models the conditions of the measurement pretty well. Now, each day that passes, the beak of a given bird grows a little bit. Here is the key point: the amount that it grows is given by a certain (unknown) probability distribution, which is the same for all birds and at all times $t<b$. We now set $n$ to be the number of days in $b$ years, and $X_i$ is the beak growth in day $i$. With this model in mind, we see that the beak length of a bird is simply $X_1+...+X_n$ and we know that this follows a normal distribution.

Another case

And what about IQ distribution, for example? Can it be a "growth process" like this one, but with things other than beak length?

• – supinf Aug 11 '17 at 10:25
• This is not the model. The length of beaks of birds is affected by genetic factors, which differ from bird to bird; if they didn't it would be impossible for natural selection to act on beak lengths. The $X_i$ are various genetic and environmental factors. – Qiaochu Yuan Aug 11 '17 at 22:11

It's not an unreasonable model, but I think a slightly more realistic one would consider the sources of error to be (1) from the noise at the molecular level (i.e. you can look at systems biology models of growth), due to the randomness of diffusion, environmental effects (e.g. chemicals), etc..., and (2) genetic differences between birds (most likely multiple genes are involved in the beak growth, and each one can be mutated in a large variety of ways, i.e. anywhere in the genetic sequence of each of the genes could be mutated). These sources of error could be considered to combine together each day to give rise to the $X_i$ you are talking about (per day).
For IQ, no doubt a growth process is present, but it is hard to imagine a process of iid additive random factors are responsible! However, the CLT works even when the $X_i$ are not identically distributed (e.g. see here), as long as they are independent and have some reasonable level of boundedness. So maybe we could model IQ like: $Z = X_1 + \ldots + X_n$, where each $X_i$ is a genetic or environmental factor with a distribution $P(X_i|\mu_i,\sigma_i^2)$. For instance, $X_i$ might be a function of the expression level and activity of some gene $A$. Note that many factors can influence both the expression level and molecular efficacy of a gene (and those many factors can be influenced by many other factors); hence, the $X_i$ may not be entirely independent. However, the number of factors per $X_i$ is presumably sufficiently large that this doesn't matter. All the differences in these factors (genetic or environmental) between people are captured in the $\sigma_i$'s.
Or, from a different perspective, you could assume that IQ (as a test score) is instead a function of a large number of separate processes in the brain: e.g. hand-eye coordination, spatial visualization, long and short term memory, reasoning ability, linguistic skills, etc... The IQ score is thus some kind of weighted average of these capabilities. Since each is noisy (not only for a single person across even short time-scales, but also between people), in a similar way to the mechanism above, the CLT says the histogram of IQs that results will be roughly normal given enough people. This is perhaps not fully satisfying to some though, since it does not explain from where the differences ($\sigma_i$s) actually arise.