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I was searching to learn which traits (e.g., hair color, intelligence, weight, etc) are and which are not normally distributed. In the lecture notes for some course on a university website (unfortunately I don't have the link), a college professor had stated that complex traits are normally distributed.

But then I came across an answer on Quora to someone asking what besides height and intelligence is normally distributed https://quora.com/What-traits-besides-IQ-and-height-does-a-normal-distribution-describe-in-a-population

A supposed statistician had replied that "Actually, nothing is described by a normal distribution.It approximately describes heights of people so long as they are all male or all female and not a mixture of East African Negros, West African Pygmies and Europeans." He went on to say that a large sample, however, with no outliers can approximate normal distribution.

So I am confused. So can anything, if the sample is large enough, approximate normal distribution? And yet nothing is normally distributed to begin with?

ironman
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Yobay
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    None of the variables you mentioned are drawn from Gaussian populations. Not one. Even intelligence (which is typically *designed* to be Gaussian) can't be (e.g, can you score lower than 0 Intellligence?) – Glen_b Jun 16 '18 at 06:14

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None of the variables you mentioned are drawn from Gaussian populations. Not one. Even intelligence (which is typically designed to be Gaussian) can't be (e.g, can you score lower than 0 Intellligence?).

The size of the sample has nothing to do with the shape of the population from which the same is drawn so having a "large sample" has nothing to do with it.

I'd say they are both trying to make a point but both stumble in doing so.

Glen_b
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    Why wouldn't you be able to have an IQ less than 0? If indeed the definition of IQ being used is designed to be Gaussian (hadn't heard that before), that implies it *is* possible to have an IQ less then 0 (though that might be so unlikely that no person on earth has it); I don't see a contradiction there. If you choose to define/design it some other way, then perhaps not. – Don Hatch Jun 16 '18 at 10:17
  • In practice it depends on the precise methodology followed and the precise instruments used, but no answers or perfectly random answers with typical approaches that are used would tend to score something around 40. (With modifications it would of course be possible to make a test that would have scores less than 0 but I've never heard of any IQ scale that actually did so). If you administer a methodology that can only yield integers or you administer a methodology that can only yield a finite range, the scores you get when you do are (clearly) not a random sample from a normal population. – Glen_b Jun 16 '18 at 10:24
  • Can you please explain your answer a little more? What is a Gaussian population? Can you give an example? thanks – Yobay Jun 16 '18 at 16:46
  • Sorry, it's not clear to me what you're after -- I can't tell whether you don't understand what a population is or whether you don't understand what a distribution is. The Gaussian distribution is detailed [here](https://en.wikipedia.org/wiki/Normal_distribution). An infinite population has a Gaussian distribution when $X$, a randomly selected member of that population has $P(X\leq x) = \Phi(\frac{x-\mu}{\sigma})$ where $\Phi$ is the standard normal cdf; i.e. $\Phi(z) = \int_{-\infty}^z {\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}dx$. ... ctd – Glen_b Jun 16 '18 at 23:23
  • ctd... If you're after a real world example of a Gaussian population, there isn't one. It's a model, an approximation of reality (as George Box said, *all models are wrong, some are useful*). Some things may come quite close to it. [I expect that response wasn't much help to you but perhaps if you can reformulate what you're after we can give you what you need.] – Glen_b Jun 16 '18 at 23:24
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This supposed statistician is mistaken. Many, many naturally occurring phenomena tend towards Gaussian. No sample will perfectly conform but when compared against other competitor distributions, the Gaussian will tend to win for things like heights and weights and IQ scores and things. More so when the sample size increases.

But, if you change the situation this isn’t true. The arrival times of radioactive alpha particles from Uranium is exponential. The number of heads coin tosses out of N throws is binomial.

This all said, the central limit theorem states that the distribution of means is nearly always Gaussian, regardless of the form of the parent distribution.

Here’s an example: consider the distribution of heights (or weights) for all people inside day care centers. While human height is normally distributed in general, at any one day care you’ll get a lot of short and light people and a couple taller heavier people. Very skewed. Now take the mean height or weight for each day care center in the United States. That distribution of means will be normal.

HEITZ
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I support @Glen_b's answer (+1)

I will just add the following:

If you imagine the evolutionary process as a random process, then when the process becomes stationary, the traits at the stationary point may exhibit Gaussian-like behavior because of the culling of the remainder of the population due to its inability to survive. However, this statement also assumes that the environment experienced by the entire population is identical. But clearly, this is not the case (for e.g. equatorial regions are hotter than polar regions). Different traits may end up surviving in different parts of the world. So at a global scale, you will never see a Gaussian or even Gaussian-like distribution. However, for the local homogeneous population, you may see Gaussian-like distribution for certain traits but not for all traits, and that depends on what factors affect that trait.

TenaliRaman
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