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A lot has been said about importance of normal distributions in nature. A lot of measurements like height or weight are distributed approximately normal. But none of them are exactly normal, as far as I understand.

Considering normal distribution is one of the maximum entropy distributions, it seems plausible that nature should "like it". But after some thinking I couldn't come up with any examples of "really" normal random variables.

My question is what good examples of exactly normally distributed random variables out there?

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    $\begingroup$ @mpiktas Brownian motion is a model; is there any evidence that any observed process is actually exactly Gaussian? I'd be quite surprised, because there's always going to be physical limitations that contradict properties of the normal. $\endgroup$ – Glen_b -Reinstate Monica Oct 16 '15 at 10:24
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    $\begingroup$ Define "exactly". $\endgroup$ – Eoin Oct 16 '15 at 10:29
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    $\begingroup$ @Glen_b is it possible to prove that any observed random quantity has an exact distribution? $\endgroup$ – mpiktas Oct 16 '15 at 11:07
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    $\begingroup$ @mpiktas That's what the OP appears to be asking for, though - exactly normally distributed variables; I'd have thought the only possible answer is that there can't be any. $\endgroup$ – Glen_b -Reinstate Monica Oct 16 '15 at 12:18
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    $\begingroup$ I think this is a bit like asking for an example of a perfectly straight line. They don't exist in nature, but they are still a useful concept. $\endgroup$ – Dikran Marsupial Oct 16 '15 at 15:25
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If you mean "exactly" precisely enough, then I think the answer is "no" because any natural event has a limited population (even if the population is very large) so no probability will be exactly correct.

Also the normal distribution applies to continuous variables and nothing is really continuous. Even weight, if you get down to the subatomic level, is a count (how much does Peter weigh? Please answer in terms of protons).

Perhaps more interesting, many variables that are assumed to be normally distributed may not even be roughly normal, in typical populations.

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    $\begingroup$ I think you would have lots of problems if you were just protons. We prefer you the way you are. $\endgroup$ – Nick Cox Oct 16 '15 at 11:43
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    $\begingroup$ Weight is an interesting example, because it's about as continuous as anything can get. It depends not only on a proton count, but also on neutrons (which have a different mass than protons), electrons, energies of binding and interaction, and altitude, among other things. Such considerations suggest that the question ought to be addressed by distinguishing "nature" from our theories of and models about natural objects. Moreover, the random variables of greatest interest are derived from others: they are sampling distributions of statistics. $\endgroup$ – whuber Oct 16 '15 at 13:05

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