Are there any examples of a variable being normally distributed that is *not* due to the Central Limit Theorem? The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
 A: To an extent I think this this may be a philosophical issue as much as a statistical one.
A lot of naturally occurring phenomena are approximately normally distributed. One can argue
whether the underlying cause of that may be something like the CLT:


*

*Heights of people may be considered as the the sum of many smaller causes (perhaps independent, unlikely identically distributed): lengths of various bones, or results of various gene expressions, or results of many dietary
influences, or some combination of all of the above.

*Test scores may be considered as the sums of scores on many individual test questions (possibly identically distributed, unlikely entirely independent).

*Distance a particle travels in one dimension as a result of Brownian motion in a fluid: Motion may be considered abstractly as a random walk resulting from IID random hits by molecules.
One example where the CLT is not necessarily involved is the dispersion of shots around a bull's eye: The distance from the bull's eye can be modeled as a Rayleigh
distribution (proportional to square root of chi-sq with 2 DF) and the counterclockwise angle from the the positive horizontal axis can be modeled as uniform on $(0, 2\pi).$ Then after changing from polar to rectangular coordinates, distances in horizontal (x) and
vertical (y) directions turn out to be uncorrelated bivariate normal. [This is the essence of the Box-Muller transformation, which you can google.] However, the normal x and y coordinates might be considered as the sum of many small inaccuracies in targeting, which might justify a CLT-related mechanism in the background.
In a historical sense, the widespread use of normal (Gaussian) distributions instead of double-exponential (Laplace) distributions to model astronomical observations may be partly due to the CLT. In the early days of modeling errors of such observations, there was a debate between Gauss and Laplace, each arguing for his own favorite distribution. For various reasons, the normal model has won out. One can argue that one reason for the eventual success of the normal distribution was mathematical convenience based on normal limits of the CLT. This seems to be true even when it is unclear which family of distributions provides the better fit. (Even now, there are still astronomers who feel that the "one best observation" made by
a meticulous, respected astronomer is bound to be a better value than than the average of many observations made by presumably less-gifted observers. In effect, they would prefer no intervention at all by statisticians.)
A: Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies? 
