Normal distribution and monotonic transformations I've heard that a lot of quantities that occur in nature are normally distributed. This is typically justified using the central limit theorem, which says that when you average a large number of iid random variables, you get a normal distribution. So, for instance, a trait that is determined by the additive effect of a large number of genes may be approximately normally distributed since the gene values may behave roughly like iid random variables.
Now, what confuses me is that the property of being normally distributed is clearly not invariant under monotonic transformations. So, if there are two ways of measuring something that are related by a monotonic transformation, they are unlikely to both be normally distributed (unless that monotonic transformation is linear). For instance, we can measure the sizes of raindrops by diameter, by surface area, or by volume. Assuming similar shapes for all raindrops, the surface area is proportional to the square of the diameter, and the volume is proportional to the cube of the diameter. So all of these ways of measuring cannot be normally distributed.
So my question is whether the particular way of scaling (i.e., the particular choice of monotonic transformation) under which the distribution does become normal, must carry a physical significance. For instance, should heights be normally distributed or the square of height, or the logarithm of height, or the square root of height? Is there a way of answering that question by understanding the processes that affect height?
 A: Very good question. I feel that the answer depends on the whether you can identify the underlying process that gives rise to the measurement in question. If for example, you have evidence that height is a linear combination of several factors (e.g., height of parents, height of grandparents etc) then it would be natural to assume that height is normally distributed. On the other hand if you have evidence or perhaps even theory that the log of height is a linear combination of several variables (e.g., log parents heights, log of grandparents heights etc) then the log of height will be normally distributed.
In most situations, we do not know the underlying process that drives the measurement of interest. Thus, we can do one of several things:
(a) If the empirical distribution of heights looks normal then we use a the normal density for further analysis which implicitly assumes that height is a linear combination of several variables.
(b) If the empirical distribution does not look normal then we can try some transformation as suggested by mbq (e.g. log(height)). In this case we implicitly assume that the transformed variable (i.e., log(height)) is a linear combination of several variables.
(c) If (a) or (b) do not help then we have to abandon the advantages that CLT and an assumption of normality give us and model the variable using some other distribution.
A: The rescaling of a particular variable should, when possible, relate to some comprehensible scale for the reason that it helps make the resulting model interpretable.  However, the resulting transformation need not absolutely carry a physical significance.  Essentially you have to engage in a trade off between the violation of the normality assumption and the interpretability of your model.  What I like to do in these situations is have the original data, data transformed in a way that makes sense, and the data transformed in a way that is most normal.  If the data transformed in a way that makes sense is the same as the results when the data is transformed in a way that makes it most normal, I report it in a way that is interpretable with a side note that the results are the same in the case of the optimally transformed (and/or untransformed) data.  When the untransformed data is behaving particularly poorly, I conduct my analyses with the transformed data but do my best to report the results in untransformed units.
Also, I think you have a misconception in your statement that "quantities that occur in nature are normally distributed".  This only holds true in cases where the value is "determined by the additive effect of a large number" of independent factors.  That is, means and sums are normally distributed regardless of the underlying distribution from which they draw, where as individual values are not expected to be normally distributed.  As was of example, individual draws from a binomial distribution do not look at all normal, but a distribution of the sums of 30 draws from a binomial distribution does look rather normal.
A: I must admit that I do not really understand your question:


*

*your raindrops example is not very satisfying since this is not illustrating the fact that the Gaussian behaviour comes from the "average of a large number of iid random variables".

*if the quantity $X$ that you are interested in is an average $\frac{Y_1+\ldots+Y_N}{N}$ that fluctuates around its mean in a Gaussian way, you can also expect that $\frac{f(Y_1)+\ldots+f(Y_N)}{N}$ has a Gaussian behaviour.

*if the fluctuation of $X$ around its mean are approximately Gaussian and small, then so are the fluctuation of $f(X)$ around its mean (by Taylor expansion)

*could you cite some true examples of (real life) Gaussian behaviour coming from averaging: this is not very common! Gaussian behaviour is often used in statistics as a first rough approximation because the computations are very tractable. As physicists uses the harmonic approximation, statisticians uses the Gaussian approximation.
A: Vipul, you're not being totally precise in your question. 

This is typically justified using the
  central limit theorem, which says that
  when you average a large number of iid
  random variables, you get a normal
  distribution.

I'm not entirely sure this is what you're saying, but keep in mind that the raindrops in your example are not iid random variables. The mean calculated by sampling a certain number of those raindrops is a random variables, and as the means are calculated using a large enough sample size, the distribution of that sample mean is normal. 
The law of large numbers says that the value of that sample mean converges to the average value of the population (strong or weak depending on type of convergence). 
The CLT says that the sample mean, call it XM(n), which is a random variable, has a distribution, say G(n). As n approaches infintity, that distribution is the normal distribution. CLT is all about convergence in distribution, not a basic concept. 
The observations you draw (diameter, area, volume) don't have to be normal at all. They probably won't be if you plot them. But, the sample mean from taking all three observations will have a normal distribution. And, the volume won't be the cube of the diameter, nor will the area be the square of the diameter. The square of the sums is not going to be the sum of the squares, unless you get oddly lucky. 
A: I think you missunderstood (half of) the use statistician make of the normal distribution but I really like your question. 
I don't think it is a good idea to assume systematically normality and I  admit it is done sometime (maybe because the normal distribution is tractable, unimodal ...) without verification. Hence your remark about monotonic map is excellent ! 
However the powerfull use of normality comes when you construct yourself new statistics such as the one that appears when you apply the empiriral counter part of expectation: the empirical mean. Hence empirical mean and more generally smoothing is what makes normality appear everywhere...  
A: Simply CLT (nor any other theorem) does not state that every quantity in the universe is normally distributed. Indeed, statisticians often use monotonic transformations to improve normality, so they could use their favorite tools.
A: Both a random variable and many transformations of it can be approximately normal; indeed if the variance is small compared to the mean, it can be that a very wide variety of transformations look pretty normal.
> a<-rgamma(10000,1000,1000)
> hist(a)
> hist(1/a)
> hist(a^2)
> hist(a^(3/2))


(click for larger version)
