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I have been playing around with datasets for the past while for practice. I've noticed that a distribution that looks something like the following appears:

enter image description here

This shape appears frequently! I can guess it is the log-normal (because I am familiar with the tip-skew of the distribution). I do not understand why this distribution appears in nature so often.

Is there some intuition to why this comes up frequently?. My guess is that there is a natural skew in the data. The distribution above was for the prices of lunch bills. The positive skew here would be due to people wanting to pay less for their meals.

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    $\begingroup$ The lognormal can arise through multiplicative processes acting on Gaussian (normal) inputs. Skew here is the name of the problem, not the name of the solution, but many processes have lower bounds at or very near zero but no well defined upper limit $\endgroup$
    – Nick Cox
    Commented Mar 30, 2016 at 17:47
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    $\begingroup$ @Nick For the same reasons Gaussian distributions may appear "in nature" as sums of many small near-independent perturbations, none of which need be Gaussian, Lognormal distributions appear as products of many small near-independent perturbations: and again they needn't be Gaussian. $\endgroup$
    – whuber ♦
    Commented Mar 30, 2016 at 20:26
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    $\begingroup$ @whuber Naturally I agree. Indeed, how does the Gaussian arise? $\endgroup$
    – Nick Cox
    Commented Mar 30, 2016 at 20:44
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    $\begingroup$ A lognormal is always right-skewed (except in some parameterisations the normal is a limiting case). What you describe is left-skewed and thus some other distribution and not lognormal. Indeed I understand SAT to be bounded, so that's inevitable for that reason alone. The existence of other distributions shapes arises for other reasons. I have never heard grading described as a "natural" cause before. $\endgroup$
    – Nick Cox
    Commented Mar 30, 2016 at 21:04
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    $\begingroup$ From far being concrete, the best rationale is abstract, some kind of multiplicative spin on the central limit theorem. I don't know a general theory otherwise. You need to separate out reasons for skewness and reasons for lognormality, which your comments seem to conflate. Why not ask, why are many distributions gamma-like or Weibull-like, which their enthusiasts would want to underline? The selection process is that theorists propose distributions as more or less tractable functions with total probability 1 and practical people push those easy to fit and which seem to work some of the time. $\endgroup$
    – Nick Cox
    Commented Mar 30, 2016 at 21:35

1 Answer 1

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As observed in comments, the lognormal distribution can arise as a kind of multiplicative central limit process. Formalizing, assume $Z_1, Z_2, \dotsc, Z_n$ are iid positive random variables such that the expectation and variance of $\log Z_i$ exists. We are interested in the (limiting) distribution of $Z=\prod_1^n Z_i$. Write $Y = \log Z = \sum_1^n \log Z_i$. Now we can apply the usual central limit theorem for the sum of $\log Z_i$, obtaining an (limiting) normal distribution for $Y$. It follows that $Z=e^Y$ will have a (limiting) lognormal distribution.

A paper with many good examples is https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

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