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I am excluding the case where someone generates a normal distribution by a Human, non Nature process.

Specifically to determine if humans are managing outcomes to achieve some result, say a higher score, wouldn't the distribution be positively skewed? Where a normal distribution might imply that either no human interference is involved or it is ineffective.

Edit: Sorry for the confusion. By natural I meant Non Human. I should have asked: what information can I glean from the lack of a skewed distribution (Normal Distribution) OR from a + of - skewed distribution? Is the only implication of a skewed distribution that my data sample is not from a single homogenous population?

burt
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    What do you mean by a non-natural process? This is not entirely clear to me. Also... Why do you believe that non-natural processes can't result in normal distributed variables (and as a consequence a normal distribution implies a natural process)? – Sextus Empiricus Sep 16 '20 at 20:42
  • Re "only implication:" skewness does not, of itself, imply a population is "inhomogeneous." – whuber Sep 18 '20 at 16:42

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Great question. There are many natural processes that exist, and that is a reason why the normal distribution is so popular and abundant in statistics, life-sciences, etc. However, broad statements like that require caution.

For example, I would simply add that the reverse is not true. That is, there are many 'natural' processes that are non-normal.

In genomics, gene expression is measured when transcription occurs inside the cell, producing RNA that can be measured by the total amount of transcripts produced by that gene. This results in count data, which is usually modeled by Poisson or Negative Binomial regression. There are transformations that can help lead to normality, but the data itself is non-normal.

There are plenty of other examples, but just wanted to point out natural, non-normal processes exist, thus the implication is one-sided and not both-sided.

Samir Rachid Zaim
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  • To claim there may be exceptions, when focusing on a single gene RNA as noted in human-based experiments (contrary to the text of the question) does not deny my comment of at least consistency. – AJKOER Sep 16 '20 at 23:59
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Yes, the normal distribution is somewhat 'natural' due to its finite variance, but perhaps more of a general averaging construct. To quote a source:

The average of a sequence of values from any distribution will approach the normal distribution, provided the original distribution has finite variance... The condition of finite variance is true for almost all systems of practical interest.

And further:

What is the engineering significance of this averaging process (which is really just a weighted sum)? Many of the quantities we measure are bulk properties, such as viscosity, density, or particle size. We can conceptually imagine that the bulk property measured is the combination of the same property, measured on smaller and smaller components. Even if the value measured on the smaller component is not normally distributed, the bulk property will be as if it came from a normal distribution.

So, per the question, 'Does a Normal distribution imply a natural process?', my answer is that [EDIT], at least,[END EDIT] it is certainly consistent with naturally occurring bulk processes.

[EDIT] Further, I would argue that the question is clear, not even a repeated human orchestrated homogeneous reaction system with varying parameters, but, as occurs in nature, so-called heterogeneous chemistry. So particle sizes, for example, are further averaged over many compounds.

To claim such a natural heterogeneous system is not at least consistent with a so-called normal distribution associated averaging process is suspect in my view.

AJKOER
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    The implicit assumption, which is incorrect, is that "naturally occurring bulk process" will produce normally distributed values. – whuber Sep 16 '20 at 16:49
  • Correction on Whuber assertion use of the verbs "will produce". Clearly stated in my last sentence, the phrase "is certainly consistent" with naturally occurring bulk processes. Being consistent is hardly arguing any definiteness as mis-asserted by the word "producing". I will edit further adding the words "at least". – AJKOER Sep 16 '20 at 20:35
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    This is probably a matter of opinion, but plenty of people have found--through experience--that many (if not most) "natural processes" don't produce anything like Normal data. I'm sure the sense of "most" is rather personal and tailored to one's specialty and experiences: but at the least, so that you do not mislead many readers, that implies you need to be much more specific about your statement and what you mean by "naturally occurring bulk processes." – whuber Sep 16 '20 at 20:39
  • The bulk property being normal distributed is only the case when the bulk stems from a sum of many little components. There are possibilities that break this situation. The bulk property might stem from a product of variables (in [which case](https://stats.stackexchange.com/questions/480967) the distribution will be log-normal distributed). Another way that you do not get a normal distribution is when you have a mixture distribution. For instance the bulk property might be a mixture of components, e.g. the strength of chocolate depends on the state of the crystals and will be multimodal. – Sextus Empiricus Sep 16 '20 at 20:52
  • Well, one can disagree with the claimed engineering-related conceptualization of "naturally occurring bulk processes" as an "averaging process", per my cited source, not myself. The latter bulk processes are only briefly enumerated, for example, as "viscosity, density, or particle size". – AJKOER Sep 16 '20 at 20:53
  • Another way is when the bulk property is some function of another bulk property. That's how spectral line intensity can follow a Lorentz distribution (Cauchy distribution for statisticians). In the case of particle size distributions one often finds distributions that vary a lot from a normal distribution (because the steps in the random growth process and the different particles are not independent, e.g. think about Ostwald ripening). – Sextus Empiricus Sep 16 '20 at 20:54
  • Sextus Empiricus: Staying with the particle size distribution, if one is manufacturing nano-particles, the particle size distribution can be a function of annealing temperature and duration of annealing. Repeating production with varying conditions, could per my opening comment derived from the CLT, to quote: "The average of a sequence of values from any distribution will approach the normal distribution, provided the original distribution has finite variance". – AJKOER Sep 16 '20 at 21:28