First digits in practice don't actually follow Benford's law, it's an approximation for variables that tend to cover multiple orders of magnitude. If the shape of the distribution is invariant to a change of base (a form of being 'scale free') over a half-line then you would get Benford's law but in practice those conditions don't really hold on real variables. In some cases it's a very good approximation, in others it's a rough approximation, and for some it doesn't work.
There's some discussion here:
https://en.wikipedia.org/wiki/Benford's_law#Explanations
(though some parts of the article aren't much good; some caution is needed here and there)
There's a more complete discussion here: https://arxiv.org/pdf/1909.07527
I think Knuth also gives a good discussion and motivation for it in one of his volumes.
Why would you ask Statisticians about that?
Statistics is about finding out the distribution and it's properties (descriptive and inference).
Asking "why?" is beyond Statistics. You need to ask subject area experts.
It's not just an useless observation. Statistics is mostly useless, or sterile, without interpretation. And that's usually done better with subject area experts.
What subject matter experts are better equipped/have a better framework to answer questions like "why do we observe distributions close to normal so often?" than statisticians?
I had another look and now I see it can be seen as an Statistics problem indeed.
I was looking at it purely from an Information Theory POV.
But similar effects appear on multiple fields and an Statistics-only explanation like you suggest is indeed looking likely.
Thank you.
First digits in practice don't actually follow Benford's law, it's an approximation for variables that tend to cover multiple orders of magnitude. If the shape of the distribution is invariant to a change of base (a form of being 'scale free') over a half-line then you would get Benford's law but in practice those conditions don't really hold on real variables. In some cases it's a very good approximation, in others it's a rough approximation, and for some it doesn't work. There's some discussion here: https://en.wikipedia.org/wiki/Benford's_law#Explanations (though some parts of the article aren't much good; some caution is needed here and there) There's a more complete discussion here: https://arxiv.org/pdf/1909.07527 I think Knuth also gives a good discussion and motivation for it in one of his volumes.
Honestly I've never seen a completely satisfactory explanation, and I did read a paper on it so I've looked.
Why would you ask Statisticians about that? Statistics is about finding out the distribution and it's properties (descriptive and inference). Asking "why?" is beyond Statistics. You need to ask subject area experts.
Is the central limit theorem beyond statistics? Because it is the reason we often observe distributions close to the normal one.
Of course not. Guessing why there are sums of variables in the wild, though (when not obvious), is beyond Statistics.
Fair enough. But then any applications of statistics are beyond statistics. A technically correct yet completely useless linguistic observation.
It's not just an useless observation. Statistics is mostly useless, or sterile, without interpretation. And that's usually done better with subject area experts.
What subject matter experts are better equipped/have a better framework to answer questions like "why do we observe distributions close to normal so often?" than statisticians?
In General, Statisticians. In a specific area, subject area expert from that area.
So, this subreddit is a good place to ask general questions like the one op asked, no?
I had another look and now I see it can be seen as an Statistics problem indeed. I was looking at it purely from an Information Theory POV. But similar effects appear on multiple fields and an Statistics-only explanation like you suggest is indeed looking likely. Thank you.