This, to me, is the most interesting part of the article.
I think this is what the German ADAC did when they falsified test results around a general "idea" what they wanted to see.
> [I]t has to do with relative growth/shrinkage and the base of the positional-numbering system you're using. If you have a random starting value (X) multiplied by a second random factor (Y), most of the time the result will start with a one.
> You're basically throwing darts at logarithmic graph paper! The area covered by squares which "start with 1" is larger than the area covered by square which "start with 9".
Most seem to match fairly closely. We accept pull requests with new datasets if anyone wants to contribute.
Assume stars are distributed roughly evenly in space. This is not true about the overall universe, they're clumped in galaxies, but fine for the dataset we're explaining, which maxes out at 3000 light-years from Earth.
Approximate the galaxy as a flat plane. This isn't a bad approximation since it's much flatter than it is wide.
Which stars are at a given distance D from earth? Those that lie on a circle of radius D centered at earth. But that count is just going to be proportional to the radius (circumference = 2pi times radius). So the count of stars does follow a power-law distribution, and that power is 1. There are KD stars at distance D, where K is a constant proportional to the overall density of stars in our neighborhood.
(I'm glossing over the fact that a circle is infinitely thin and stars are finite in number and so any given circle will probably have 0 stars on it.)
If you instead model the galaxy as a sphere, you get the same sort of result: count(stars) at distance D is proportional to the surface area of a sphere of radius D, 4pi*D^2. Still a power law. (The galaxy is about 1000 ly thick though, according to Wikipedia, and the data go out to 3000ly. So a sphere is probably not great here.)
So why do power law distributions tend to obey Benford's law? I'll punt on that: https://en.wikipedia.org/wiki/Benford%27s_law#Explanations
Correct me if I'm wrong, but I think any function with an increasing rate of change (ie. second derivative > 0) will yield a distribution with the same ordering of digits as Benford's if random numbers are taken from it.