On fractals and statistics

Just what's on my mind today:
What is the connection between the Mandelbrot set and the bell-shaped normal distribution curve, or any distribution for that matter?

This is something I have been thinking about for some time. I am surprised that fractals are not used to describe patterns.
It came to me today while on my bike to work: Fractals are an object thing, and describe the object through all scales; normal distribution or statistics need a parameter to function.
example: Stars are fractal, and will not distribute without a parameter: when we add say star size, star colour or distance, we get a distribution.
So I believe there is a very close relationship between the two - what is interesting is that distribution patterns are very fractal, absolutely universal.

It is a goal of mine to understand this more: for there is more to it.

Update Feb 2020
It was some time ago when I wrote the above; it is coming back to me know as I know more and have more questions.
What I am talking about above is the difference between power laws and normal distribution. And I think through the fractal I can unify them. Power laws are the direct stuff of fractals (scale invariance) and normal distribution, as I tried to claim, to do with parameters. I think I can explain why Pi shows up in normal distribution; it is that systems are fractal.


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