On fractals and statistics

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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?

I have been thinking about this for some time. I am surprised that fractals are not used to describe patterns.
It came to me today while I was biking to work: Fractals are objects that describe the object through all scales; normal distributions or statistics need a parameter to function.
For 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 and absolutely universal.

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

Update Feb 2020
I wrote the above some time ago, but it is coming back to me now as I know more and have more questions.
I am talking about the difference between power laws and normal distribution. And I think I can unify them through the fractal. 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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