You might be thinking I did something tricky with the math - that ON AVERAGE, most randomness does not behave this way. But what makes you so sure? In fact, 'Wild Randomness', or heavy-tailed distributions may in fact be the norm, and our thinking about what average is may be wrong. Read Mandelbrot's Misbehavior of Markets, or google Long-Tailed distributions if you want to really rock your world.
In fact, often times it is the outliers that define the norm. Maybe that's true in business. Maybe that's true in Ultimate Frisbee. Maybe it's true of quantities like Intelligence, whatever that is, which are allowed to grow and compound. Maybe it's true of income distributions.
So the takeaway is - next time you catch yourself using the term 'on average' - recognize that even Statisticians don't pin themselves down to one definition of average. And if you have the Calculus chops to investigate moments, and *prove* that the average of the Cauchy distribution does not 'settle down' like that of the normal distribution - please drop me a line with a write-up or video, and I'll shoot you something fun back.
Cheers, and Happy Math-ing!
Dean, President Deanius Solutions