First thing to understand is that there's hierarchy of statistical distributions, our high school friend Gaussian being near the friendly bottom.
Unless you understand the underlying process, like in some areas of physics, you can measure all you like, but you can't prove something is in the lower left. You can however *disprove* it, often with a single observation (after you check your instruments). Falsification at work.
As you disprove friendly distributions like Gaussian, you move up.
Which brings us to the S&P chart. If it was Gaussian, it should stay under the blue line. It's not even close, so it must a scarier distribution. But what's in the chart?
Well we can plough through 3 pages of Taleb's math, or check Wikipedia for (non nuclear war) MAD: "In other words, for a normal distribution, mean absolute deviation is about 0.8 times the standard deviation"
That's where the sqrt(π/2) blue line comes from. QED-ish; not a normal distribution.
@stevenroose maybe just start with Fooled by Randomness; no math. I read Talebs books out of order and read this one last. Funny to notice how he suddenly became much friendlier 🙂
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