Mediocristan, Extremistan, and the Turkey Problem

There are two types of randomness: Mediocristan (where extremes are bounded, like height -- no single person is so tall they change the average of 1,000 people) and Extremistan (where a single observation can dominate, like wealth -- add Bill Gates to a room and the average explodes).

Consider a turkey that is fed every day. Every single feeding will firm up the bird's belief that it is the general rule of life to be fed every day. On the afternoon of the Wednesday before Thanksgiving, something unexpected will happen to the turkey. It will incur a revision of belief.

City sizes follow a power law distribution (Zipf's law), where the largest city is roughly twice the size of the second largest, three times the third, and so on -- an example of Extremistan dynamics.

The Central Limit Theorem, which underpins much of conventional statistics, requires finite variance. When variance is infinite -- as with Cauchy or Levy-stable distributions -- the CLT breaks down entirely, and averages do not converge.

Mandelbrot discovered in 1963 that cotton price variations follow a Levy-stable distribution with parameter alpha of approximately 1.7, not the Gaussian distribution (alpha = 2), meaning infinite variance and much fatter tails than standard models assume.

Galton's original 1886 study of hereditary stature found that children's heights regress toward the population mean -- tall parents tend to have children shorter than themselves, and short parents tend to have taller children, with a regression ratio of approximately 2/3.

Wealth distribution follows Extremistan dynamics -- the Pareto principle (80/20 rule) shows that a small fraction of the population controls a disproportionate share of total wealth, and a single individual can distort the average of any group.

Taleb's Mediocristan/Extremistan distinction maps onto the mathematical distinction between thin-tailed distributions (Gaussian, exponential) where the Law of Large Numbers converges quickly, and fat-tailed distributions (power law, Cauchy) where convergence is slow or nonexistent.