Getting to the core of the Central Limit Theorem

A previous post has used this cornerstone of statistics, the Central Limit Theorem, but did not explain it fully (See my post of July 30, 2011: the Central Limit Theory.). In George G. Szpiro, Pricing the Future: finance, physics, and the 300-year journey to the Black-Scholes equation (Basic Books 2011) at 186, Szpiro writes: “The theorem says that when many independent random variables combine, the resulted is distributed according to the bell-shaped curve.”

To illustrate: invoices arrive to be paid by law departments for all kinds of unpredictable (random) reasons (independent of each other). If you graph the distribution of those invoice amounts the shape will look much like the well-known bell. A few very small, a few very large, a bulging cluster around the average amount. Slightly more than two-thirds of all the invoices will fall within one so-called standard deviation on both sides of the bell’s center, which is about where the bell starts to curve outward.

Or consider lawsuits. Their number, timing, and differences each quarter over the past five years are to a fair degree determined by random and unrelated forces. The Central Limit Theory predicts that the quarterly totals will look like a bell-curve distribution if sorted high to low, and more so if the company has lots of litigation.