Mathy Links

1. Econophysics is a real thing (learnt about it here)

2. Why \(i^i\) is real!

3. Riffing on mistakes (John D. Cook)

Central Limit Theorem and Wisdom of Crowds

There is an illuminating correspondence between the central limit theorem and the notion of wisdom of crowds.

Informally, the central limit theorem says that the average of many independent trials is normally distributed around the true mean. A similar idea that animates the wisdom of crowds: consensus estimates obtained by aggregating or averaging the estimates of a number of individuals are superior.

The central limit theorem, while powerful, rests on three premises: the trials are independent, identically distributed, and have finite variance. If any of these three conditions are violated, then all guarantees are void.

In the context of the wisdom of crowds, the three premises loosely translate as follows: independent means that each trial (or individual estimate) does not know or care about other trials or estimates; identically distributed implies they are looking at the same underlying problem; and finite variance implies estimates are not so stupidly wild, that a single bad trial or estimate has an outsized effect on the average.

Joel Greenblatt presents a fascinating experiment in which he brought a jar of jelly beans into a class, and asked students to guess the number of beans, and to write it down on a piece of paper. The average of the student estimates was surprisingly close to the true number of the jelly beans in the jar.

But before he disclosed this fact to the class, he repeated the experiment. This time, instead of independently writing down their estimates, students were instructed to announce their guesses to the class. The estimates were no longer independent, and the consensus was way off.

QuickTip: LaTeX formatting for SymPy symbols

You can use LaTeX formatting to describe SymPy variables using:

Z, a, Zbar = symbols(r'Z, \alpha, \bar{Z}')