# What Numbers Reveal -- From Sumo Wrestlers to Professors

Looking at large data sets and deriving loud conclusions from the reams of whispering numbers is often enjoyable. Herein are three quite disparate examples.

The first concerns sumo wrestlers and comes from "Freakonomics," a fascinating new book by economist Steven Levitt and writer Stephen Dubner, that employs Levitt's quirky economic insights to illuminate many everyday activities and practices. The second is simply a study I reported on in a book I wrote on the stock market, and the third comes from a simple analysis I recently made of grade distributions for a required math course at my university.

As Levitt's analysis makes clear, Sumo wrestling is a sport in which the best practitioners earn big bucks and the journeymen earn far, far less. Among the topmost tier of wrestlers, those who have a winning percentage in a set of 15 crucial tournament matches qualify for the larger pay and numerous other perks. Drawing on an extensive data set of such contests, Levitt focuses on the outcomes of certain pivotal matches in which sumo wrestlers in the elite category have a 7-7 record (and thus are on the cusp of a winning 8-7 winning percentage). In particular, they check to see what happens when these wrestlers face those with an 8-6 record in the tournament. The latter have already made the cut-off, but have no chance for the top prizes.

The outcomes of these crucial matches strongly suggest collusion. Much more often than one would expect, the matches result in the 7-7 wrestlers beating the 8-6 wrestlers. In fact, relying on the record of past matches between the wrestlers, one might reasonably estimate that the chances 7-7 wrestlers would beat 8-6 wrestlers to be slightly under 50 percent, rather than the 80 percent rate at which they do win in the crucial matches.

Furthermore, as Levitt stresses, when these same pairs of wrestlers meet in a subsequent tournament match that is not crucial, the wrestlers who had the 7-7 records in the previous tournament win only 40 percent of the time against the wrestlers with the 8-6 records in the previous tournament. That is, they win 80 percent of the time when they have much to gain and the other wrestler has little to lose and then win only 40 percent of their next matches. This and other evidence gleaned from the voluminous records and frequent matches among the wrestlers in the top tier strongly suggest that collusion among the wrestlers is not rare.

#### Earnings -- Pennies From Heaven

Another example of unlikely results in crucial cases involves the quarterly earnings that companies announce. Will they meet the estimates analysts have established for them? When companies' earnings fall short by a penny or two per share, investors sometimes react as if this were tantamount to near-bankruptcy, and when they exceed them investors are often inordinately pleased.

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