Relation between the Chi-Squared and ANOVA Tests for Testing the Equality of k Independent Dichotomous Populations

This analysis may help to explain the "dark horses" of some tournaments whose playing ability had never previously been noted because they had been knocked out of play early in the competition by better players. It may also help to explain the lack of consistency in performance by "flash in the pan" players whose final rank in a tournament may have been due to the fortuitous elimination of the better players earlier in the competition. The analysis does bring out clearly the inherent inequities of the knockout tournament as compared to a round robin tournament. In the former, the players are exposed to the vagaries of chance through both the uncertainty of the initial alignment and the uncertainty of the individual encounters, while in the latter, since each player plays each other player an equal number of times, only the second source of unpredictability is present.