A thought about contests

I was thinking of Broadway or Bust, the PBS special made about the National High School Musical Theater Awards of 2012. 50,000 high school students ranging in age from 15 to 18 took part in the contest. The winning boy and the winning girl from each of 30 regional contests went to New York to compete in the final. From these 60 kids, three boys and three girls were judged to have performed best in the preliminaries, and from them the winning boy and the winning girl were chosen.

U.S. News this year ranked 22,000 high schools in the United States, noting that they did not have data on schools in one state and that the data was not complete for the states for which they did have data. For simplicity, lets assume that there are about 25,000 high schools in the United States. That means that on average, each high school had two students enter the contest. Magnet schools for the performing arts may have entered a number of students while other high schools may not have had any students enter.

There were nearly 15 million high school students this year. Thus on the average, a high school had about 600 students. Think about the musical put on in your high school each year. How talented were the best performers in that musical? Pretty talented I bet. The odds against the best performer in a high school musical winning one of the 30 regional contests were greater than 800 to 1! The odds of winning the contest over all were 12,500 to 1. All of the 60 participants in the Broadway part of the contest were fantastically talented -- many with almost professional training on arrival.

Of course, the 15 year olds probably have not developed their performance skills as well as they will have at age 18. Moreover, the kids have chosen things to perform, based on those that they have performed previously; there is luck in the draw in terms of its impact on the judges. Moreover, there is chance in how well the rehearsals and actual performances went, and chance in the accuracy of the judge's evaluation of the performances. The likelihood that the most talented kid would actually win seems very small to me -- better than one chance in 12,500 but how much better?

This is borne out by the fact that Joshua Grosso, who was one of the two winners, did not win his regional contest. He came in second in the regional, and got to New York when the regional winner was unable to make the trip.

This is not to take anything away from the achievement of the winners, who managed to perform at a fantastic peak repeatedly, and did that which most impressed the judges on the day.