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# Sports Seasons Based on Score Distributions

How does your sports team's success over a season vary according to the statistical distribution of its scores and those of its opponents? This Demonstration from the field of sabremetrics lets you select two parameters for your team and for its adversary that describe the statistical distribution of scores. It then determines for a user-selected number of sample seasons the number of wins and (where permitted) the number of ties of your team over all of those seasons. It produces histograms showing the distribution of scores for your team and its opponent. It also produces histograms of the number of wins and ties over each of the sample seasons. You can select the type of two-parameter distribution from which scores will be drawn, the number of games in a season, and whether games are permitted to end in a tie. It is worth noting that the distribution of wins is likely to have a standard deviation that will prove significant in a competitive league, that is, there may be a fair amount of "luck" in determining your team's standings.

### DETAILS

The Demonstration is faster if the number of seasons sampled is not set to 100.
There is good evidence that the number of runs scored by an American major league baseball team is Weibull-distributed.
Baseball's "Pythagorean theorem" suggests that the percentage of wins over a season should be , where is the number of runs you score and is the number of runs your opponent scores. The exponent γ is generally estimated to be about 1.79. This Demonstration suggests that the standard deviation of the distribution of runs may matter, too.
A leading academic work on this topic is "A Derivation of the Pythagorean Won-Loss Formula in Baseball" by Steven J. Miller.
Snapshot 1: what might happen if a baseball games were doubleheaders in which each game lasted a fewer number of innings
Snapshot 2: simulating an English Premier League 38-game season; "your" team based loosely on Manchester United for 2008
Snapshot 3: in a short "season" such as a baseball World Series, the better team (measured by average score) does not always win

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