Monday, April 23, 2012

SFS Swimming: Conspiracy? Part 2

In the previous post, I introduced data showing a possible referee bias against the Toledo St. Francis de Sales (SFS) swim team.  In this post, I will use the binomial distribution to show how unlikely it is that SFS would be DQ'ed in 5 of the 61 relay races (8.2%) at the state tournament.

First, let's see how likely these 5 DQ's for one team is compared to all boy's Division 1 relay swims.  I previously showed that the frequency of DQ's for all non-SFS teams is 22/1259 (1.7%). Using the Binomial distribution with n = sample size = 61 and p = probability of DQ = 22/1259, the probability of a team being DQ'ed 5 or more times is 0.0043.  This means that if 1,000 teams would each swim 61 relay races at the state tournament, we would only expect 4 of the 1,000 to be DQ'ed at least 5 times.  This seems very improbable, so we can conclude that the probability of being DQ'ed 5 times out of 61 cannot be explained by random chance.

In the last post, I also compared SFS to the other 4 "perennial contenders", teams that have finished the in the top 3 team standings at least 3 times in the past 10 years.  I previously argued that these teams are well coached and accustomed to regularly winning, so it makes sense that these teams should be DQ'ed less frequently than all state relay teams combined.  I showed that this is true, as the frequency of a relay DQ for the non-SFS teams is 1.3%.  Using the binomial distribution again, the probability that a perennial contender would be DQ'ed 5 out of 61 swims is 0.0012. This means that if 1,000 perennial contender teams would each swim 61 relay races at the state tournament, we would only expect 1 of the 1,000 to be DQ'ed at least 5 times.  This is even more improbable than the previous analysis, so we can conclude that the probability of a perennial contender being DQ'ed 5 times out of 61 cannot be explained by random chance.

I have included the following plot to show the probability distribution of a perennial contender being  DQ'ed out of 61 relay swims.  This shows how highly unlikely it is that a perennial contender is DQ'ed even 4 times out of 61.

This analysis seems to confirm the suspicions that all SFS swimmers the past decade have had: that there is a referee bias against us.  However, the binomial distribution assumes that all trials (relay races) are independent.  This is surely not true, as many swimmers swim in multiple relays.  Additionally, each team has a different coach, so they may have different strategies and been trained differently.  I will discuss this last point in more detail in the next post.

The final part of this trilogy will try to identify other possible explanations for this large deviation from the expected number of DQ's for a perennial contender.

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