February 7, 2012

# Superbowl statistics

American football games, like many sporting events, start with a coin toss, in this case to decide which team is playing in which direction.   At the last 14 Superbowls, the team from the National Football Conference has won the toss (via).  In a standard test of the hypothesis that the coin was fair, the p-value would be 0.0001.  So, does this mean the NFC is cheating? Well, no.  We have overwhelmingly good reasons to believe that coin tosses are very close to fair, and a mere 1 in 8000 coincidence shouldn’t change our minds.   As Tom Stoppard put it in  Rosencrantz and Guildensten Are Dead: “A spectacular vindication of the principle that each coin, spun individually, is just as likely to come up head as tails, and should cause no surprise each individual time it does.”

The generalization of this principle to studies purporting to find small, but statistically significant, benefits of homeopathy is left as an exercise to the reader.

Thomas Lumley (@tslumley) is Professor of Biostatistics at the University of Auckland. His research interests include semiparametric models, survey sampling, statistical computing, foundations of statistics, and whatever methodological problems his medical collaborators come up with. He also blogs at Biased and Inefficient See all posts by Thomas Lumley »

• The probabilities given are wrong. Instead of calculating the odds of 13 heads in a row you should be calculating the probability that a sequence of 46 coin tosses has a run of length 14 in it. That will be less than 1 in 2^13.

The more Superbowls you have, the more likely it is that you get a long run of tosses all the same.

• Thomas Lumley

Tony,

1. By simulation, the probability based on the run length is about 0.002, so it’s larger but still impressive.

2. On what principle can you extend this to all Superbowl coin tosses, but not all NFC vs AFC coin tosses, or all sporting grand final coin tosses, or more sensibly, all coin tosses not performed by professional magicians?

• “We have overwhelmingly good reasons to believe that coin tosses are very close to fair, and a mere 1 in 8000 coincidence shouldn’t change our minds.”

Yep – we know a lot about coin tosses, not just this particular data, and we should take all the information into account. Also, this is obviously cherry-picked data, and there’s no guarantee that you’ll get good inferences by choosing arbitrary subsets of the information and then performing the inference as if it was all the information that you had.