March Madness: It Didn't Happen Yet Again

It didn't happen yet again. In last month's NCAA "March Madness" men's college basketball championship, a number of 16 seeds have again failed to best the number 1 seeds. The histogram above shows the score differences in such matchings since 1985. It closely matches a normal distribution, allowing for us to estimate the probability that such an upset could happen as the area under the approximating bell-curve that falls beyond zero. Our estimated probability that a number 16 seed would beat a number 1 seed is 0.0208. It has risen a bit since our last view.

Deflater Mess

The statistical rumble over the fumble continues. Did the first analysis use the wrong statistic (similar to miles per gallon versus the gallons per mile discussions of a few years ago), the wrong comparison distribution, or did it not account for the myriad ways and opportunities for fumbling the football? Above is the histogram of fumbles per play (the reciprocal of plays per fumble in the first analysis). The New England Patriots are still separated, now on the left, from the rest of the league, but now they do not look as extreme. The histogram looks much more like what might be obtained from a normal distribution. Now the team is 2.45 standard deviations below the mean (as compared to 3.9 standard deviations above the mean using the plays per fumble statistic). Such results would occur slightly less than 1% of the time. Certainly not the one in 16,000 chance that the plays per fumble would suggest. What we have here is a statistical "teachable moment."

Super Hold

As the build-up to next Sunday's NFL Super Bowl continues, more evidence is surfacing about the amazing ability of the New England Patriots to hold onto the football. First, there was the football deflation controversy, termed InflateGate by some and Ballghazi by others. Somehow 11 of the 12 balls that the Patriots used in the first half of their rainy game against the Indianapolis Colts were under-inflated by as much as 2 psi. The twelfth ball was also under-inflated but not by as much. It is claimed that this makes the ball easier to hold onto, especially in the rain.

Now, from football data analyst Warren Sharp we get the histogram above. This shows data from 2010-2014 on the number of offensive plays run divided by the number of fumbles lost. This ratio of Plays per Fumble for the Patriots was 187, far out in the tail of the histogram. Based on Sharp's data on the 32 NFL teams, the mean number of Plays per Fumble was just over 105 and the standard deviation was just over 21. This makes the Patriots' value of 187 over 3.9 standard deviations above the mean. Assuming a normal distribution for this measure, as Sharp reported, such results have only a one chance in over 16,000 of occurring. A clear outlier.

Have they practiced and honed the skill of holding onto the football or is possibly deflation also at work here?

Probability WONK

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Robert Jernigan WONK Challenge from American University on Vimeo.
I finally saw my American University WONK Challenge Spot on the Jumbotron at the Washington Nationals game on August 27. Here's me pointing, and the Nationals won!

Foul balls have really hit the news lately with a fan in Cleveland catching 4 in one game this last month! As I mention in the spot some put the probability of catching a foul ball in any game at about  1 in 1000. This, of course, varies with where you sit. Defending or attacking this figure was not possible in such a short spot, so if we accept it, we compute the probability that you catch at least one foul ball in say, n, games. We can compute this probability by first finding the probability of its complement.  The complementary event of catching at least one foul ball is catching no foul balls. In one game our chance of not catching a foul ball is 1-0.001=0.999. If our catching a foul ball is independent from game to game, then our chance of not catching a foul in n games is (0.999)ⁿ. Subtracting this from one, we get the probability of catching at least one foul ball in n games: 1- (0.999)ⁿ. If we want this result to be at least 50-50 (that is, 0.50) we need to find the value of n so that: 0.50 < =1- (0.999)ⁿ. You can do this by trial and error on a calculator or by using logarithms to solve for n. This will be the number of home games you must attend to increase your chances of catching at least one foul ball to at least 50-50. Now convert this to seasons of home games. There are 162 games in a season, but only 81 are home games. You should get an answer of 8 home seasons plus about half of a ninth season, hence choice B in the video.

This was fun to do.

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Via Deadspin,

March Madness: It Didn't Happen Again!

Since 1985, when the NCAA Division I Men's Basketball Tournament expanded to 64 teams, a team seeded (or ranked) 16 (last in each of the four regions) has never beaten a team seeded first. Over the 27 years from 1985 to 2011, (108 pairings) the number 1 seed has won by as much as 58 points (in 1998, Kansas(1) over Prairie View(16) 110-52). But twice over that time, both times in 1989, the number 1 seed narrowly edged out the number 16 seed by only 1 point (Georgetown(1) over Princeton(16) 50-49 and Oklahoma(1) over East Tennessee State(16) 72-71.

The histogram above shows the winning point margins. It closely fits a normal distribution with a mean of about 25 and a standard deviation of about 12. This gives us an estimate of the probability that a 16 seed could beat a number 1 seed, as just the probability that this normal distribution is less than zero. Here, we get a value of about 2%.

Wait 'til next year!

P.S. (3/20/2011)
A colleague observed that since a 16 seed has never won over a 1 seed in 108 trials of men's basketball (although it has happened in women's in 1998, 16 seed Harvard over 1 seed Stanford 71-67) that a 2% probability of an upset is perhaps too high and the probability should be less than 1% (1/108). But this treats a trial as only a success or failure and ignores how close the score was to an upset. A value near 1% is likely too small.

Cycle Man

On Friday, June 29, 2007 Aubrey Huff first baseman for the Baltimore Orioles hit for the cycle. In one game, he had a single, a double, a triple, and a homerun. This has happened 276 times in major league history, only three times for the Baltimore Orioles and it is the first time it has happened at home in Baltimore and at Oriole Park at Camden Yards. In the major leagues he is the third player to hit for the cycle in 2007. So what is the distribution of such an event?

Interestingly, just this year Huber and Glen have modeled the distribution of such rare events in their article, “Modeling Rare Baseball Events – Are They Memoryless?” in the Journal of Statistics Education. They model three baseball events as a Poisson Process: no-hit games, triple plays, and hitting for the cycle. Using data from 1901 through 2004 they get the distribution shown above of how many cycles are seen in a given season. The observed results are a bit short on seasons with exactly 2 players hitting for the cycle, and there are a few more seasons than expected with exactly 1 player hitting for the cycle. The expected results are shown in red: a Poisson distribution with the observed mean of 2.19.

Over the time period examined, nearly 160,000 games were played, and only 0.14% of these games had a player hit for the cycle. As Huber and Glen found the cycle is rarer than a triple play but not as rare as a no-hitter.

So how closely does this match a Poisson process? One measure is a Poissonness Plot developed by Hoaglin (American Statistician 1980). If f[k] represents the observed frequency of a random count k then the log of the Poisson density would essentially be -λ+klog(λ)-log(k!). So that in a plot of log(f[k])+log(k!) against k, a straight line would indicate a good fit to a Poisson distribution. The slope of this line is an estimate of log(λ). Here log(λ) is 0.9583, so that an estimate of λ is 2.607. Of course the maximum likelihood estimate is simply the mean number of cycles per season 2.19. Since the variance of a Poisson distribution is also equal to λ, this would provide another estimate of 2.90. Note the Poissonness plot estimate falls in between the two.

Another way of checking for a Poisson process is to check the inter-arrival times, that is, the number of games between any two players hitting for the cycle. These inter-arrival times should follow an exponential distribution. The figure below shows the cumulative relative frequency of the observed time between cycles for the period 1901 through 2004. Also shown is the theoretical exponential cumulative probability distribution with a mean equal to the observed mean of 719.533 games. This indicates that the cycle process is memoryless. Even if there have been 1000 games without any major league player hitting for a cycle, you would still have to wait on average over 719 games more before one does.

For Aubrey Huff it was certainly a special night. Not only did he hit for the cycle he also had the 1000th major league hit and the 200th double of his career. Even with all that the Orioles still lost the game to the LA Angels 9 to 7.