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Are This Year’s Home Runs Really That Different?
*December 22, 2010*

*Posted by tomflesher in Baseball, Economics.*

Tags: Carlos Pena, Carlos Quentin, home run distributions, home runs, Jose Bautista, kurtosis, Mark Teixeira, Miguel Cabrera, Paul Konerko, R, skewness, statistics

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Tags: Carlos Pena, Carlos Quentin, home run distributions, home runs, Jose Bautista, kurtosis, Mark Teixeira, Miguel Cabrera, Paul Konerko, R, skewness, statistics

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This year’s home runs are quite confounding. On the one hand, home runs per game in the AL have dropped precipitously (as noted and examined in the two previous posts). On the other hand, **Jose Bautista** had an absolutely outstanding year. How much different is this year’s distribution than those of previous years? To answer that question, I took off to Baseball Reference and found the list of all players with at least one plate appearance, sorted by home runs.

There are several parameters that are of interest when discussing the distribution of events. The first is the mean. This year’s mean was 5.43, meaning that of the players with at least one plate appearance, on average each one hit 5.43 homers. That’s down from 6.53 last year and 5.66 in 2008.

Next, consider the variance and standard deviation. (The variance is the standard deviation squared, so the numbers derive similarly.) A low variance means that the numbers are clumped tightly around the mean. This year’s variance was 68.4, down from last year’s 84.64 but up from 2008’s 66.44.

The skewness and kurtosis represent the length and thickness of the tails, respectively. Since a lot of people have very few home runs, the skewness of every year’s distribution is going to be positive. Roughly, that means that there are observations far larger than the mean, but very few that are far smaller. That makes sense, since there’s no such thing as a negative home run total. The kurtosis number represents how pointy the distribution is, or alternatively how much of the distribution is found in the tail.

For example, in 2009, **Mark Teixeira** and **Carlos Pena** jointly led the American League in home runs with 39. There was a high mean, but the tail was relatively thin with a high variance. Compared with this year, when Bautista led his nearest competitor (**Paul Konerko**) by 15 runs and only 8 players were over 30 home runs, 2009 saw 15 players above 30 home runs with a pretty tight race for the lead. Kurtosis in 2010 was 7.72 compared with 2009’s 4.56 and 2008’s 5.55. (In 2008, 11 players were above the 30-mark, and **Miguel Cabrera**‘s 37 home runs edged **Carlos Quentin** by just one.)

The numbers say that 2008 and 2009 were much more similar than either of them is to 2010. A quick look at the distributions bears that out – this was a weird year.