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
add a comment
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. Th
e 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.
Teixeira and Cano: Picking up slack? August 5, 2010
Posted by tomflesher in Baseball, Economics.Tags: A-Rod, Alex Rodriguez, binomial distribution, Mark Teixeira, probability, Robinson Cano, statistics, Yankees
add a comment
Michael Kaye, the YES broadcaster for the Yankees, often pointed out between July 22 and August 4 that the Yankees were turning up their offense to make up for Alex Rodriguez‘s lack of home run production. That seems like it might be subject to significant confirmation bias – seeing a few guys hit home runs when you wouldn’t expect them to might lead you to believe that the team in general has increased its production. So, did the Yankees produce more home runs during A-Rod’s drought?
During the first 93 games of the season, the Yankees hit 109 home runs in 3660 plate appearances for rates of 1.17 home runs per game and .0298 home runs per plate appearance. From July 23 to August 3, they hit 17 home runs in 451 plate appearances over 12 games for rates of 1.42 home runs per game and .0377 home runs per plate appearances. Obviously those numbers are quite a bit higher than expected, but can it be due simply to chance?
Assume for the moment that the first 93 games represent the team’s true production capabilities. Then, using the binomial distribution, the likelihood of hitting at least 17 home runs in 451 plate appearances is
The cumulative probability is about .868, meaning the probability of hitting 17 or fewer home runs is .868 and the probability of hitting more than that is about .132. The probability of hitting 16 or fewer is .805, which means out of 100 strings of 451 plate appearances about 81 of them should end with 16 or fewer plate appearances. This is a perfectly reasonable number and not inherently indicative of a special performance by A-Rod’s teammates.
Kaye frequently cited Mark Teixeira and Robinson Cano as upping their games. Teixeira hit 18 home runs over the first 93 games and made 423 plate appearances for rates of .194 home runs per game and .0426 home runs per plate appearance. From July 23 to August 3, he had 5 home runs in 12 games and 54 plate appearances for rates of .417 per game and .0926. That rate of home runs per plate appearance is about 8% likely, meaning that either Teixeira did up his game considerably or he was exceptionally lucky.
Cano played 92 games up to July 21, hitting 18 home runs in 400 plate appearances for rates of .196 home runs per game and .045 per plate appearance. During A-Rod’s drought, he hit 3 home runs in 50 plate appearances over 12 games for rates of .25 and .06. That per-plate-appearance rate is about 39% likely, which means we don’t have enough evidence to reject the idea that Cano’s performance (though better than usual) is just a random fluctuation.
It will be interesting to see if Teixeira slows down as a home-run hitter now that Rodriguez’s drought is over.
