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The Choke Index August 1, 2010

Posted by tomflesher in Baseball.
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It’s been quite a while since Alex Rodriguez hit Home Run #599 – nine days since July 22, but more quantifiably, 42 plate appearances. Just how much of a slump is he in? I’d like to propose a quantifiable answer: the Choke Index.

From 2000 to 2009, A-Rod was hitting approximately .064 home runs per plate appearance. In 2008 he hit .059 and in 2009 he hit .056, so it’s probably much fairer to use a slightly lower rate. I’m going to make the assumption that Rodriguez’s true production is about .055 home runs per plate appearance, since he exhibited a downward trend and his 2010 production has been very low. (It also cuts him some additional slack in the Choke Index.)

Simply, we should assume that A-Rod’s failure to produce is merely the result of chance, and not due to choking or media distraction or even Rodriguez’s discomfort with the special chipped baseballs. (A better man than I would call this the Numbered Ball Effect.) Then, we should see how likely that is.

At .055 home runs per plate appearance, the likelihood of going 42 plate appearances without a home run is (1-.055)^{42} or approximately .093. The Choke Index is simply 1-(likelihood) or, in this case, .907. As it becomes progressively less likely that Rodriguez will go another plate appearance without hitting a home run, the Choke Index number rises. A theoretical Choke Index of 1 would indicate that the player’s lack of home run hitting is nearly impossible to describe by chance alone.

A-Rod’s Choke Index between #499 and #500 was about .877. This is a man who doesn’t handle milestones well.

Another example was Gary Sheffield in 2009, when he was attempting to hit his 500th home run. In the previous two years, he hit approximately .041 home runs per plate appearance. Much was made of Sheffield’s trouble hitting #500, but since he was hitting almost exclusively as a pinch hitter, he simply didn’t have many opportunities. Between his final plate appearance on September 26 of 2008 and his only plate appearance on April 17 of 2009, Sheffield went 21 plate appearances without hitting a homer. That gives him a choke index of .556.

Barry Bonds, meanwhile, was hitting .065 home runs per plate appearance in the seasons prior to his record-breaking home run #756. #755 was hit in Bonds’ first plate appearance on August 4, 2007. Bonds made 3 more plate appearances, all walks, in that game. He hit #756 in his third plate appearance only three days later on August 7.  He had August 5 off and made 4 plate appearances on August 6, meaning that Bonds went 9 plate appearances between home runs, giving him a choke index of .454.

Rodriguez will hit his 600th home run eventually, but it’s getting painful to watch.

The 600 Home Run Almanac July 28, 2010

Posted by tomflesher in Baseball, Economics.
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People are interested in players who hit 600 home runs, at least judging by the Google searches that point people here. With that in mind, let’s take a look at some quick facts about the 600th home run and the people who have hit it.

Age: There are six players to have hit #600. Sammy Sosa was the oldest at 39 years old in 2007. Ken Griffey, Jr. was 38 in 2007, as were Willie Mays in 1969 and Barry Bonds in 2002. Hank Aaron was 37. Babe Ruth was the youngest at 36 in 1931. Alex Rodriguez, who is 35 as of July 27, will almost certainly be the youngest player to reach 600 home runs. If both Manny Ramirez and Jim Thome hang on to hit #600 over the next two to three seasons, Thome (who was born in August of 1970) will probably be 42 in 2012; Ramirez (born in May of 1972) will be 41 in 2013. (In an earlier post that’s when I estimated each player would hit #600.) If Thome holds on, then, he’ll be the oldest player to hit his 600th home run.

Productivity: Since 2000 (which encompasses Rodriguez, Ramirez, and Thome in their primes), the average league rate of home runs per plate appearances has been about .028. That is, a home run was hit in about 2.8% of plate appearances. Over the same time period, Rodriguez’ rate was .064 – more than double the league average. Ramirez hit .059 – again, over double the league rate. Thome, for his part, hit at a rate of .065 home runs per plate appearance. From 2000 to 2009, Thome was more productive than Rodriguez.

Standing Out: Obviously it’s unusual for them to be that far above the curve. There were 1,877,363 plate appearances (trials) from 2000 to 2009. The margin of error for a proportion like the rate of home runs per plate appearance is

\sqrt{\frac{p(1-p)}{n-1}} = \sqrt{\frac{.028(.972)}{1,877,362}} = \sqrt{\frac{.027}{1,877,362}} \approx \sqrt{\frac{14}{1,000,000,000}} = .00012

Ordinarily, we expect a random individual chosen from the population to land within the space of p \pm 1.96 \times MoE 95% of the time. That means our interval is

.027 \pm .00024

That means that all three of the players are well without that confidence interval. (However, it’s likely that home run hitting is highly correlated with other factors that make this test less useful than it is in other situations.)

Alex’s Drought: Finally, just how likely is it that Alex Rodriguez will go this long without a home run? He hit his last home run in his fourth plate appearance on July 22. He had a fifth plate appearance in which he doubled. Since then, he’s played in five games totalling 22 plate appearances, so he’s gone 23 plate appearances without a home run. Assuming his rate of .064 home runs per plate appearance, how likely is that? We’d expect (.064*23) = about 1.5 home runs in that time, but how unlikely is this drought?

The binomial distribution is used to model strings of successes and failures in tests where we can say clearly whether each trial ended in a “yes” or “no.” We don’t need to break out that tool here, though – if the probability of a home run is .064, the probability of anything else is .936. The likelihood of a string of 23 non-home runs is

.936^{23} = .218

It’s only about 22% likely that this drought happened only by chance. The better guess is that, as Rodriguez has said, he’s distracted by the switching to marked baseballs and media pressure to finally hit #600.

Quickie: Kiss the Sheff April 22, 2009

Posted by tomflesher in Baseball.
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Why is Gary Sheffield employed for the league minimum when Barry Bonds can’t get a job?

Sheffield had a Batting Average, On-Base Percentage, Slugging Percentage and On-Base Plus Slugging of .225/.326/.400/.725 in 2008; Bonds was last active in 2007 and hit .276/.480/.565/1.045 (with the .480 OBP leading the National League). Clearly, something’s wrong. Collusion?

What’s wrong, in my estimation, is still that Bonds represents a negative externality on his team’s production, reputation, and revenue; Sheffield, meanwhile, is less of a threat to ticket sales. Despite being unpopular and saying bizarre things, Sheffield has not yet to my knowledge irritated fans to the extent that Bonds has, nor is he quite the clubhouse menace Bonds is said to be.

Of course, time will tell whether Sheffield produces $400,000 worth of runs for the ailing Mets.

Barry Bonds (with bonus Collusion discussion) March 25, 2009

Posted by tomflesher in Academia, Baseball, Economics.
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Sorry about the infrequent updates. It’s a busy time in the semester.

Barry Bonds is, without a doubt, one of the most controversial figures in baseball. He’s currently trying, again, what he tried last year – shopping himself around for the league’s minimum salary. (Thanks to the Sports Law Blog for the link.) Inside, I’d like to briefly discuss collusion and look at the incentives involved with this situation.

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