If you haven't read the previous posts in this series, we have been using the binomial distribution to gauge the likelihood that A-Rod's 2012 performance would occur for a hitter whose true talent level is a HR/AB ratio of 6.2%. 6.2% is used because that was A-Rod's HR rate in 2009-2010, when he hit 30 HRs each year (in less than full seasons).
In 2012, A-Rod ended up with 18 HRs in 463 ABs, for a 3.89% HR/AB ratio. 18 HRs is below the expected range of a player with a 6.2% HR/AB ratio: the binomial distribution indicates that 95% of the time, a player with a talent level corresponding to A-Rod's 2009-2010 performance should have hit somewhere in the range of 20-38 HRs.
So where does A-Rod stand according to this method? The binomial distribution indicates there is a 1.9% probability that A-Rod's true talent level is a 6.2% HR/AB ratio despite his poor 2012 performance. In other words, the binomial distribution indicates there is a 1/20 chance that A-Rod is still really a slugger.
Obviously 1/20 are poor odds. And since A-Rod's under contract for 5 years at an average of $22,800,000 (plus some possible bonus money), the question of whether or not A-Rod can still be a middle-of-the-order hitter is critical to the Yankees' success for the foreseeable future.
I don't want to pretend the numbers are other than they are and make a fool of myself a la the comical escapades of math-phobic right-wing partisans regarding the Presidential election. However, since there has been plenty of understandable fork-sticking into A-Rod, I want to explore the alternate perspective, which is that 1/20 odds are not astronomical. So let's look at a couple of historical precedents for aging players:
- In 2010 (and the first part of 2011), people were sticking a fork in Derek Jeter's career. A 36-year-old Jeter hit .270 in 2010, well below his then-career average of .317. In 2011-2012, Jeter rebounded to hit .308. So let's use .308 as a measure of Jeter's true talent level in 2010. The odds of Jeter hitting .270 despite having a .308 talent level, according to the binomial distribution, were 1.8% - similar to A-Rod's.
- Darrell Evans had an unusual aging pattern. In 1981-1982, at ages 34-35, his power seemed to slightly decline - he hit 28 HRs in 882 ABs for a HR/AB ratio of 3.41%. Over the next 5 years, he slugged 149 HRs in 2435 ABs, for a 6.12% HR/AB ratio. If we use 6.12% as Evans' true talent level, the odds of his 1981-1982 performance were 0.03% - in other words, the odds were 3 out of 10,000!
An implication is that the Yankees would have been highly irresponsible if they were considering what media hacks had been speculating upon in October, which is that the Yankees would pay some other team to take A-Rod. These hacks were writing as though the odds of A-Rod returning to his slugging days were 1/100,000, not around 1/20!
They also were ignoring the fact that even if A-Rod's cleanup days are behind him, he still was an above-average hitter as a 3B - for example, his .342 wOBA ranked 8th among MLB 3rd basemen.