How reasonable is it to pass judgement on Ichiro after 77 PAs in 22 games? This is a rhetorical question, as it's obviously unreasonable. However, for fun let's use the binomial distribution to estimate the probability that Ichiro's performance to date indicates his true talent level.
To use the binomial distribution, we need to estimate Ichiro's likely talent level. If we look at his Marcel projections, Ichiro's projected batting stats are as follows:
- BA: .273
- OBP: .311
- XBH/AB (Extra-base hits per AB)*: .056
- BA: .300
- OBP: .341
- XBH/AB*: .054
Not that this is surprising, but the probabilities indicate it is unlikely that Ichiro's first 77 PAs of the season reflect how he will perform in 2013. Moreso if you assume the Yankees' projections of his performance significantly exceed Marcels'.
Let's look at one other set of numbers, which are the expected range of these statistics after 77 PAs. The following table shows the range of values that the binomial distribution would expect, with 90% confidence, after 77 PAs:
|Marcels||.186 - .357||.221 - .403||.014 - .100|
|2010-based estimate||.214 - .386||.260 - .429||.014 - .100|
In other words, after 77 PAs a player's batting statistics could be either those of a superstar or those below a replacement-level player.
* To gauge Ichiro's projected power, I used extra-base hits per AB (XBH/AB) instead of slugging percentage because the latter cannot be used to estimate probability. This is because the binomial distribution requires that each trial have only 1 of 2 outcomes, and slugging percentage allows for 5 outcomes: out, single, double, triple, homer. So for today's purpose we'll just look at the probability that Ichiro gets an extra-base hit or not in HIS at-bats.