In this case, standard error is relatively easy to calculate. According to linear weights, each plate appearance is a multinomial event, which I presume are independent and identically distributed. After n events, we weight each event and find the sum. The variance of an iid weighted sum is a weighted sum of the variances, where the weights are squared.
For example, let us assume a batter only has two possibilities: get on base or produce an out. The latter is good for +0.5 runs, the latter for – 0.25 runs. Now pretend a player gets on base at a rate of 0.350 of the time. Over 500 plate appearances, we expect this player to accrue
175*.5 – 325*.25 = 6.25 runs.
The variance of a single event is n*p*(1-p), which is equivalent to
175*.650*.25 + 325*.350*.1025 = 40.1
Take the square root to get a standard error of 6.33
Note the variance will increase as n increases as we are finding the variance of a sum rather than a mean. Furthermore, power hitters will have considerably more variance, as their more probable outcomes are associated with large weights. A small change in HRs will have a much bigger change than a small change in singles, leading to more variance.
Comment by guesswork — December 12, 2011 @ 4:38 pm
The multinomial actually has a non-trivial VC matrix, so you have to take that into account when calculating the variance of a projection of the vector of outcomes onto the reals (such as wOBA).
Comment by Barkey Walker — December 13, 2011 @ 12:38 pm
Good point. That actually simplifies things a lot now that I think about it. The variance of the multinomial is n(Ip – ppT) where I is the identity matrix, p is a column vector of probabilities, and T just means transpose. Then let w be our column vector of weights, so the variance of linear weights would be
wT(n(Ip – ppT))w
For my example, that comes out as 10.859, so our standard error is 3.295.
Comment by guesswork — December 13, 2011 @ 1:39 pm
guesswork, you used a binomial example where the VC matrix is rank1 (i.e. you don’t need a matrix).
Comment by Barkey Walker — December 14, 2011 @ 2:30 am