Sample Samples, Evolving Narratives

We all know better than to overreact to small samples, and we all overreact to small samples anyway. It's human nature, and we're only human.

The narrative on Justin Smoak is a nice illustration. The former Cape Code League MVP was the most widely owned minor league hitter in fantasy baseball on April 1 of last year, which illustrates how huge expectations were. He arrived with Texas, showed good plate discipline but not much power, and when he arrived in Seattle, he was thought a blue chip prospect, if not quite the prospect Jesus Montero was. Smoak was horrible with Seattle in his first month or so there, and returned to AAA until late September. His work in September was dead sexy, but it was only a couple of weeks. To start the season, M's fans opinions ranged from hopeless to cautiously optimistic. Fans Projections at Fangraphs gave him a .352 wOBA, but you couldn't find a sensible Mariners fan willing to endorse that expectation in print.

I doubt you can find a Mariners fan that doesn't think Smoak is the most exciting thing on the team right now, with the possible exception of Michael Pineda. Just yesterday, Dave Cameron said that Justin Smoak makes the M's line up "not atrocious." That sounds like faint praise, but it's pretty telling that a guy who was inches from bust is now a difference making bat. How quickly the pendulum of our expectations swings.

What should our expectations be? When you're not day dreaming in Seattle, or being an impatient fantasy baseball manager, or just being realistic but recognizing your cognitive limits to be realistic precisely, what number should you stick on Smoak now?

In walks some good Sabermetrics. This article by Pizza Cutter tells us what we need to know to shape our expectations. There's a handy equation, if you know your correlation coefficient and the number of observations you've made, that tells you how much to regress your observations to the mean; it is r = x / (x + c). X is the number of observations, r is the coefficient, and c is constant. Pizza cutter gives us the values of x for r = .7 for most of the numbers you like to know.

A little math gets us to c = 86 for BB%. This means that if you take 86 of your pre-season expected PA and weight that against your observations this season, you get your properly regressed rest-of-season expectation, i.e., a true-talent level. Or, (86 * BB% + observed BB)/(86 + observed PA) = true talent BB%.

.372

That's the wOBA you should pin on Smoak right now. Or, if you were the typical fan, that's your new number. When you're being rational. I figured that number out by deriving c for BB%, K%, 1B%, 2B%,* and HR%, and then crunching his wOBA from there.