# Origin of the formula for weighted stolen base runs (wSB)

I am considering the formula for weighted stolen base runs (wSB), which is given as

wSB = a * SB + b * CS - lgwSB * (1B + uBB + HBP)

and a = 0.2 and b = (-2 * RunsPerOut + 0.075).

I have two questions:

1.) does anyone know the original paper/inventer of wSB? I have been trying to find out about the history of this statistic but have not been able to

2.) Does anyone know why there is a multiple of 2 in runs per out for the coefficient of caught stealing? I thought wSB was supposed to quantify the contributions to the run count of the stealing attempts of a specific player. Since every time being caught stealing contributes one out, I would have thought the the coefficient b of CS would be more or less RunsPerOut; I don't understand why we then scale by 2.

Can anybody explain this apparent contradiction to me?

The oldest reference to wSB that I'm familiar with is this Fangraphs article, which is from 2013, and I think in 2012 perhaps is when they actually introduced the stat. It's possible it originated elsewhere, but the formula you provide there is Fangraphs' formula, and I imagine another source would use different parameters.

As for why it's -2*R/O ... that's what they determined it should be. Coefficients like that (and the 0.2 for SB) are usually determined through linear regressions or similar statistical modeling methods; don't try to assign too much direct meaning to them. According to their models, the contribution of a base stealer to his team's overall win total increased by 0.2 times the number of stolen bases, and decreased by -2 times the league runs per out [for reference, this number is around 0.15 to 0.2, I believe]. It's not saying "a caught stealing is in some way causatively related to the league runs per out", by the way; it's just using the league runs per out as a scaling mechanism - in seasons where league runs per out are higher, a CS hurts a little more than in a season where there's a lot less scoring.

Outs, remember, are basically a fixed number - every season has the same number of them, given the same number of games and teams, except for small adjustments for home teams skipping the 9th half the time and extra innings - so R/O is really just another way of controlling for the scoring environment (league runs scored), adjusted to a more useful size number. They could as easily have used "R/G" times 0.225 or something like that, maybe that would've been less confusing?

• yes The Fangraphs article was the earliest reference I could find as well, which is pretty disappointing as there is no accompanying code or work to indicate where anything come from, or any validation to show that wSB corresponds to anything meaningful. Is wSB actually a commonly used stat in SABRmetrics? Jan 9 at 18:15
• I'm pretty sure it's a Fangraphs original. Tom Tango refers to it obliquely as theirs, and I found several other comments/etc. that credit it to Fangraphs, and nothing prior to about 2013. They give some details of how their models are made, but ultimately anything linear weight based is down to just modeling... not that much else you can do. Tom Tango does talk about an alternative in that article you could look into more, there's more detail there.
– Joe
Jan 10 at 0:08