## Wednesday, February 29, 2012

### BA/OBA/SLG Algebra, pt. 1

This post is mathematical in nature and wholly unoriginal. All it does is walk through the algebraic relationships between BA, OBA, SLG and some other related metrics, with a little bit of my commentary sprinkled in. I believe there is nothing here that I haven’t written about before, let alone others, but I thought it would be nice to have it all in one place for easy reference. There is also a certain quickie method that involves two of the big three slash stats (other than OPS!) that drives me nuts, and occasionally I feel compelled to fire off a missive against it.

I will assume here that BA = H/AB, OBA = (H + W)/(AB + W), and SLG = TB/AB--in other words, ignoring the presence of HB and SF in the OBA formula. If you prefer, just think of HB as being combined with walks in a single category, so that walks in the formulas actually represent W + HB. Since they are essentially the same and none of the metrics here draw any distinction between them, such treatment won’t cause any distortion and will clean up the clutter. Sacrifices are a mess, but don’t have a major effect in any event.

This post would look much better with numerators written over denominators, but it is so much more time effective to just type everything out.

Isolated Power (ISO)

ISO is very easily calculated as SLG - BA:

SLG - BA = TB/AB - H/AB = (TB - H)/AB

Which is equivalent to (S + 2D + 3T + 4HR - (S + D + T + HR))/AB = (D + 2T + 3HR)/AB

At Bat Percentage (AB%)

That is, at bats as a percentage of plate appearances (AB + W). This is not a particularly interesting statistic on its own, but it is the complement of W/(AB + W), and knowing how to calculate it makes it easier to do later manipulations:

AB% = (1 - OBA)/(1 - BA)

(1 - OBA)/(1 - BA) = (1 - (H + W)/(AB + W))/(1 - H/AB)

Anytime you see a one, you can get rid of it by replacing it with denominator/denominator of the fraction you’re interested in:

= ((AB + W)/(AB + W) - (H + W)/(AB + W))/(AB/AB - H/AB)
= ((AB - H)/(AB + W))/((AB - H)/AB)

Of course, dividing by something is the same as multiplying by the reciprocal:

= (AB - H)/(AB + W)*AB/(AB - H) = AB/(AB + W)

OBA - BA

You don’t see OBA - BA used in many formal situations, but it is sometimes used as a quick indicator of plate discipline. It’s easy to see why people might use it that way--it resembles isolated power, and since it is walks that cause OBA to differ from BA, it seems fairly sensible.

However, the reason ISO works so nicely is because both SLG and BA have the same denominator. That is not the case of OBA and BA, which makes the subtraction messy:

OBA - BA = (H + W)/(AB + W) - H/AB

In order to get a common denominator:

(H + W)/(AB + W)*(AB/AB) - (H/AB)*(AB + W)/(AB + W)

= ((H + W)*AB - H*(AB + W))/(AB*(AB + W))

= (W*AB - W*H)/(AB*(AB + W))

= W*(AB - H)/(AB*(AB + W))

= W/(AB + W)*(AB - H)/AB

Since (AB - H)/AB = 1 - BA, we can write the above equation as:

= W/(AB + W)*(1 - BA)

In other words, OBA - BA is walk rate times outs/at bat. This chart shows OBA - BA for three walk rates at four different BA levels. As you can see, the lower the BA, the higher OBA - BA is for a given walk rate. The distortion increases as walk rate increases. The differences are not enormous except for extreme cases, but they’re still plenty large enough for me to avoid using OBA - BA as a stand-in for walk rate in any circumstance.

Some people may object to my discussion here and counter that OBA - BA is not intended to be a measure of walk rate so much as it is a measure of the amount of on base ability the player brings above and beyond his batting average. My objections to this position are many, but one that deserves mention here is that it is a position that starts with the inferior, less fundamental statistic (BA) and attempts to build up to the statistic that pretty much defines a fundamental baseball measure (assuming runs and wins are off the table). Being forced to use OBA - BA to measure “additional” on base ability just points out the folly of making BA the building block rate in the first place.

Walks to At Bat Ratio (W/AB)

The ratio of walks to at bats is not a particularly meaningful ratio, but it will produce the same rank order list as walks per plate appearance given the assumptions of this post (ignoring SF and treating HB as zero or walks, walks per plate appearance by definition would simply be W/AB divided by itself plus one). It is also a component of secondary average, and it’s instructive to see how it can be pulled out of OBA and BA.

We know from above that OBA - BA = W*(AB - H)/(AB*(AB + W)). We also know (or can easily demonstrate) that 1 - OBA = (AB - H)/(AB + W). Divide by (OBA - BA) by (1 - OBA) and you have:

(OBA - BA)/(1 - OBA) = W*(AB - H)/(AB*(AB + W))*(AB + W)/(AB - H)

= W/AB

So if you want an acceptable (in that it orders players by the frequency with which they actually draw walks) stand-in for walk rate from OBA and BA, you can just figure (OBA - BA)/(1 - OBA). But there’s an equally easy way to get to the more useful and properly denominated walks per plate appearance.

Walks per Plate Appearance (W/PA)

Knowing the formulas for OBA - BA and 1 - BA, we can quickly see:

(OBA - BA)/(1 - BA) = W*(AB - H)/(AB*(AB + W))*(AB - H)/AB = W/(AB + W)

So please scrap the silly, unitless, distortive OBA - BA and replace it with (OBA - BA)/(1 - BA), which actually is walk rate.

Secondary Average (SEC)

Secondary Average without considering steals is (TB - H + W)/AB. This is easily recognizable as ISO + W/AB, and from the equations here it can be easily converted to a BA/OBA/SLG equivalent:

SEC = SLG - BA + (OBA - BA)/(1 - OBA)