THE BOOK--Playing The Percentages In Baseball

Thursday, May 31, 2007

Converting OBP (or wOBA) to Runs

This is a pure numbers post.

Can we convert OBP to runs in a straightforward manner?

In this article:
http://www.tangotiger.net/rc3.html

I show an image of score rate relative to OBP:
http://www.tangotiger.net/rc3oba.jpg

As you can see, it’s fairly straightforward, according to BaseRuns.  Is it really?

Using my basic Markov model:
http://www.tangotiger.net/markov.html

I plugged in various batting lines, from an OBP of .050 to .995.  As it happens, BaseRuns is correct.  There’s a fairly straight line correlation of OBP (times on base per PA) to score Rate (runs per times on base), on the order of r=.999.

Trying to best-fit an equation, while acknowledging we care about the OBP of .300 to .400 the most, we end up with this equation:
scoreRate = 1.087 * OBP - .055

Obviously, this breaks down at an OBP level of under .050 and over .970.

From an OBP level of .200 to .400, we get an almost perfect match.  At the other levels, we get an extremely strong match.

Since we know that
R = scoreRate * ROB + HR
where ROB is runners on base

We can convert OBP to R fairly easily.

Let’s assume that HR rate = OBP * .08

PA = OBP/(1-OBP)*27+27
PA = 27*(OBP/(1-OBP) + 1)
ROB = PA*(OBP - HRrate)
ROB = PA*(.92*OBP)

R = scoreRate * ROB + HR
R = (1.087*OBP-.055) * (.92*OBP*PA) + (.08*OBP*PA)
R = PA*(OBP^2 - .05*OBP) + PA*(.08*OBP)
R = PA*(OBP^2 + .03*OBP)
R = 27*(OBP/(1-OBP) + 1) * (OBP^2 + .03*OBP)

So, if you have OBP = .333 (and HR rate is 8%), R=4.905.

If you are looking for a very quick shorthand:

R = ( (OBP/(1-OBP)) ^1.5)*14

For OBP = .333, we get R=4.950, which is pretty close to the above

At OBP = .250, the long equation gets us to 2.50 runs, the shorthand is 2.69, and Markov says 2.53. 

At OBP = .500, the long equation gets us to 14.3 runs, the shorthand is 14.0, and Markov is 14.4 runs.

Here’s the full results:

wOBA	 Markov	1.50 	long


 0.050 	 0.17 	 0.17 	 0.11 


 0.100 	 0.47 	 0.52 	 0.38 


 0.150 	 0.93 	 1.04 	 0.85 


 0.200 	 1.59 	 1.75 	 1.54 


 0.250 	 2.53 	 2.69 	 2.50 


 0.300 	 3.80 	 3.93 	 3.80 


 0.350 	 5.49 	 5.53 	 5.50 


 0.400 	 7.72 	 7.62 	 7.71 


 0.450 	 10.63 	 10.36 	 10.57 


 0.500 	 14.42 	 14.00 	 14.27 


 0.550 	 19.37 	 18.92 	 19.10 


 0.600 	 25.90 	 25.72 	 25.47 


 0.650 	 34.66 	 35.43 	 34.04 


 0.700 	 46.73 	 49.90 	 45.93 


 0.750 	 64.02 	 72.75 	 63.11 


 0.800 	 90.43 	 112.0 	 89.55 


 0.850 	 134.9 	 188.9 	 134.5 


 0.900 	 224.4 	 378.0 	 225.8