# a power metric, using an analogy to physics

In physics, power is dE/dt, energy per unit time. The goal of this post is to extend this idea to create a “power” metric for baseball.

The added energy, dE is advancement toward scoring runs. I will use the linear weights values of an event, but instead of referencing them to an out (as in wOBA), I will reference them to a BB. This is somewhat arbitrary, but it fits the intuitive notion that a BB contains no power, and extra-base hits contain progressively more power. Note that this means a single does contribute to power which might be different than other formulations of power, and in particular is distinct from ISO. Now what about dt, the change in time? In the most general sense, time is tricky to actually define. One way to put it is that time is a monotonic parameterization of casually connected events that goes from “start” to “finish”. The key point here is that in baseball, time is outs.

With all of that in mind, I will write down my power metric. Let me index an event by i, and let the number of events be $n_i$, and the wOBA weight for an event be $w_i$. Then wOBA is

$\mathrm{wOBA} = \frac{\Sigma_i w_i n_i}{\mathrm{PA}}$

My power metric on the other hand is,
$\Sigma_i (w_i - w_{\mathrm{BB}}) n_i/\mathrm{outs}$

outs is (PA – non-outs), so 1/outs = 1/(PA-non-outs) = 1/PA x 1/(1-OBP). Using this, my power metric is,

$\frac{1}{1-\mathrm{OBP}} \times (\frac{\Sigma_i w_i n_i}{\mathrm{PA}} - \frac{\Sigma_i w_{\mathrm{BB}} n_i}{\mathrm{PA}}) = \frac{1}{1-\mathrm{OBP}} \times (\mathrm{wOBA} - \mathrm{wBB} \cdot \mathrm{OBP})$

So you start with wOBA, but then for every positive event, subtract off the value of a BB. Then, divide by 1-OBP, which accounts for the fact that “time” passes more slowly for players that create less outs.

I don’t have a clever name for this metric. The best I could come with was “Work Aggregated over Time, per unit Time”, or WATT (rhymes with dot). So let’s look at the top seasons in WATT. To do this I take data from the Fangraphs leader boards, and use the Fangraphs guts database to find the value of BB for each particular year.

Here are the top 5 for 2014.
 0.2586 2014 Jose Abreu 0.2437 2014 Mike Trout 0.2403 2014 Anthony Rizzo 0.2370 2014 Andrew McCutchen 0.2283 2014 Edwin Encarnacion 

Here is the top 10, 1901-2014,
 0.4531 1920 Babe Ruth 0.4115 1921 Babe Ruth 0.3807 1927 Lou Gehrig 0.3793 2001 Barry Bonds 0.3730 1923 Babe Ruth 0.3702 1927 Babe Ruth 0.3699 1941 Ted Williams 0.3641 1926 Babe Ruth 0.3620 1925 Rogers Hornsby 0.3580 1924 Babe Ruth 

Here is the correlation coefficient of WATT with various other offensive metrics, for the period 2005-2014.

 quantity r wOBA 0.887 ISO 0.889 SLG 0.947 OPS 0.905