I have been thinking a lot about the Pythagorean Formula lately. Plenty has been written on the subject in the past, so I had some great resources to accompany me late at night. For now, I can’t really say I have any new research or insights to add to the subject, but I thought I would make some new graphs that may lend a new perspective on the subject.

In this article, I’ll be looking at Bill James’ original Pythagorean equation, his updated equation where the power in the formula changed from 2 to 1.83 and the Pythagenpat formula, where a team’s run environment is a factor in determining the projected win percentage of a team. I’ll be referring to these as Pyth1, Pyth2, and Pyth3, respectively.

### Theoretical graphs

Allow me to introduce some pythago-raphs, where the color of an area on the graph is based on the Pyth1 projected wins percentage. If you think of a runs scored (R), runs allowed (RA) combination for a hypothetical team, you will get a general idea of that team’s win percentage from the graph. Pretty straightforward; it won’t give you an exact number, but it can help in visualizing how the formula works more easily.

I also included another graph where only the winning percentage interval of 0.350 to 0.650 is shown, since that’s the usual range most ball clubs are in. This graph is easier on the eyes, as you can see the differences in win percentages more clearly due to the higher change in colors.

Also, the shape (it looks a little like a sword from Zelda) of the colored area shows that as more R and RA are observed, the formula estimates win percentage in a more narrow interval (in this case, 0.350 to 0.650).

While this graph gives you a general sense of where your win percentage will be (based on R and RA), the format of this graph won’t really help us look for differences between the three formulas. Thus, I decided not to add the other two, and included something a bit different for that discussion.

Below you will see three graphs that depict the projected win percentage differences between the three formulas. The first is the difference between Pyth1 and Pyth2, the second is between Pyth1 and Pyth3 and the last depicts the difference between Pyth2 and Pyth3.

For example, a positive difference in an area on the graph means Pyth1 projects a higher win percentage than Pyth2 (use the same idea for the other graphs, and also reverse the logic when seeing a negative difference). The white area depicts no real difference in the formulas in projecting win percentage.

A quick look at the first graph shows how Pyth1 expects a higher win percentage when a team’s run differential is positive, and vice versa for a negative run differential. Since Pyth2 was an update to Pyth1 by Bill James, it looks like the difference in the formulas was to correct the over and under expectations of Pyth1: they were simply a bit too extreme, and James found his second formula fit better to the patterns in projecting wins from R and RA.

The other two graphs (graph 2 and graph 3) have interesting kite-shaped outlines. The difference between the two is where the short diagonal is located; the second graph’s is higher in the upper right area of the graph, while the third graph’s is closer to the middle area. Its cross diagonal seems more defined than the second graph’s as well.

I like how in the second graph, the minimal differences in the two formulas (the white area) has the same sword-like shape as the the second graph I posted. It’s telling us the real differences between Pyth1 and Pyth3 come from extreme cases of R and RA (as in, when a very wide run differential is observed from a team, the two formulas vary more in expected win percentage).

### Sample graphs using past season data

In this section, I compiled for each team that played in 1960-2010 (using Baseball Databank), their respective runs scored and runs allowed totals and their season record. I also calculated their Pythagorean win percentage, using all three formulas mentioned previously.

Below is a graphic that displays the difference between each team’s actual and expected win percentage. I animated all three to see the difference in how each formula performs in predicting win percentage. The points are a little lighter than desired, but going from pyth1 to pyth2, many of the points on the outskirts get lighter (the difference gets closer to zero).

Overall, these sample graphs show the general decreasing trend in differences in actual and expected win percentage from one formula to the next. We see over time and through many analyst’s research, these formulas have gotten better at predicting win percentage, little by little. The changes are subtle, but they are there.

**References & Resources**

Data used in the sample graphs comes from Baseball Databank.

For more information about what sorts of factors can make a team’s actual win percentage vary from its Pythagorean formula, read these two articles by THT’s Studes and Gassko.

Dave Studeman said...

Cool stuff, Kevin. On the RS and RA graphs I’ve typically done for this site and the THT Annual, I’ve flipped the RA axis (so that more runs allowed is on the bottom). It think it’s more intuitive that way, cause then the best winning percentages are in the upper right-hand corner of the graph.

Just FYI.

Kevin Lai said...

Thanks for the tip, Dave. I guess I would have failed a THT Annual pop quiz eh?