One of the fundamental concepts of sabermetrics is the Pythagorean Theorem. Invented by Bill James in the late 1970s, the Pythagorean Theorem is a formula that shows the relationship between a team's runs scored (RS), runs allowed (RA) and winning percentage (PCT). It is as follows:
The formula works reasonably well in most years for most teams. This indicates that winning games is essentially a function of how many runs a team scores and allows and when the runs were scored is not so important in most cases.
For example, the 2009 Tigers scored 743 runs and allowed 745. Plugging those numbers into the above formula yields a winning percentage of .499 which translates into 81 predicted wins. In reality, they won 86 games so they over performed their pythagorean estimate by five wins.
PCT=(RS squared)/(RS squared + RA squared)
The formula works reasonably well in most years for most teams. This indicates that winning games is essentially a function of how many runs a team scores and allows and when the runs were scored is not so important in most cases.
For example, the 2009 Tigers scored 743 runs and allowed 745. Plugging those numbers into the above formula yields a winning percentage of .499 which translates into 81 predicted wins. In reality, they won 86 games so they over performed their pythagorean estimate by five wins.
Interestingly, the Tigers scored 36 runs fewer than they allowed in 2008 and had an estimated win total of 78. So, according to their runs scored and runs allowed, they were only three wins better this year compared to last year. But they won 86 games this year and 74 in 2008 so what happened?
When a team under performs like the Tigers did in 2008, it often means they did poorly in close games. Last year the Tigers were 16-25 in one run games so that explains part of the discrepancy between their 2008 predicted wins and actual wins.
Conversely, when a team over performs as the Tigers did in 2009, it typically means that they performed well in close games. in 2009, the Tigers were 28-22 in one run games and 24-11 in two run games so they were indeed very good in tight games. In fact their 52 wins in games decided by two runs or fewer was the best in MLB and their 52-33 record was second best to the Yankees (48-25).
The Pythagorean estimates for all American and National League teams are presented in tables 1 and 2 below. The Mariners (10 more wins than expected wins) and the Yankees (+6) had the out did the estimates more than any other teams. On the other hand, the Blue Jays (-9), Indians (-7) and Athletics (-6) fell on the wrong side of their expected win total. In the National League, the Padres (+9) and Astros (+7) over performed and the Braves (-6) under performed.
Getting back to the Tigers, one of the chief reasons for their good record in close games was Fernando Rodney's 37 out of 38 save conversion rate. Another reason was their seven walk off wins and numerous other come back victories this year. As much as we agonize over the fact that one more victory would have given them the AL central division title, it's a lot easier to find games where they hung on to win or stole a victory late in the game. They took more than they gave away in 2009.
The Pythagorean estimates for all American and National League teams are presented in tables 1 and 2 below. The Mariners (10 more wins than expected wins) and the Yankees (+6) had the out did the estimates more than any other teams. On the other hand, the Blue Jays (-9), Indians (-7) and Athletics (-6) fell on the wrong side of their expected win total. In the National League, the Padres (+9) and Astros (+7) over performed and the Braves (-6) under performed.
Getting back to the Tigers, one of the chief reasons for their good record in close games was Fernando Rodney's 37 out of 38 save conversion rate. Another reason was their seven walk off wins and numerous other come back victories this year. As much as we agonize over the fact that one more victory would have given them the AL central division title, it's a lot easier to find games where they hung on to win or stole a victory late in the game. They took more than they gave away in 2009.
The bad news is that teams who out do their Pythagorean estimates tend to regress in the following season. The reason is because performance in close games is not consistent from year to year. Thus, I think the Tigers will have a hard time repeating their 86 wins unless they play significantly better next year. Specifically, I think they'll need to score a lot more runs.
Table 1: Pythagorean estimates for American League teams in 2009
| runs | runs allowed | actual wins | pred wins | difference | % difference | |
| Mariners | 640 | 692 | 85 | 75 | 10 | 13.3 |
| Yankees | 915 | 753 | 103 | 97 | 6 | 6.2 |
| Tigers | 743 | 745 | 86 | 81 | 5 | 6.2 |
| Angels | 883 | 761 | 97 | 93 | 4 | 4.3 |
| Rangers | 784 | 740 | 87 | 86 | 1 | 1.2 |
| Red Sox | 872 | 736 | 95 | 95 | 0 | 0.0 |
| Royals | 686 | 842 | 65 | 65 | 0 | 0.0 |
| Twins | 817 | 765 | 87 | 87 | 0 | 0.0 |
| White Sox | 724 | 732 | 79 | 80 | -1 | -1.3 |
| Rays | 803 | 754 | 84 | 86 | -2 | -2.3 |
| Orioles | 741 | 876 | 64 | 68 | -4 | -5.9 |
| Athletics | 759 | 761 | 75 | 81 | -6 | -7.4 |
| Indians | 773 | 865 | 65 | 72 | -7 | -9.7 |
| Blue Jays | 798 | 771 | 75 | 84 | -9 | -11 |
Table 2: Pythagorean estimates for National League teams in 2009
| Team | runs | runs allowed | actual wins | pred wins | difference | % difference |
| Padres | 638 | 769 | 75 | 66 | 9 | 13.6 |
| Astros | 643 | 770 | 74 | 67 | 7 | 10.4 |
| Marlins | 772 | 766 | 87 | 82 | 5 | 6.1 |
| Reds | 673 | 723 | 78 | 75 | 3 | 4.0 |
| Brewers | 785 | 818 | 80 | 78 | 2 | 2.6 |
| Rockies | 804 | 715 | 92 | 90 | 2 | 2.2 |
| Giants | 657 | 611 | 88 | 87 | 1 | 1.1 |
| Phillies | 820 | 709 | 93 | 93 | 0 | 0.0 |
| Cardinals | 730 | 640 | 91 | 92 | -1 | -1.1 |
| Mets | 671 | 757 | 70 | 71 | -1 | -1.4 |
| Cubs | 707 | 672 | 83 | 85 | -2 | -2.4 |
| Pirates | 636 | 768 | 62 | 65 | -3 | -4.6 |
| Diamondbacks | 720 | 782 | 70 | 74 | -4 | -5.4 |
| Dodgers | 780 | 611 | 95 | 100 | -5 | -5.0 |
| Nationals | 710 | 874 | 59 | 64 | -5 | -7.8 |
| Braves | 735 | 641 | 86 | 92 | -6 | -6.5 |
nice job Lee. Doesn't bullpen/defense also play a role in pythagorean variance? I've always heard that. One thing I've always thought about in regard to those walk off wins is to what degree were the walk offs necessary due to declining performance from Granderson, Magglio, Guillen, and the other "Big Boys"? I wonder if improvements from them would negate the pythagorean effect and it's expected decline next year.
ReplyDeleteoblong, it's been proven that teams with good bullpens tend to out perform their pythagorean estimates and that teams with bad bullpens tend to under perform. I don't know about defense.
ReplyDeleteAs for your other point, teams that don't score or allow a lot of runs will likely be involved in more close games. That probably gives them more opportunities for walk-offs. It also gives them more opportunities to blow games. So, you may be right that better hitting might make their actual wins closer to their estimated wins.
Lee