Pythagorean Wins (PW) is a concept invented by Bill James which numerically states the obvious--teams that score more runs than their opponents typically win more games. This is the formula:
Runs Scored ^ 2 / ((Runs Scored ^ 2) + (Runs Allowed ^ 2))
It's a retrospective formula in that it can only describe what already occurred, but it still has value. It's common to relegate major differences between the formula and actual records to "luck." For example, if a team wins nine games by one run and loses the tenth 10-1, the PW formula would suggest a .500 record instead of the actual 9-1 performance. This is an extreme example but describes some of the variability involved.
Can PW be used to evaluate managerial effectiveness? I'll use the manager with the most wins (regular season only) in baseball history, Connie Mack, as an example:
Mack had an actual career winning percent of .486, whereas the PW formula suggests his teams should only have won at a .483 rate, suggesting his teams outperformed expectations by 23 wins. In 53 years as a manager that worked out to around one-half a win per year.
The chart below shows managers of more recent vintage:
PWs dropped dramatically as these managers managed far fewer games than Mack. But this is only half the story--let's look at Tony LaRussa season-by-season to see how PWs move (playoff years in bold):
|Chicago White Sox||1980||70||90||.438||587||722||.398||6.3|
|Chicago White Sox||1981||54||52||.509||476||423||.559||-5.2|
|Chicago White Sox||1982||87||75||.537||786||710||.551||-2.2|
|Chicago White Sox||1983||99||63||.611||800||650||.602||1.4|
|Chicago White Sox||1984||74||88||.457||679||736||.460||-0.5|
|Chicago White Sox||1985||85||77||.525||736||720||.511||2.2|
|Chicago White Sox||1986||26||38||.406||280||324||.428||-1.4|
This is the real story--what luck gave in one year was taken away in another, as PWs fluctuated from as many as 7.8 in 2007 to as few as -6.1 in 2010. There's a randomness that had a tendency to regress to the mean, suggesting it really is luck, the kind of luck seen in having a better record in one-run games or having fewer blowout games, but it's difficult to credit this to the manager.
If PWs fall short, what can be used to measure manager effectiveness? I'll throw out a couple of ideas, neither of which is perfect. The first is record in one-run games (1RGw and 1RGl) and the second the number of games in first (G1st). Here are these values for the recent managers:
I suspect one of the best predictors of a good manager is having good players. Tony LaRussa was considered a bright young manager but it took the Oakland Athletics of the late 1980s to propel him to among the best in the game (along with George Will's book "Men at Work"). When the Athletics regressed in the early 1990s, LaRussa's genius could do nothing except perhaps soften the slide. Joe Torre was average, at best, in his three stops prior to joining the Yankees, and Bobby Cox had been underwhelming in stints with Atlanta and Toronto prior to installing himself as Braves manager in 1990.
None of this is to diminish the role of a manager--it's one thing to look back retrospectively and entirely another to make in-game decisions, especially with the game on the line. But it's too much to take PWs and use them as a proxy measure of managerial skill. I've introduced the Mistake Index before (in this Beyond the Box Score post and more completely here), which suggests the number of mistakes correlates well with a team's record. If a manager can decrease mistakes, he certainly can play a role, but that's not luck--that's skill and the ability to teach, two highly desired and valuable qualities.
There's a reason Pythagorean Wins aren't listed on Baseball-Reference manager pages--it's not an accurate measure of managerial effectiveness. Like many other numbers, it's fun to look at to see if it increases our understanding, but in this case it appears to add little value. The next big thing in measuring managerial effectiveness is right around the corner as MLB Advanced Media (MLBAM) installs cameras that can track each play. When that is fully implemented the things a manager can really control, like fielder positioning can be measured and evaluated. Until then, we do the best we can with the tools at hand, and Pythagorean Wins for managers doesn't appear to be the best suited metric.
All data from Baseball-Reference.com