If you're a devotee of The Stats That Matter Most, you're already familiar with the principles behind baseball's Pythagorean Theorem.  In the extraordinarily unlikely event that you haven't read every word of the Stats That Matter Most feature, here's a brief description of the theorem:

Many moons ago, trendsetting sabermetrician Bill James demonstrated a relationship between runs scored, runs allowed and wins which, because of the necessity of squaring the elements, he dubbed the Pythagorean Theorem (of baseball).  The formula is that the square of runs scored divided by the sum of the square of runs scored and the square of runs allowed equals a team's expected winning percentage.  From there, you simply multiply the winning percentage and games played and you have a projected win total.  The measure works remarkably well, and, obviously, the more games a team plays, the better the relationship ordinarily is.

That there is a relationship between the number of runs a team scores and allows--essentially, its run differential--and its record is intuitive.  That the relationship is close to perfect when the elements are squared, added and divided in the particular manner described above is less obvious, but there it is.

When analyzing the number for The Stats that Matter Most, it's been commonplace to make statements about win differentials--that is, the difference between a club's actual wins and the wins that the Theorem projects it should have based on the runs it has scored and allowed.  I thought it might be interesting to take a look at how well the Theorem holds up when applied to the past and what sorts of things we can say about various win differentials.

The inspiration to run the Pythagorean numbers for every team in the modern (post-1900) history of baseball was the performance of the 2004 New York Yankees who finished with twelve more wins in real life than the Theorem projected.  I'd been calculating the Stats That Matter Most numbers for about five years and that was easily the largest win differential I'd seen.  I had a feeling it was one of the largest ever, but I really didn't know for certain.  So, I decided to take a look.

Disclaimer

There are no wondrous conclusions to draw from what follows.  It's worth mentioning, I suppose, that this is pretty well-trodden ground.  The Theorem is, I would say, a reliable, useful tool for quickly identifying whether a team is performing significantly above or below what one could reasonably expect, which is pretty much the way we've been using it all along.  I have found that the Theorem is notably more predictive of baseball in the divisional era than it is of the more distant past, as you'll see below, but I don't think that's worth getting particularly excited about.  The information in this piece is more of the whimsical "isn't that interesting" variety than the epiphanic "omigosh" sort.  (Well, I hope it's interesting anyway.)

The Numbers

Here are some basic stats to chew on, covering all teams since 1900:

What this tells us is that the Theorem is slightly overestimating the value of run differential.  Teams are coming up, on average roughly 2/5 of a win less than the Theorem projects.  The standard deviation for WDIFF is less than 4.2 wins.

After eyeballing the entire set of statistics, I decided to see if the numbers changed when broken down into a set of distinct eras.  Here's what I found:

What you can is that the Theorem becomes far more accurate in the post-expansion era of baseball.  It still overestimates wins, but by the time you come to the era of divisional play, beginning in 1969, the discrepancy is significantly below 1/10 of a win per season per team.  The model is, by far, least predictive during the 1946-60 era.  The greater number of games played after the 1961/1962 major league expansion would be a possible partial explanation, but the number of contests per season was the same (154 games) for virtually all of the 1901-45 era as the 1946-60 era.  Also note that the WDIFF standard deviation has oscillated over time as well, but has gradually shrunk since the 1946-60 era to the point where it has dropped just below four in the era of divisional play.  

Of course, there's the obvious question: why does the model do so much "better" with these more recent seasons?  I'm not certain, but I think it has to do with the relative level of parity that marks the last 35+ seasons of big league ball relative to prior eras.  For all the carping in recent years about the dominance of teams like the Yankees and Red Sox, the fact is that the 1970s, 1980s and at least the first half of the 1990s represented as evenly matched a period as has ever been seen in the modern history of baseball.  Evenly distributed talent pools presumably mean fewer of the blowout games that most dramatically confound the predictive computations of the Theorem.  The flip side of course implies that the so-called "Golden Age of Baseball" (from 1946 or '47 through the late 1950s) coincided with an unprecedented era of stratification in terms of the distribution of talent across major league teams, which led to larger margins of victory in individual games which led to a less predictive model.

So, we know that the average team from 1960 on had a -.0659 WDIFF and the standard deviation for this segment of the population is almost exactly four wins.  If the sample is normally distributed--which is a pretty reasonable expectation--we'd expect roughly 2/3 of the cases to fall within one standard deviation of the mean and approximately 95% of them to fall within two standard deviations of the mean.  712 of the cases in this segment are within four games--plus or minus--of zero in the WDIFF category, which is 74%.  925 of the cases in this segment are within plus or minus eight games of the mean; that's roughly 97% of the sample.  It's not perfect but for all practical purposes, this segment resembles a normal distribution and can be treated as such.

Given that the Theorem is so much more predictive since divisional play began and given that there have been 958 "team seasons" since 1969 (a nice sample), I'll focus the lion's share of the analysis on this era.

Negative Outliers:  The Underachievers

18 teams since 1969 have finished more than eight games below their projected Pythagorean win totals:

This is a fairly balanced list of clubs.  Only one (the 1990 Mets) projects as an extremely good team; the Theorem had them winning 100 games and they settled for 91 in real life.  There are some other pretty good clubs on this list--the 1970 Cubs, the 1990 Brewers, the 1997 Astros.  There are some high scoring clubs (the 2001 Rockies, the 1999 Royals, the 1980 Brewers, the 1970 Cubs, etc.) and some low scoring teams (the 1975 Yankees, the 1984 Pirates, the 1972 Orioles, the 1974 Angels, etc.) 

What is missing is a list of truly dreadful clubs...and that shouldn't be a surprise.  Imagine a team that projected to lose 105 games; to make this list, that club would have to actually lose 115-odd games.

Positive Outliers:  The Overachievers

15 teams since 1969 have finished more than eight wins above their projected win totals:

This isn't merely the flip side of the positive group.  There are several really genuinely lousy clubs on this list--the 1974 Padres, for instance, are one of the worst teams of all time in terms of run differential.  Only two teams since 1969 (the 1996 Tigers and the 2003 Tigers) have had a worse negative run differential than the '74 Padres.  (22 pre-1969 clubs had run differentials worse than -289; only four of those clubs were post-WWII).  The 1998 Royals, the 1978 Reds, the 2004 Reds and the 1970 Phillies were all truly poor clubs.  There were a couple of pretty good teams--the 1970 National League champion Reds--in fact the Reds are on this list four separate times and the Mets three times--and the 2004 American League East champion Yankees, the only true really high scoring team on the list.  There are numerous low scoring clubs--the '74 Padres, the incredibly offensively challenged 1972 Angels (since the beginning of divisional play, only the '69 Padres have scored fewer runs per game than the Angels of 1972), the '72 Mets, the 1978 Athletics, the 2001 Mets.

Not surprisingly, in line with what we saw with the underachievers, there are no exceptionally good teams on this list (no club above projected to win more than 91 games).  Imagine a team that really won 105 games; how often is a squad so good that it "should" have won 115?

Regressing Toward the Mean

Bill James always speculated that clubs that were dramatically worse (or better) than their Pythagorean projection would regress toward the mean in the succeeding season.  Everyone, it would be expected, would not only see their WDIFF be far closer to zero the following season, but also see their record go in the direction of the previous season's projection.  For instance, on balance, teams that finished far below their win expectations would win more games the following season.  Overachieving teams, naturally, could be expected to sink back in the direction of their projected number of wins.

Obviously this is an imperfect system because franchises don't simply remain static from year to year.  They make personnel changes, and so do the teams they compete with.  Players develop.  Players become injured.  Players heal.  All of these things, and others, impact the performance of clubs.

Still, the expectation is that underachieving teams--particularly dramatically underachieving teams--will perform better in succeeding years and the overachieving teams--especially substantially overachieving teams--will perform worse in succeeding seasons.  There's a sense that, historically, clubs have been easily fooled by biases; teams playing in good offensive ballparks have a tendency to overrate their offense and underrate their pitching, for instance.  There's an analogous notion that teams are easily misled by their record.  A 90-win club with a +10 WDIFF, for example, is probably going to regard itself as just a move or two away from pushing 100 wins rather than as a .500 club in need of an overhaul.

How does this play out with these outlying clubs, the ones who should be most likely to regress?

First, the underachievers

*--because of the work stoppage in 1994, the Mets only played 113 games; the team's Win% went from .364; .487
**--because of the work stoppage in 1981, the Cardinals played only 103 games; the team's Win% went from .457 to .578
***--because of the work stoppage in 1994, the Padres played only 117 games; the team's Win% went from .377 to .402
****--because of the work stoppage in 1981, the Brewers played only 109 games; the team's Win% went from .531 to .569

14 of the 18 teams above improved in their actual winning percentage from year 1 to year 2.  Only a few teams clearly got worse--the 1984-85 Pirates and the 1990-91 Mets.  The 2001-02 Rockies also slid back in terms of overall play if not wins and, to a more modest extent, so did the 1970-71 Cubs, the 1984-85 Astros and the 1974-75 Angels.

No clubs had consecutive outlying seasons, though a few (most notably the 1972-73 Orioles) came pretty close.

The overachievers:

*--because of the work stoppage in 1981, the Reds played only 108 games; the team's Win% went from .611 to .377

Two of the 15 teams in the above list are incomplete since we don't yet have a follow-up season...though it's highly likely that both the Reds and the Yankees will win fewer games in 2005 than they did in 2004.  If that happens, nine of the clubs on this list will have regressed in the expected direction, but several improved to a surprising degree (the '74 Padres and the '84 Mets in particular).  The overall trend is in the anticipated direction but the consistency of that tendency is less professed than expected.

The 1998-99 Royals are one of the more intriguing teams I've seen, going from a +10 to a -11 WDIFF in one season.  They are the only club in the segment to have consecutive outlying seasons, even if they are in opposite directions.  Very odd.  The Royals were almost certainly a significantly better team in 1999 when they went 64-98 than in 1988 when they went 72-90.  What is the explanation for this?  This is a unique case out of a sample of nearly 1000.  Chalk it up to randomness.

A Wider Historical View

The Theorem may be less accurate when applied to the pre-divisional years of Major League Baseball, but I thought it would be interesting to take a brief look at some of pre-1969 outlying clubs.

Underachievers