Fun With Pythagoras #3: Ratios, Runs, and Wins | Disciples of Uecker

Disciples of Uecker

We'd like to go to the Playoffs, that would be cool.

Years ago, I wrote a round-up for Bernie’s Crew titled, “Fun With Pythagoras.” As the Brewers’ disappointing 2010 campaign was winding down, I asked, “By how many runs do the Brewers need to improve to field a competitive team in 2011?”

The question connected to my realization that, in order to win in 2011, the Brewers could conceivably improve their offense, or improve their pitching; my inclination at the time was (a) that there was not a right answer for turning a disappointing club into a contender, and (b) that scoring runs was equally as important to allowing runs (and, of course, (c) that there was no way on earth the Brewers would improve their pitching staff by approximately 100-to-150 runs in one offseason. I was wrong about that). Due to assumption (b), I entered the following experiment:

“Given the choice between a 866 RS / 865 RA Brewers team and a 766 RS / 765 RA Brewers team, I felt confident that I would take either team in order to start a template for a competitive 2011 Brewers team.”

I began studying the actual relationship between runs scored and runs allowed. I realized that by using the Pythagorean W-L calculation, even when separating different models by the same number of runs on the same league average scale, the actual proportion of the runs scored to runs allowed influenced a team’s win total. I wrote:

“For instance, if the Brewers score 155 more runs next year, and finish the season 921 RS / 865 RA, they will be on pace to win approximately 85.6 games. Not bad! However, if they allow 155 fewer runs next year, and finish the season 766 RS / 710 RA, they will be on pace to win 86.6 games.”

In hindsight, I shouldn’t have been so surprised that even when teams shared the same distance between their runs scored and runs allowed, the team that allowed fewer runs could be expected to win more games. The most basic mathematical reason for that phenomenon is that “Runs Allowed” only serves in the denominator when calculating an expected W-L record: [(RS^1.8) / ((RS^1.8)+(RA^1.8))]. A smaller runs allowed total in the denominator changes the ratio between runs scored and runs allowed, even when the basic difference is the same (i.e., 56 runs between 766 RS/710 RA and 921 RS/865 RA).

Beyond stats and numbers, I should have simply thought about it from a sheer baseball perspective. A team that scored 56 runs while allowing 0 runs in ten games would not lose once, but a team that scored 1,056 runs while allowing 1,000 runs over 162 games could lose as many as 80 (or more). The key point is that the balance between a team’s runs scored and runs allowed is crucial to determining its expected win-loss record (or, quite simply, its ability to win as many games as possible with its players’ performances).

Upon analyzing some teams with shared expected W-L records from 2004-2009 seasons, I hypothesized the following rule:

“Where two teams share the same Pythagorean W-L, the team that allows fewer runs will tend to win more games unless the other team manages to match or surpass the proportion of RS to RA of the first team.”

After analyzing the run differentials of teams with the same expected W-L records from 2004-2009 seasons, I found that the hypothetical rule was more of a general guide than a categorical principle; exceptions to the rule jumped out in certain cases. Given the myriad approaches to winning a baseball game, it should never be surprising that a rule about run differentials would have exceptions.

For instance, some teams have great bullpens and win a high percentage of close games, while losing bunches of blowouts. Such a team might win more games than expected because even five 2-1 victories have a tough time catching up to two 11-4 losses. A club that went 18 RS / 27 RA over seven games might be expected to lose a bunch, but since there isn’t a rule about how many runs a team must score to beat their opponent (2-1 is as good as 10-1 in that regard), that 18 RS / 27 RA could shock fans and win five of seven. You might scoff, and it is not common for teams that allow more runs than they score to post winning records, but it has happened — the 2007 Arizona Diamondbacks and 2012 Baltimore Orioles are two examples.


Recently, I began compiling run differentials from clubs with the same expected W-L records over the 2011 and 2012 seasons. I compiled these differentials for the same reason I looked into the issue of run differentials in 2010: the Brewers have a losing record, and need to improve their club. Yet, it’s not simply as easy as saying, “the Brewers need to improve pitching by X runs,” or, “the Brewers need to improve hitting by X runs.” The Brewers need to improve their ratio between RS / RA.

By analyzing the relationship between runs scored and runs allowed through different seasons, we can learn the different ways teams manage to win (or lose) ballgames.

2011 and 2012 Successful Rules

Between 2011 and 2012, approximately sixteen groups of teams shared Pythagorean W-L records. Of those groups, four groups featured successful applications of the hypothetical Pythagorean W-L / Runs Ratio rule.

Team Expected Wins Runs Scored / Allowed Actual Wins
2012 Atlanta 92 700 RS / 600 RA 94
2012 Oakland 92 713 RS / 614 RA 94
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Detroit 88 787 RS / 711 RA 95
2011 Arizona 88 731 RS / 662 RA 94
2011 St. Louis 88 762 RS / 692 RA 90
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Atlanta 85 641 RS / 605 RA 89
2011 Angels 85 667 RS / 633 RA 86
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Washington 78 624 RS / 643 RA 80
2011 Kansas City 78 730 RS / 762 RA 71

The Atlanta and Washington cases are rather straightforward. In each case, the club that allowed fewer runs won more (or, the same number of) games than their Pythagorean W-L partner. So, these three cases are clear examples of the first part of the rule: “Where two teams share the same Pythagorean W-L, the team that allows fewer runs will tend to win more games.” (Even in the 2012 Braves’ case, the Athletics technically surpassed the Braves’ RS/RA ratio, but since both teams won the same amount of games, that kind of disqualifies the example in the first place. So, I thought I’d present it with the successful cases just for fun).

The Detroit / Arizona / St. Louis trio follows the “unless” step in the rule: “…unless the other team manages to match or surpass the proportion of RS to RA of the first team.” I love this example because between these three teams, we can see each mechanism of the rule at work:

(1) Arizona allowed fewer runs than St. Louis and won more games than St. Louis.

(2) St. Louis failed to match or surpass the ratio between RS and RA, and they did not win as many games as Arizona or Detroit.

(3) Detroit allowed significantly more runs than Arizona, but they surpassed Arizona’s ratio between RS / RA, and fittingly won the most games of the trio.

Obviously, there are a lot of dynamics within these teams’ seasons that can help to explain why these W-L records occurred, but in these cases, the hypothetical rule fits their relationship between their expected wins, ratios between RS/RA, and actual wins.

Matched Ratios

These cases are tough, and show some ambiguity to the initial rule: in some cases, when two teams share an expected win total, the team that allows more runs also matches the RS/RA ratio without necessarily winning more games; in other cases, the rule is successful.

Team Expected Wins Runs Scored / Allowed Actual Wins
2012 Dodgers 86 637 RS / 597 RA 86
2012 Arizona 86 734 RS / 688 RA 81
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Oakland 77 645 RS / 679 RA 74
2011 Colorado 77 735 RS / 774 RA 73
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Cleveland 75 704 RS / 760 RA 80
2011 White Sox 75 654 RS / 706 RA 79
Team Expected Wins Runs Scored / Allowed Actual Wins
2012 San Diego 75 651 RS / 710 RA 76
2012 Mets 75 650 RS / 709 RA 74
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Baltimore 67 708 RS / 860 RA 69
2011 Seattle 67 556 RS / 675 RA 67

The success rate of these examples is split down the middle. Cleveland, San Diego, and Baltimore allowed more runs than their Pythagorean partner, but they won more games and matched the RS/RA ratio of their partner. Colorado and Arizona matched the RS/RA of their Pythagorean partners but won fewer games. In this regard, we see that simply matching the RS/RA ratio doesn’t necessarily mean more wins in each and every case.

Exceptions / Oddities / Failures

At the end of the day, we shouldn’t be surprised that by studying a hypothetical rule about baseball teams and their run differentials, there are arguably as many oddities, exceptions, and failures as successful cases for the rule. Really, for just about every general rule one can build about baseball, there are exceptions to that rule, down to the most basic rule: “A team that allows more runs than they score should be expected to lose more than they win.” This might be a rule with relatively rare exceptions, but the exceptions occur nonetheless. So too, with this hypothesis about how many wins can be expected when comparing two teams with the same expected wins total.

Team Expected Wins Runs Scored / Allowed Actual Wins
2012 Yankees 95 804 RS / 668 RA 95
2012 Tampa Bay 95 697 RS / 577 RA 90

This Yankees/Rays exception is straightforward. The Yankees allowed more runs than the Rays, but they did not match or surpass the RS/RA threshold (which was 807 RS, in this case). For what it’s worth, this exception cannot be explained by head-to-head play; not only did the Rays allow notably fewer runs than the Yankees, but they also beat the Yankees 10-8 (76 RS/74 RA).

Team Expected Wins Runs Scored / Allowed Actual Wins
2012 San Francisco 88 718 RS / 649 RA 94
2012 Angels 88 767 RS / 699 RA 89
2012 White Sox 88 748 RS / 676 RA 85
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Toronto 79 743 RS / 761 RA 81
2011 Mets 79 718 RS / 742 RA 77
2011 San Diego 79 593 RS / 611 RA 71
Team Expected Wins Runs Scored / Allowed Actual Wins
2012 Toronto 74 716 RS / 784 RA 73
2012 Kansas City 74 676 RS / 746 RA 72
2012 Boston 74 734 RS / 806 RA 69

Now, three tricky trios, which perhaps suggests that this rule about expected W-L records might only be applicable to two team comparisons, rather than three team comparisons. Analyzing these trios of teams devolves into some kind of gossip about RS/RA, and adds difficult cases of which RS/RA ratios are most important (I still used the lowest RA total as the basis for analysis under this rule).

(1) Giants / Angels / White Sox: If this example only included the Giants and Angels, the rule holds: San Francisco allowed fewer runs and the Angels failed to match the RS/RA ratio of the Giants, which fits the rule of the Angels sharing an expected W-L record while failing to win as many games as the Giants. However, the White Sox matched the RS/RA ratio of the Giants, which means that even though they allowed nearly 30 more runs than the Giants, they also had a stronger ratio of runs scored to runs allowed.

What a strange team were the 2012 White Sox. They underplayed their run differential despite (a) winning more one-run games than they lost, and (b) destroying their opponents in blow outs. 89 of their 162 games were blow outs or one-run games, and the White Sox went 52-37 in those contests (425 RS / 345 RA). This means that in 73 remaining games, the White Sox went 33-40 despite scoring 323 runs and allowing 331 runs.

(2) Blue Jays / Mets / Padres: At what point does a team’s low runs allowed total become inconsequential? Take the 2011 Padres, who allowed 611 runs. That’s an extremely low runs allowed total, but the club also plays in an extreme pitcher’s park, and, of course, their offense managed to score 593 runs. Yes, a 593 RS / 611 RA differential is worth 79 Pythagorean Wins, but at some point the league context must kick-in, and the simple difficulty of winning games with less-than-four runs scored per game kicks in.

Nevertheless, once again, one team fits the rule: the Blue Jays surpassed the RS/RA ratio of the Padres, and they won 10 more games than the Padres despite allowing 150 more runs (they also had a park rating on the other end of the spectrum). So, the Mets are the oddity here, as they failed to match the Padres RS/RA ratio, but still managed to win more games despite allowing more runs. This is where I think the Padres’ basic disadvantage of scoring 593 runs kicks in. In some regard, with their extreme park environment, it makes comparisons to their run differential, expected wins, and actual wins difficult and less meaningful.

(3) Blue Jays / Royals / Red Sox: Here we find a degree of rules and ratios between these clubs. Although the Royals allowed the fewest runs of this trio, the Blue Jays surpassed the Royals’ RS/RA ratio (and, oddly enough, won more games than the Royals). The Red Sox are the oddity, but they also allowed the most runs of the group. In this case, even though they surpassed the Royals RS/RA ratio, they did not surpass the Blue Jays’ RS/RA.

Perhaps a more straightforward explanation: in 2012, the Blue Jays went 11-7 against the Red Sox, and 6-2 against the Royals. By contrast, the Red Sox scored 44 runs and allowed 35 against the Royals, but only managed to turn that differential into a 4-3 record; of course, they also had that losing record against the Blue Jays. This is one case where head-to-head records might help explain some of the discrepancy between overall W-L records and run differentials.

Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Pittsburgh 70 610 RS / 712 RA 72
2011 Cubs 70 654 RS / 756 RA 71
Team Expected Wins Runs Scored / Allowed Actual Wins
2012 Miami 68 609 RS / 724 RA 69
2012 Minnesota 68 701 RS / 832 RA 66
Team Expected Wins Runs Scored / Allowed Actual Wins
2011 Minnesota 62 619 RS / 804 RA 63
2011 Houston 62 615 RS / 796 RA 56

I don’t mean to be rude to Marlins, Astros, Twins, Cubs, and Pirates fans, but these examples disproved my hypothesis without providing excitement. The Cubs surpassed the Pirates’ RS/RA ratio but won one less game than the Pirates. The 2011 Twins and 2012 Twins suffered exactly opposite fates; in 2011, the Twins did not surpass the Astros’ RS/RA ratio, but still managed to win seven more games. However, in 2012, the Twins did surpass the Marlins RS/RA ratio without winning more games than the Miami Nine.

With the exception of the 2011 Orioles/Mariners example above, these three duos claim the lowest Pythagorean win totals in this study. Which leads me to ask, “at what point does a losing run differential become impervious to trends and rules?” With these three final duos, we’re talking about expected win totals of 70, 68, and 62, each requiring their respective club to allow at least 100 more runs than they scored. Perhaps that outlines a future project: do clubs with run differentials greater than 100 experience more extreme shifts between their expected and actual win totals?


A common refrain: the Brewers need to improve their pitching. The Brewers need to develop pitching. The Brewers would be a better club with better pitchers. We will hear this throughout the 2013 season, the 2013 offseason, and into spring training 2014. I promise you, it will not go away.

Yet, we should not commit ourselves to such stark analyses. Pitching does not win championships categorically, but only where hitting fails to win championships. In this regard, we should always keep in mind the importance of the ratio between runs scored and runs allowed; more than improving pitching, more than maintaining hitting, more than improving hitting, more than developing a strong farm system, a team’s balance between their runs scored and runs allowed will influence their ability to win games. We may now understand that there are exceptions to this rule, but this is more of a lesson about making generalizations about baseball; baseball scoffs at general rules.

Still, the fastest way for the Brewers to improve their fortunes is through the twists and turns of Pythagoras’s universe: improving pitching sounds great, if the club maintains their hitting; absent that, the club had better hit like hell.

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Tell us what do you think.

  1. Luke says: June 13, 2013

    But on the bright side the development of pitchers is looking pretty good even as guys are moving through the system. They are either maintaining or improving their performances. Nelson, Pena, Goforth, Jungmann, Hellweg, Magnifico, Bradley….even Gagnon and Thornburg putting up nice peripherals without the spiffy ERAs to show for it. Bucci, Scarpetta and Burgos perhaps will contribute if they overcome injuries. Hell, at least one of these guys has to turn into a halfway decent #3 right? RIGHT?

    • Nicholas Zettel says: June 13, 2013

      We can certainly hope so! Let’s hope for a strong second half of 2013 for that group. That could help the organization make some decisions about trades, rebuilding, offseason acquisitions, etc., sooner rather than later.

  2. Andyroo says: June 13, 2013

    I appreciate the time and work you put in to this, but isn’t it all a bit overly technical? The Brewers have the seventh worst run differential in baseball right now. They rank 17th in runs scored and 26th in runs allowed. They have one of the worst starting pitcher units in all of baseball, as measured by ERA and BAA. This would suggest there is more room for improvement, and improvement more needed (relatively speaking), in runs allowed, wouldn’t it? Which is to say, while the ratio is undoubtedly important, as a practical matter the Brewers will be more likely to improve their ratio by expending energy and resources to improve their pitching, right?

    • Nicholas Zettel says: June 13, 2013

      I’d certainly agree that it’s technical, but I think it’s important to think about the various elements to constructing a ballclub that could, say, win 90 games. A club can only win as many games as their RS / RA in many cases, but where they diverge from their RS / RA, there are tons of lessons about the game. This is why I focus on comparing clubs with the same expected W-L and how their actual RS / RA and actual W totals fared.

      If you’re a GM and you aim to win 90 games based on your run estimates for your roster, isn’t it important to understand the factors that could cause your club’s actual W total to diverge from expectations?

      I absolutely agree with the fact that the Brewers need to improve their pitching; I think that’s beyond true for every Brewers analyst, fan, and writer. What is important is to keep other lessons about the relationship between RS / RA in mind while thinking about roster improvements.

      So, yes, indeed technical. But, hopefully some interesting lessons, too.


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