Major League Baseball's Pythagorean Expectation - jmik58's Blog

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Major League Baseball's Pythagorean Expectation Stuck

Posted on August 20, 2012 at 10:41 AM.

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When the recently-busted Melky Cabrera made the decision to use a banned substance in hopes of boosting his on-field performance, perhaps his error was actually a simple one that many humans make.

Maybe while walking down the hall of his local high school he slipped up and took a left when he should have turned right -- ending up in chemistry class instead of his scheduled mathematics course. Surely that's it. Cabrera just wanted to help his team win. He heard about this great thing called Pythagorean Expectation -- or "P.E." as the kids affectionately refer to it -- and he had to get some.

Unfortunately for Cabrera, it seems as if a strange coincidence occurred where a letter-grade "D" was written inconveniently on his paper next to a note about "P.E." -- leading the troubled youngster to instead chase the truth on P.E.D's straight into the chemistry lab and onto MLB's suspension list.

Had he made it to the correct place he would have been opened up to the world of the more beneficial -- and legal -- Pythagorean Expectation.

Formulated by baseball statistician guru Bill James, P.E. was created to help determine how many games each MLB team should win.

The formula incorporates a team's runs scored, runs allowed, and games played. James' goal: to create a mathematically superior doppelganger for the almighty "win."

To arrive at our assumed win total for a team, the formula requires that a ratio (expressed in decimal form) be multiplied by the total number of games a team has played. Roughly stated, the ratio is formed when the numerator (runs scored) is divided by the denominator (runs scored + runs allowed). Each variable within the formula is also multiplied by an exponent (originally to the power of "2"), but a definitive value has not been reached for what the exponent should be. Nonetheless, many agree on a more complex and statistically accurate method called the Pythagenpat formula to determine the value of the exponent (click here for more details).

But enough of the geeky baseball sabermetric lingo, let's look at how it applies to the teams of the current MLB season.

To use an example from standings on 8/16/12; the Washington Nationals had scored 525 runs, allowed 420 runs, and played 118 games. Incorporating the formula using the Pythagenpat formula (developed by David Smyth) for the exponent value, we arrive at a value of .55556. That value is then multiplied by the number of games the Nationals had played (118) to arrive at our expected win total of 73.81.

By comparison, the Nationals had 73 wins going into 8/16/12, only .81 wins lower than their Pythagorean Expectation (73.81) suggests. In other words, Washington has performed within the expected correlation based on their win total in comparison to James' formula.

As is with any sample size, there will always be outliers at each end of the spectrum. In the realm of P.E. that means clubs will win more than expected while others will under-perform, or lose more than they should.

Sticking with the same date as before (8/16/12), the teams with the largest advantage between actual vs. P.E. wins were: Baltimore (+11.37), Cleveland (+10.60), Miami (+7.02), Pittsburgh (+3.49), and Minnesota (+3.41).

Conversely, the worst offenders that had struggled in the win column versus what their Pythagorean Expectation showed were: St. Louis (-9.97), Boston (-6.73), Arizona (-4.94), Milwaukee (-4.14), and the Chicago White Sox (-3.82).

Before judging the numbers, it's important to note that the average margin of error for the formula is approximately three games over the course of a full 162-game season. In other words, a team with a Pythagorean Expectation of 100 wins should finish with between 97 and 103 wins. On 8/16/12 the league-wide average was approximately -3.98 -- the equivalent of losing four less games than what the Pythagorean Expectation would suggest. This under-performance, however, is still relatively close to the accepted margin of three games.

Based on this information it would be a fair guess to expect that number to shift closer to -3.00 (at the lowest) before the season is finished. For this to happen, teams will have to -- on average -- win more games than their Pythagorean Expectation would suggest for the remainder of the season. In practical terms, expect quality teams with P.E. scores below -3.00 to win more by season's end, or win at the same rate while slacking on their run differential. This means St. Louis (-9.97), Boston (-6.73), Arizona (-4.94), the Chicago White Sox (-3.82), NY Yankees (-3.56), and LA Angels (-3.25) all should win at a higher rate than they have up to this point of the season -- assuming they continue to score runs and allow runs at the same rate.

Conversely, it's realistic to expect teams at the top end to fall as well. If the numbers hold true and the law of averages play out, this spells disaster for Baltimore (+11.37) and could mean that Pittsburgh's (+3.49) recent dance with playoff contention could start to fade. It could also mean these teams continue to win while their run differential shrinks at a faster rate (a situation that seems difficult to accept as realistic).

For a nice example that we can focus on for the remainder of the season, the NL Central provides a great case study in the making. Considering the Pirates and Cardinals were tied in the standings on 8/16/12, James' Pythagorean Expectations suggest that St. Louis has nowhere to go but up and Pittsburgh should slide a little, or hold where they are at best.

As a side note, it's also somewhat comical (and painful) to see teams like Cleveland and Miami have such high P.E. ratings while still sporting unsavory records. Not only have the Indians and Marlins struggled to win games this year, but according to the Pythagorean Expectation each has actually over-performed. Ouch.

As a final observation, it's important to remember that P.E. isn't necessarily a predictive tool as much as it is an after-the-fact assessment measure. Like many of Bill James' statistical creations, Pythagorean Expectation helps to tell a more accurate story of a team's success rather than wins alone -- much the way OPS is widely accepted as a more accurate measure of player skill than that of batting average.

Regardless of best use, the formula has evolved to a point where the margin of error can be used loosely to monitor the law of averages and the subsequent ebb and flow as I did earlier. Whether P.E. accomplishes anything in the way of predicting World Series teams or champions is yet to be studied extensively to my knowledge.

The baseball season will soon climb out of the dog days and into the home stretch for the playoff run. To get there, teams obviously have to win more games than the competition, but keep in mind that the win itself doesn't always tell the story.

A handful of teams on the cusp of the postseason hope this to be true as the season winds down. They also hope their players stay away from chemistry class.

[Click HERE to download the Excel file with Pythagorean Expectation data through 8/16/12]

Justin Mikels is a staff writer for Operation Sports. Follow him on Twitter @long_snapper or send him a PM with your comments, questions, or future blog ideas.