Explaining a Team's W-L Record

According to Baseball-Reference.com:

The Pythagorean Theorem of Baseball is a creation of Bill James which relates the number of runs a team has scored and surrendered to its actual winning percentage, based on the idea that runs scored/runs allowed is a better indicator of a team's (future) performance than a team's actual winning percentage. This results in a formula which is referred to as Pythagorean Winning Percentage....

There are two ways of calculating Pythagorean Winning Percentage (W%). The more commonly used, and simpler version uses an exponent of 2 in the formula.

W%=[(Runs Scored)^2]/[(Runs Scored)^2 + (Runs Allowed)^2]

More accurate versions of the formula use 1.81 or 1.83 as the exponent.

W%=[(Runs Scored)^1.81]/[(Runs Scored)^1.81 + (Runs Allowed)^1.81]

Expected W-L can then be obtained by multiplying W% by the team's total number of games played, then rounding off....

The rationale behind Pythagorean Winning Percentage is that, while winning as many games as possible is still the ultimate goal of a baseball team, a team's run differential (once a sufficient number of games have been played) provides a better idea of how well a team is actually playing. Therefore, barring personnel issues (injuries, trades), a team's actual W-L record will approach the Pythagorean Expected W-L record over time, not the other way around. Expected W-L is almost always within 3 games of actual W-L at the end of a season (although a recent exception is the 2005 and 2007 Arizona Diamondbacks, who both beat their expected W-L by 11 games). Deviations from expected W-L are often attributed to the quality of a team's bullpen, or more dubiously, "clutch play"; many sabermetrics advocates believe the deviations are the result of luck and random chance.

I agree with those who say that deviations reflect the quality of a team's bullpen. A more precise formula can be obtained by regressing winning percentage on two explanatory variables: RFA (runs scored/[runs scored + runs allowed]) and saves recorded by a team's bullpen. The result for the American League in 2008:

W-L percentage (expressed as a decimal fraction) = -0.44595 + 1.66556 x RFA + 0.002747 x saves

Adjusted R-squared: 0.899; standard error: 0.022 (i.e., 2.2 percentage points); t-statistics on the intercept and coefficients: -4.246, 7.319, 3.763 (all significant above the 0.99 level).

That is, the average American League team (RFA = .506, saves = 41) compiled a W-L percentage of .510. (The AL beat the NL in interleague play, thus enabling the AL as a whole to compile a better-than-.500 average.)

According to the Pythagorean formula, the LA Angels were the lucky recipients of 11 extra wins in 2008; that is, the formula underestimates the Angels' 2008 wins by 11. The regression equation, on the other hand, underestimates the Angels' 2008 wins by only 2. Generally, the regression equation (indicated by blue) gives much better results than the Pythagorean formula (indicated by black):


"Luck" is a catch-all term for unexplained variance. It shouldn't be thrown around as if it has real meaning. In this case, the evidence suggests that a decisive factor in a team's W-L record is the quality of its bullpen -- especially the quality of its closers.

Checking In

UPDATED, 08/11/10

About four years ago (at Liberty Corner) I drew on the archives of Dead or Alive? to list a number of erstwhile celebrities who were then alive at the age of 90 or older. Here's how the list looks today:

Charles Lane 102, George Kennan 101, George Beverly Shea 101, Max Schmeling 99, Eddie Albert 99, Michael DeBakey 99, Luise Rainer 100, Gloria Stuart 100, Dale Messick 98, John Wooden 99, Mitch Miller 99, John Kenneth Galbraith 97, Ernest Gallo 97, John Mills 97, Estée Lauder 97, Al Lopez 97, Karl Malden 97, Art Linkletter 97, Risë Stevens 97, Fay Wray 96, Kitty Carlisle 96, Jane Wyatt 96, Tony Martin 96, Kevin McCarthy 96, Irwin Corey 96, Henri Cartier-Bresson 95, Peter Rodino, Jr. 95, Joseph Barbera 95 ,Jack LaLanne 95, Harry Morgan 95, Herman Wouk 95, Byron Nelson 94, Constance Cummings 94, Lady Bird Johnson 94, Robert Mondavi 94, Sammy Baugh 94, Les Paul 94, Sargent Shriver 94, Eli Wallach 94, Olivia de Havilland 94, Artie Shaw 93, Frankie Laine 93, Ruth Hussey 93, Richard Widmark 93, Robert McNamara 93, Ernest Borgnine 93, Zsa Zsa Gabor 93, Vera Lynn 93, Oleg Cassini 92, Ralph Edwards 92, Lena Horne 92, Ernie Harwell 92, Herbert Lom 92, Patti Andrews 92, William Westmoreland 91, Frances Langford 91, John Profumo 91, Geraldine Fitzgerald 91, Archibald Cox 91, Julia Child 91, Bob Feller 91, Billy Graham 91, Monte Irvin 91, Jane Wyman 90.

For many, many more names, go to "People Alive Over 85" at Dead or Alive?

Baseball in the Nation's Capital, Revisited

Way back in September 2004, before the Montreal Expos became the Washington Nationals, I wrote:

To succeed financially, the new Washington team must draw well from the Maryland and Virginia suburbs. Attendance will be high for a few years, because the closeness of major-league baseball will be a novelty to fans who've had to trek to Baltimore to see the increasingly hapless Orioles. But suburbanites' allegiance to the new Washington team won't survive more than a few losing seasons -- and more than a few seem likely, given the Expos' track record. As the crowds wane, suburbanites will become increasingly reluctant to journey into the city. And, so, the taxpayers of D.C. (and perhaps the taxpayers of the nation) are likely to be stuck with an expensive memento of false civic pride.

I'm not sure about this year's attendance, but the trend is almost certain to be downward, given the Nats steady dive toward the bottom of the National League. Here are the Nats' W-L records from 2005 through yesterday:

2005 - .500 (5th of 5 in their division; tied for 9th in their 16-team league)

2006 - .438 (5th of 5 in their division; 14th in the league)

2007 - .451 (4th of 5 in their division; tied for 11th in the league)

2008 - .366 (5th of 5 in their division; last in the league)

2009. - .345 (5th of 5 in their division; last in the league)

As I've said before, D.C. isn't a baseball town. The teams are jinxed by their non-fans.

Timely Trivia Question

One person administered the presidential oath of office nine times (a record). Who was that person, and to which presidents did he administer the oath? Scroll down for the answer.



















John Marshall, Chief Justice of the United States from 1801 to 1835, administered the oath to Thomas Jefferson in 1801 and 1805, James Madison in 1809 and 1813, James Monroe in 1817 and 1821, John Quincy Adams in 1825, and Andrew Jackson in 1829 and 1833.

Roger B. Taney, Marshall's successor as Chief Justice (1836 to 1864), administered the oath of office seven times. Warren E. Burger (Chief Justice from 1969 to 1986) administered the oath six times.

For more trivia about inauguration day, go here.

Math Puzzler

Here is the problem (from Misha Lemeshko, via Eugene Volokh):

8809 = 6
7111 = 0
2172 = 0
6666 = 4
1111 = 0
3213 = 0
7662 = 2
9312 = 1
0000 = 4
2222 = 0
3333 = 0
5555 = 0
8193 = 3
8096 = 5
7777 = 0
9999 = 4
7756 = 1
6855 = 3
9881 = 5
5531 = 0

2581 = ?

I found the general and specific solutions to the problem after pondering it for about 15 minutes. Can you do it?

If you've given up, or want to check your answers against mine, scroll down.



















Specific solution: 2581 = 2, because...

General solution: The value of a string of numbers comprising the integers 0, 1, 2, 3, 5, 6, 7, 8, 9 is equal to the sum of the values of the integers contained in the string, where the value assigned to each integer is equal to the number of closed curves contained in it. Thus: 0 = 1, 2 = 0, 3 = 0, 5 = 0, 6 = 1, 7 = 0, 8 = 2, and 9 = 1. Therefore, for example, 0000 = 4 because each integer in the string has 1 closed curve; that is, 1 + 1 + 1 + 1 = 4.

Note that the preceding general solution omits the integer 4. Why? There is no way of determining the value of 4 because it doesn't occur in Lemeshko's list of strings. If, however, the value of 4 were known to be 0 (e.g., 8884 = 6, 1114 = 0), the general solution would be as follows: The value of a string of numbers comprising the integers 0 through 9 is equal to the sum of the values of the integers contained in the string, where the value assigned to each integer is equal to the number of closed curves contained in it. Thus: 0 = 1, 2 = 0, 3 = 0, 4 = 0, 5 = 0, 6 = 1, 7 = 0, 8 = 2, and 9 = 1. Therefore, for example, 4444 = 0 (0 + 0 + 0 + 0 = 0) because 4 (in standard typography) contains a closed area but not a closed curve.

If, however, the value of 4 were known to be 1 (e.g., 8884 = 7, or 1114 =1), the general solution would be as follows: The value of a string of numbers comprising the integers 0 through 9 is equal to the sum of the values of the integers contained in the string, where the value assigned to each integer is equal to the number of closed areas contained in it. Thus: 0 = 1, 2 = 0, 3 = 0, 4 = 1, 5 = 0, 6 = 1, 7 = 0, 8 = 2, and 9 = 1. Therefore, for example, 4444 = 4 because each integer in the string has 1 closed area; that is, 1 + 1 + 1 + 1 = 4.

A Logical Fallacy

The sub-hed of an article at City Journal asks "If human beings are naturally risk-averse, then what the heck happened on Wall Street?" The question can be expressed in the following syllogism:

Major premise: All humans are risk-averse.

Minor premise: Humans work on Wall Street (i.e., financial markets).

Conclusion: The humans who work on Wall Street are risk-averse.

It should be obvious to the casual observer that both the major premise and conclusion are false.

The article, by the way, is spot-on. Don't be deceived by its flawed sub-hed.