Tag Archives: Pythagorean expectation

Just a little bit of luck

A couple days ago, I showed that True Wins is an able replacement for Pythagorean wins as a simple, “luck-free” measure of team quality in the NFL. True Wins gives full credit for blowout wins and half credit for close wins and losses (defining close games as games within 7 points). Heading into week 15 tonight, I thought it would be fun to see how teams stack up this season. First, let’s look at teams almost certain to make the playoffs (4 of the 7 have already clinched):

True Wins agrees that the Packers are the clear best team, and there’s not much difference among the six teams at 10-3. All these teams have gotten a little lucky in close games (i.e., their wins are higher than their True Wins), but only the Packers and Saints True Win differences are especially high. These teams are all legitimate playoff contenders.

Next, consider teams still in the hunt (I’ll be generous and include 6-7 teams plus the Eagles): Continue reading

Pythagorean Part 2 — True Wins Rises

Last week, I wrote about the use of the Pythagorean expectation for measuring NFL team quality. I didn’t have much nice to say. I think that the Pythagorean has a catchy name, but it’s more complicated than it needs to be. The Pythagorean uses points for and against to capture the intuition that close games are decided by luck, not skill. I proposed True Wins as a way to frame this more transparently:

True Wins =  Blowout wins + Close wins/2 + Close losses/2 + Ties/2

True Wins gives half credit for all close games, whether won or lost, since teams should win half their close games in the long run if they are truly decided by luck. For this post, Continue reading

Pythagorean expectations, Pythagenports, and Pythagenpats – a bunch of mumbo jumbo

It seems like everywhere you turn in sports statistics, someone is talking about Pythagorean expectations. The Pythagorean expectation is a formula created by Bill James to estimate team quality in baseball, with runs scored and runs allowed as inputs. Proponents argue that luck plays a big part in close games, making point totals a better measure of team quality than wins. The formula is

Pythagorean = Runs scored^/ (Runs scored^c + Runs allowed^c),

where c is some exponent (usually greater than one) that can be calibrated. This kinda looks like the old Pythagorean formula from grade school (hence the name), though not really. The Pythagorean rises (at a decreasing rate) as you score more and drops (at a decreasing rate) as your opponents score more. In other words, the Pythagorean rewards teams for blowouts and punishes them for getting spanked, since these scoring outcomes may reflect team quality. If you score the same amount as your opponents, your Pythagorean is 0.5. The max is one and the min is zero, like a winning percentage. “Pythagorean wins” are given by the Pythagorean multiplied by the number of games.

Over time, this formula has been exported to basketball and football, and probably other sports. My buddy Adrian the Canadian sent me this week’s DVOA update at Football Outsiders, in which Aaron Schatz informs us that Football Outsiders has upgraded from the Pythagorean expectation to the “Pythagenport,” and Baseball Prospectus (the original Pythagenporters) has moved on to the Pythagenpat! These are identical to the Pythagorean, but allow the exponent c to depend on the number of runs scored and/or the teams involved.

Maybe this sounds reasonable to you, but these formulas make my head hurt. The Pythagenport even gets a logarithm involved. Where do these crazy functional forms come from? The intuition for these stats is simple: Continue reading