Introduced by baseball sabermetrics creator Bill James, the Pythagorean Winning Expectation has become one of the most widely used innovations in the world of sports analytics. The concept is simple, the formula predicts how successful a team is by looking at the number of runs they score and allow. Of course, scoring lots of runs or/and giving up few runs yields a better record.
The formula got its name because of its similarity with the original Pythagorean Theorem, it is as follows:
$latex frac{RS^2}{RS^2+RA^2}&s=3$
Just like how the two legs of a triangle affect its hypotenuse, the legs in this formula are the runs scored and runs allowed, and the hypotenuse is the winning percentage. Generally, this formula 1) measures a team’s luck or 2) tells if a team tanked a season on purpose. Many people have also used formula mid-season to predict the outcome of a team’s finish.
So I bet you’re wondering by now, what does a baseball sabermetric formula have to do in hockey? A lot actually. The formula can be used in almost any pro team sport with adjustments to its exponents, which is the main factor in predicting the correct results, where for example, the exponent in baseball is 2. Previous studies by mathematicians have shown this formula is equally effective in hockey. The ideal exponent in hockey is actually somewhere just above 2, but for the sake of time we’ll move on.
Hockey’s Pythagorean Expectation in the NHL is a bit different, as the league uses a points system other than ranking its teams with wins. The tweaks to the formula is as follows:
$latex frac{GF^2}{GF^2+GA^2}times frac{2.23}{2}&s=3$
With the points system present in the NHL, we’re not calculating a team’s winning percentage but rather its points percentage in terms of regular season stats. So taking overtime and shootout losses into account, the NHL averages 2.23 points given out every game, and since both teams in a game may earn a point, we multiply the initial formula by 2.23 then divide it by 2. In baseball, since only one team may win a game, the second part of the formula is redundant.
If you plug in the numbers to the formula, here are the projected point percentages for the top 16 teams in the league this year.
| TEAM | GF | GA | xPct | aPct | |
|---|---|---|---|---|---|
| 1 | BOS | 261 | 177 | .764 | .713 |
| 2 | STL | 248 | 191 | .700 | .677 |
| 3 | ANA | 266 | 109 | .689 | .707 |
| 4 | SJS | 249 | 200 | .678 | .677 |
| 5 | CHI | 267 | 220 | .664 | .652 |
| 6 | PIT | 247 | 204 | .663 | .659 |
| 7 | LAK | 206 | 174 | .651 | .610 |
| 8 | COL | 250 | 220 | .628 | .683 |
| 9 | NYR | 218 | 193 | .625 | .585 |
| 10 | TBL | 240 | 215 | .619 | .616 |
| 11 | CBJ | 231 | 216 | .595 | .567 |
| 12 | MTL | 215 | 204 | .588 | .601 |
| 13 | DAL | 235 | 228 | .574 | .555 |
| 14 | MIN | 207 | 206 | .560 | .598 |
| 15 | PHI | 236 | 235 | .560 | .573 |
| 16 | WSH | 235 | 240 | .546 | .543 |
According to the Pythagorean Expectation, Boston actually still underachieved this season and has potential to go even further. Detroit shouldn’t even have made the playoffs as Washington took the last Wild Card spot in the Eastern Conference.
But as you can see, the formula predicts all of the teams’ records with surprising accuracy, and over the course of the last five years, has recorded a mean absolute deviation of .249, which means that the formula’s average margin of error is only 2.5%.
If you plot the pairings on the graph the results are even more stunning, the correlation between the predicted points and actual points is almost perfect at .967.
So are we going to see a Stanley Cup Final between Boston and St Louis this year? There’s a high chance that’s gonna happen, but lately I’ve been experimenting with the formula that might just give us a better picture of the Cup Finalists this year.
Like most people, I’ve always believed that teams do go through significant changes over the course of the year, for the better and for the worse. Would taking the Pythagorean Expectation and applying it to a team’s final 20 games of the year better predict the playoff upsets? Why not try it this year?
Since there are no points nor shootouts involved in the playoffs, we’ll revert back to the standard aforementioned formula used in baseball. At the same time we have to omit any extra goals from shootouts as well. Plugging in the data from every team’s past 20 games, the results I got are shown on this table:
| TEAM | GF | GA | xPct | |
|---|---|---|---|---|
| 1 | BOS | 66 | 36 | .771 |
| 2 | NYR | 56 | 35 | .719 |
| 3 | LAK | 54 | 39 | .657 |
| 4 | SJS | 61 | 47 | .627 |
| 5 | ANA | 65 | 55 | .583 |
| 6 | PHI | 62 | 53 | .577 |
| 7 | TBL | 60 | 52 | .571 |
| 8 | DAL | 59 | 52 | .563 |
| 9 | CBJ | 62 | 53 | .534 |
| 10 | COL | 56 | 53 | .528 |
| 11 | MTL | 54 | 52 | .519 |
| 12 | MIN | 53 | 52 | .510 |
| 13 | DET | 57 | 57 | .500 |
| 14 | CHI | 53 | 54 | .491 |
| 15 | PIT | 48 | 51 | .470 |
| 16 | STL | 40 | 49 | .400 |
After new calculations, there’s a high chance we’ll be seeing the Bruins, Rangers, Kings, and Ducks (Kings and Sharks play in Round 1) remain in the Conference Finals. St Louis’s 6-game slide to end the season has put them down in the worst position in the standings, Chicago and Pittsburgh also has absymal percentages due to their recent struggles.
This table definitely won’t correctly predict all playoff results, and one will probably have to consider the records of two teams when they play each other. However, it sure sheds more light to insightful fans who are going for their perfect brackets these year too. But just how effective my experiment will prove to be, only time will tell.
ABOVE FEATURED IMAGE: A stickless Slava Voynov celebrates after scoring (NHL.com/Getty Images)<
