*What follows is a post from my old hockey analysis site*

**puckerings.com**(later hockeythink.com). It is reproduced here for posterity; bear in mind this writing is over a decade old and I may not even agree with it myself anymore. This post was originally published on August 1, 2001 and was updated on April 10, 2002.

**Isolating Team Defence***Copyright Iain Fyffe, 2002*

Probably the most difficult aspect of hockey to quantify is defence. Among official stats, we basically only have goals-against average (team) and plus-minus (individual). I will not discuss plus-minus, as my emphasis here is on team defence. Another way of looking at defence is by using marginal goal analysis, which forms the basis for the Point Allocation system. This essay details another way of examining team defence, Defensive Winning Percentage.

Goals-against average is not sufficient for the evaluation of team defence, for two reasons. First, it varies with the level of goal-scoring in general, making direct comparisons across years difficult. Second, and most important, goals-against average is made up of two things: team defence and goaltending. What we need is a method for separating the two.

The DWP method stems from the Neutral Winning Percentage system for goaltenders. In that essay, we arrive at a winning percentage for goaltenders that is (for the most part) attributable only to the goaltender, not to the team he plays for. And since offence is relatively easy to quantify, that leaves only one facet of the game not measured: defence.

What we need now is a formula that combines all of the following elements: team winning percentage, offence, defence, and goaltending. No such formula exists, but we can adapt one for our use. Bill James developed a Pythagorean Winning Percentage formula for baseball, which uses offence and defence together to calculate an expected winning percentage. It works quite well. Marc Foster of the HRA adapted this concept to hockey, and the formula (for modern times) looks something like this:

TWP = GF^2 / (GF^2 + GA^2)

Where TWP is the team’s (expected) winning percentage, GF is goals for and GA is goals against. The exponent is more accurately set at 2.03, but for our purposes, we will use the simpler exponent of 2.

Goals against are determined by two things: defence and goaltending. For our purposes (creating a measure of team defence), we will make an assumption on how to separate the two. Since at any time there are (usually) five players on the ice who are responsible for defence, and only one for goaltending, we will assume that defence is five times more important than goaltending in determining goals against.

To keep the presentation constant, we will work only with winning percentages. This has the additional benefit of making numbers from one year directly comparable with numbers from another. We have team winning percentage (TWP) and goaltender (neutral) winning percentage (GWP) so far. In order to calculate defensive winning percentage (DWP), we will first need to calculate offensive winning percentage (OWP).

To calculate OWP, we use a method similar to the one used in calculating GWP. First, we compile the records of all teams in the league for each level of goals scored (0, 1, 2, 3, etc.). This gives us a winning percentage determined by the number of goals you score, independent of how many goals you allow. Thus, it measures only offence.

Second, we compile the number of times each team scores each number of goals. However, since we want this to be free of any bias created by the opponent’s defence, we must make an adjustment. For each opponent, we calculate a multiplier based on their goals-against average relative to the league goals-against average. Thus, if a team allows 3.00 goals per game, and the league average is 2.50, this team will have a multiplier of 0.83 (2.50/3.00), reflecting the relative ease with which you can score on this opponent. So if you score 5 goals against this opponent, you get credit for only 4.15 (5 x 0.83), which is rounded off to 4.

Now, using the winning percentages for each number of goals and the number of times each team scores each number of goals, we can calculate a weighted-average winning percentage. This is the OWP, and it (in theory) reflects only the quality of team offence.

Now, we can express the Pythagorean Formula as follows, bearing in mind the assumptions we have made thus far:

TWP = OWP^2 / {OWP^2 + [1- (5/6 x DWP + 1/6 x GWP)]^2}

With some simple algebraic manipulation, the formula for DWP is:

DWP = 1.2 - .2 x GWP - 1.2 x OWP x (1/TWP - 1)^.5

One of the nice things about using the Pythagorean Formula is that it captures the synergies that exist in hockey. Winning is not linear; the results of above-average goaltending, offence, and defence is greater that the sum of the individual parts. With this method, having a .500 OWP, GWP and DWP will produce a .500 TWP. However, if several of these factors are above-average, the TWP will not increase linearly. Results for 2000/01 are included at the end of this essay; examine them to see what I mean.

DWP is really only a rough measure. When examining it, please remember the following points:

(a) it absorbs all the error present in the Pythagorean Formula,

(b) it assumes defence is fives times as important as goaltending in preventing goals,

(c) it assumes the GWP and OWP calculations are accurate, and

(d) it indicates only how defence was played, not the potential for defence; it can be influenced by strategy.

**2000/01 NHL Results**

Team | TWP | OWP | GWP | DWP |

Anaheim | .372 | .404 | .487 | .473 |

Atlanta | .354 | .446 | .485 | .380 |

Boston | .488 | .498 | .467 | .494 |

Buffalo | .591 | .483 | .585 | .600 |

Calgary | .421 | .445 | .492 | .475 |

Carolina | .518 | .451 | .512 | .576 |

Chicago | .402 | .461 | .459 | .433 |

Colorado | .650 | .609 | .536 | .557 |

Columbus | .396 | .406 | .523 | .494 |

Dallas | .634 | .535 | .559 | .600 |

Detroit | .652 | .566 | .531 | .598 |

Edmonton | .549 | .554 | .484 | .501 |

Florida | .348 | .436 | .526 | .379 |

Los Angeles | .543 | .557 | .507 | .485 |

Minnesota | .384 | .375 | .563 | .517 |

Montreal | .390 | .465 | .499 | .402 |

Nashville | .470 | .418 | .579 | .552 |

New Jersey | .659 | .610 | .517 | .570 |

NY Islanders | .299 | .394 | .438 | .388 |

NY Rangers | .433 | .555 | .435 | .351 |

Ottawa | .640 | .595 | .536 | .557 |

Philadelphia | .591 | .521 | .521 | .576 |

Phoenix | .530 | .458 | .561 | .570 |

Pittsburgh | .567 | .591 | .491 | .482 |

St.Louis | .598 | .564 | .519 | .541 |

San Jose | .561 | .474 | .586 | .580 |

Tampa Bay | .329 | .428 | .495 | .368 |

Toronto | .518 | .490 | .503 | .532 |

Vancouver | .506 | .531 | .429 | .485 |

Washington | .561 | .508 | .552 | .550 |

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