KRACH Ratings for D1 College Hockey (2002-2003)

© 1999-2003, Joe Schlobotnik (archives)

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Game results taken from US College Hockey Online's Division I composite schedule

Up-to-the-minute KRACH On NEW!

Starting this season, US College Hockey Online is featuring the current KRACH, calculated from the latest scores, so if you just want the latest KRACH rankings, you should go there. For Joe Schlobotnik's geeky analysis of the system, with ratings recalculated daily, read on.

Today's KRACH (including games of 2003 March 23)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
CO College 1 1105 .8607 2 29-6-5 3.706 8 298.2
Cornell 2 1023 .8523 1 28-4-1 6.333 28 161.5
New Hampshire 3 892.3 .8368 5 25-7-6 2.800 5 318.7
Minnesota 4 787.4 .8218 8 24-8-9 2.280 3 345.4
Boston Coll 5 706.3 .8080 10 23-10-4 2.083 4 339.0
Maine 6 698.8 .8067 6 24-9-5 2.304 7 303.3
Boston Univ 7 664.6 .8001 13 24-13-3 1.759 1 377.9
Michigan 8 558.0 .7764 4 28-9-3 2.810 21 198.6
North Dakota 9 526.6 .7682 9 26-11-5 2.111 16 249.4
Ferris State 10 494.4 .7592 3 30-9-1 3.211 30 154.0
MSU-Mankato 11 480.6 .7551 14T 20-10-10 1.667 12 288.3
Providence 12 417.0 .7341 20 19-14-3 1.323 6 315.3
Denver U 13 396.5 .7264 18 21-14-6 1.412 13 280.8
St Cloud 14 386.8 .7227 27T 17-15-5 1.114 2 347.1
Harvard 15 366.9 .7146 7 22-9-2 2.300 29 159.5
Ohio State 16 350.7 .7075 11 25-12-5 1.897 25 184.9
Minn-Duluth 17 329.8 .6978 19 22-15-5 1.400 18 235.5
Mass-Amherst 18 323.8 .6949 27T 19-17-1 1.114 11 290.5
Mich State 19 297.1 .6812 16 23-14-2 1.600 24 185.7
Northern Mich 20 239.3 .6460 23 22-17-2 1.278 23 187.3
Merrimack 21 207.7 .6225 40 12-18-6 .7143 10 290.7
Dartmouth 22 197.5 .6141 17 20-13-1 1.519 33 130.0
Notre Dame 23 181.1 .5996 30T 17-17-6 1.000 26 181.1
Mass-Lowell 24 175.1 .5940 45T 11-20-5 .6000 9 291.9
Miami 25 161.9 .5808 25 21-17-3 1.216 32 133.2
AK-Fairbanks 26 158.0 .5766 29 15-14-7 1.057 31 149.4
Yale 27 155.4 .5739 22 18-14 1.286 34 120.9
Wisconsin 28 150.8 .5689 45T 13-23-4 .6000 15 251.4
Western Mich 29 147.5 .5651 39 15-21-2 .7273 20 202.8
Northeastern 30 130.7 .5447 51 10-21-3 .5111 14 255.7
NE-Omaha 31 120.7 .5314 42 13-22-5 .6327 22 190.8
Brown 32 119.9 .5302 26 16-14-5 1.121 39 106.9
Michigan Tech 33 113.9 .5217 53 10-24-4 .4615 17 246.8
Colgate 34 75.16 .4532 35 17-19-4 .9048 43 83.07
Vermont 35 72.27 .4469 41 13-20-3 .6744 38 107.2
Clarkson 36 70.64 .4433 43 12-20-3 .6279 35 112.5
Union 37 66.76 .4343 37 14-18-4 .8000 42 83.45
Bowling Green 38 58.48 .4134 57 8-25-3 .3585 27 163.1
Wayne State 39 56.40 .4078 21 21-16-2 1.294 46 43.58
St Lawrence 40 55.52 .4053 47 11-21-5 .5745 41 96.64
AL-Huntsville 41 54.51 .4025 24 18-14-3 1.258 47 43.33
RPI 42 52.20 .3958 52 12-25-3 .5094 40 102.5
Bemidji State 43 44.32 .3711 30T 14-14-8 1.000 44 44.32
Niagara 44 39.78 .3552 36 15-17-5 .8974 45 44.32
AK-Anchorage 45 32.14 .3247 60 1-28-7 .1429 19 225.0
Lake Superior 46 29.80 .3142 58 6-28-4 .2667 36 111.8
Mercyhurst 47 19.53 .2588 12 22-12-2 1.769 50 11.04
Princeton 48 16.16 .2357 59 3-26-2 .1481 37 109.1
Quinnipiac 49 13.91 .2183 14T 22-13-1 1.667 57 8.346
Findlay 50 13.76 .2170 49T 10-21-4 .5217 48 26.37
Sacred Heart 51 9.382 .1757 33 14-15-6 .9444 52 9.934
Holy Cross 52 8.310 .1636 32 17-18-1 .9459 55 8.785
Air Force 53 7.928 .1590 56 8-24-3 .3725 49 21.28
Army 54 7.046 .1478 34 15-16 .9375 59 7.515
Canisius 55 6.160 .1356 44 12-21-4 .6087 51 10.12
Bentley 56 5.635 .1279 38 15-19 .7895 60 7.137
Iona 57 4.973 .1174 49T 11-22-2 .5217 54 9.531
American Intl 58 4.439 .1084 48 10-20-2 .5238 56 8.474
Connecticut 59 3.732 .0954 54 8-23-3 .3878 53 9.625
Fairfield 60 2.849 .0772 55 8-23-2 .3750 58 7.596

Explanation of the Table

Ken's Rating for American College Hockey is an application of the Bradley-Terry method to college hockey; a team's rating is meant to indicate its relative strength on a multiplicative scale, so that the ratio of two teams' ratings gives the expected odds of each of them winning a game between them. The ratings are chosen so that the expected winning percentage for each team based on its schedule is equal to its actual winning percentage. Equivalently, the KRACH rating can be found by multiplying a team's PF/PA (q.v.) by its Strength of Schedule (SOS; q.v.). The Round-Robin Winning Percentage (RRWP) is the winning percentage a team would be expected to accumulate if they played each other team an equal number of times.
A multiplicative analogue to the winning percentage is Points For divided by Points Against (PF/PA). Here PF consists of two points for each win and one for each tie, while PA consists of two points for each loss and one for each tie.
Sched Strength
The effective measure of Strength Of Schedule (SOS) from a KRACH point of view is a weighted average of the KRACH ratings of its opponents, where the relative weighting factor is the number of games against each opponent divided by the sum of the original team's rating and the opponent's rating. (Each team's name in the table above is a link to a rundown of their opponents with their KRACH ratings, which determine each opponent's contribution to the strength of schedle.)

The Nitty-Gritty

To spell out the definition of the KRACH explicitly, if Vij is the number of times team i has beaten team j (with ties as always counting as half a win and half a loss), Nij=Vij+Vji is the number of times they've played, Vi=∑jVij is the total number of wins for team i, and Ni=∑jNij is the total number of games they've played, the team i's KRACH Ki is defined indirectly by

Vi = ∑j Nij*Ki/(Ki+Kj)

An equivalent definition, less fundamental but more useful for understanding KRACH as a combination of game results and strength of schedule, is

Ki = [Vi/(Ni-Vi)] * [∑jfij*Kj]

where the weighting factor is

fij = [Nij/(Ki+Kj)] / [∑kNik/(Ki+Kk)]

Note that fij is defined so that ∑jfij=1, which means that, for example, if all of a team's opponents have the same KRACH rating, their strength of schedule will equal that rating.

Finally, the definition of the KRACH given so far allows us to multiply everyone's rating by the same number without changing anything. This ambiguity is resolved by defining a rating of 100 to correspond to a RRWP of .500, i.e., a hypothetical team which would be expected to win exactly half their games if they played all 60 Division 1 schools the same number of times.


KRACH has been put forth as a replacement for the Ratings Percentage Index because it does what RPI in intended to do, namely judge a team's results taking into account the strength of their opposition. It does this without some of the shortcomings exhibited by RPI, such as a team's rating going down when they defeat a bad team, or a semi-isolated group of teams accumulating inflated winning percentages and showing up on other teams' schedules as stronger than they really are. The two properties which make KRACH a more robust rating system are recursion (the strength of schedule measure used in calculating a team's KRACH rating comes from the the KRACH ratings of that team's opponents) and multiplication (record and strength of schedule are multiplied rather than added).


The strength-of-schedule contribution to RPI is made up of 2 parts opponents' winning percentage and 1 part opponents' opponents' winning percentage. This means what while a team's RPI is only 1 parts winning percentage and 3 parts strength of schedule, i.e., strength of schedule is not taken at face value when evaluating a team overall, it is taken more or less at face value when evaluating the strength of a team as an opponent. (You can see how big an impact this has by looking at the "RPIStr" column on our RPI page.) So in the case of the early days of the MAAC, RPI was judging the value of a MAAC team's wins against other MAAC teams on the basis of those teams' records, mostly against other MAAC teams. Information on how the conference as a whole stacked up, based on the few non-conference games, was swamped by the impact of games between MAAC teams. Recently, with the MAAC involved in more interconference games, the average winning percentage of MAAC teams has gone down and thus the strength of schedule of the top MAAC teams is bringing down their RPI substantially. However, when teams from other conferences play those top MAAC teams, the MAAC opponents look strong to RPI because of their high winning percentages. (In response to this problem, the NCAA has changed the relative weightings of the components of the RPI from 35% winning percentage/50% opponents' winning percentage/15% opponents' opponents' winning percentage back to the original 25%/50%/25% weighting. However, this intensifies RPI's other drawback of allowing the strength of an opponent to overwhelm the actual outcome of the game.)

KRACH, on the other hand, defines the strength of schedule using the KRACH ratings themselves. This recursive property allows games further down the chain of opponents' opponents' opponents etc to have some impact on the ratings. Games among the teams in a conference are very good for giving information about the relative strengths of those teams, but KRACH manages to use even a few non-conference games to set the relative strength of that group to the rest of the NCAA. And if a team from a weak conference is judged to have a low KRACH desipte amassing a good record against bad competition, they are considered a weak opponent for strength-of-schedule purposes, since the KRACH itself is used for that as well.


One might consider bringing the power of recursion to RPI by defining an "RRPI" which was made up of 25% of a team's winning percentage and 75% of the average RRPI of their opponents. (This sort of modification is how the RHEAL rankings are defined.) However, this would not change the fact that the rating is additive. So, for example, a team with a .500 winning percentage would have an RPI between .125 and .875, no matter what their strength of schedule was. Similarly, a team playing against an extremely weak or strong schedule only has .250 of leeway based on their actual results.

With KRACH, on the other hand, one is multiplying two numbers (PF/PA) and SOS which could be anywhere from zero to infinity, and so no matter how low your SOS rating is, you could in principle have a high KRACH by having a high enough ratio of wins to losses.

The Nittier-Grittier

How To Calculate the KRACH Ratings

This definition defines the KRACH indirectly, so it can be used to check that a given set of ratings is correct, but to actually calculate them, one needs to do something like rewrite the definition in the form

Ki = Vi / [∑jNij/(Ki+Kj)]

This still defines the KRACH ratings recursively, i.e., in terms of themselves, but this equation can be solved by a method known as iteration, where you put in any guess for the KRACH ratings on the right hand side, see what comes out on the left hand side, then put those numbers back in on the right hand side and try again. When you've gotten close to the correct set of ratings, the numbers coming out on the left-hand side will be indistinguishable from the numbers going in on the right-hand side.

The other (equivalent) definition is already written as a recursive expression for the KRACH ratings, and it can be iterated in the same way to get the same results.

How to Verify the KRACH Ratings

It should be pointed out that if someone hands you a set of KRACH ratings and you only want to check that they are correct, it's much easier. You just calculate the expected number of wins for each team according to

Vi = ∑j Nij*Ki/(Ki+Kj)

And check that you come up with the actual number of wins. (Once again, a tie counts as half a win and half a loss.)

Dealing With Perfection

As described in Ken Butler's explanation of the KRACH, the methods described so far break down if a team has won all of their games. This is because their actual winning percentage is 1.000, and it's only possible for that to be their expected winning percentage if their rating is infinitely compared to those of their opponents. Now, if it's only one team, we could just set their KRACH to infinity (or zero in the case of a team which has lost all of their games), but there are more complicated scenarios in which, for example, two teams have only lost to each other, and so their KRACH ratings need to be infinite compared to everybody else's and finite compared to each other. The good news is that this sort of situation almost never exists at the end of the season; the only case in recent memory was Fairfield's first Division I season, when they went 0-23 against tournament-eligible competition.

An older version of KRACH got around this by adding a "fictitious team" against which each team was assumed to have played and tied one game, which was enough to make everyone's KRACH finite. However, this had the disadvantage that it could still effect the ratings even when it was no longer needed to avoid infinities.

The current version of KRACH does not include this "fictitious team", but rather checks to see if any ratios of ratings will end up needing to be infinite to produce the correct expected winning percentages. The key turns out to be related to the old game of trying to prove that the last-place team is better than the first-place team because they beat someone who beat someone who beat someone who beat the champions. If you can take any two teams and make a chain of wins or ties from one to the other, then all of the KRACH ratings will be finite.

If that's not the case, you need to work out the relationships teams have to each other. If you can make a chain of wins and ties from team A to team B but not the other way around, team A's rating will need to be infinite compared to team B's, and for shorthand we say A>B (and B<A). If you can make a chain of wins and ties from team A to team B and also from team B to team A, the ratio of their ratings will be a unique finite number and we say A~B. If you can't make a chain of wins and ties connecting team A and team B in either direction, the ratio of their ratings could be anything you like and you'd still get a set of ratings which satisfied the definition of the KRACH, so we say A%B (since the ratio of their ratings can be thought of as the undetermined zero divided by zero). Because of the nature of these relationships, we can split all the teams into groups so that every team in a group has the ~ relationship with every other team in the group, but not with any team outside of its groups. Furthermore if we look at two different groups, each team in the first group will have the same relationship (>, <, or %) with each team in the second group. We can then define finite KRACH ratings based only on games played between members of the same group, and use those as usual to define the expected head-to-head winning percentages for teams within the same group. For teams in different groups, we don't use the KRACH ratings, but rather the relationships between teams. If A>B, then A has an expected winning percentage of 1.000 in games against B and B has an expected winning percentage of .000 in games against A. In the case where A%B there's no basis for comparison, so we arbitrarily assign an expected head-to-head winning percentage of .500 to each team.

In the case where everyone is in the same group (again, usually true by the middle of the season) we can define a single KRACH rating with no hassle. If they're not, we need the ratings plus the group structure to describe things fully. However, the Round-Robin Winning Percentage (RRWP) can still be defined in this case and used to rank the teams, which is another reason why it's a convenient figure to work with.

See also

Last Modified: 2020 February 1

Joe Schlobotnik /

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