KRACH Ratings for D1 College Hockey (2006-2007)

© 1999-2007, Joe Schlobotnik (archives)

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

Today's KRACH (including games of 2007 March 17)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Minnesota 1 624.5 .8346 2 30-9-3 3.000 7 208.2
Notre Dame 2 495.6 .8026 1 31-6-3 4.333 32 114.4
St Cloud 3 427.1 .7800 7 22-10-7 1.889 3 226.1
Boston Coll 4 359.3 .7519 5 26-11-1 2.304 11 155.9
New Hampshire 5 358.6 .7516 4 26-10-2 2.455 15 146.1
North Dakota 6 354.8 .7497 14 22-13-5 1.581 4 224.5
Clarkson 7 301.8 .7216 3 25-8-5 2.619 31 115.2
Michigan 8 278.5 .7071 6 26-13-1 1.963 19 141.9
Boston Univ 9 268.6 .7004 9 20-9-9 1.815 14 148.0
Wisconsin 10 257.7 .6927 29 19-18-4 1.050 1 245.5
Denver U 11 251.9 .6884 19 21-15-4 1.353 10 186.2
Mich State 12 235.0 .6752 13 22-13-3 1.621 16 145.0
CO College 13 229.9 .6709 27 18-17-4 1.053 5 218.4
Michigan Tech 14 224.5 .6663 28 18-17-5 1.051 6 213.5
Maine 15 220.6 .6629 16 21-14-2 1.467 13 150.4
Mass-Amherst 16 212.2 .6554 15 20-12-5 1.552 21 136.8
Miami 17 198.4 .6420 12 23-13-4 1.667 30 119.0
St Lawrence 18 185.5 .6286 11 23-13-2 1.714 35 108.2
MSU-Mankato 19 173.8 .6154 41 13-19-6 .7273 2 239.0
Vermont 20 170.7 .6117 23 18-16-5 1.108 12 154.0
Dartmouth 21 154.4 .5910 17 18-12-3 1.444 36 106.9
Lake Superior 22 136.5 .5655 25 21-19-3 1.098 26 124.4
Minn-Duluth 23 133.9 .5614 44T 13-21-5 .6596 9 203.0
AK-Anchorage 24 133.7 .5611 46 13-21-3 .6444 8 207.5
Quinnipiac 25 132.5 .5593 18 21-14-5 1.424 43 93.06
Western Mich 26 123.6 .5446 30T 18-18-1 1.000 27 123.6
NE-Omaha 27 123.6 .5446 24 18-16-8 1.100 34 112.3
Ohio State 28 120.2 .5388 33 15-17-5 .8974 22 134.0
Northeastern 29 106.1 .5125 37T 13-18-5 .7561 20 140.3
Cornell 30 105.2 .5108 26 14-13-4 1.067 37 98.67
Harvard 31 102.6 .5053 35 14-17-2 .8333 28 123.1
Niagara 32 96.93 .4934 21 18-13-6 1.312 49 73.85
Bemidji State 33 94.80 .4888 30T 14-14-5 1.000 40 94.80
Northern Mich 34 90.82 .4798 47 15-24-2 .6400 18 141.9
Princeton 35 83.10 .4612 32 15-16-3 .9429 45 88.13
Ferris State 36 79.68 .4524 44T 14-22-3 .6596 29 120.8
Colgate 37 72.81 .4338 40 15-21-4 .7391 38 98.50
Union 38 71.62 .4304 37T 14-19-3 .7561 41 94.72
AK-Fairbanks 39 69.71 .4248 50 11-22-6 .5600 25 124.5
Providence 40 67.25 .4175 54T 10-23-3 .4694 17 143.3
Yale 41 63.88 .4071 43 11-17-3 .6757 42 94.54
RPI 42 62.27 .4019 48 10-18-8 .6364 39 97.85
Brown 43 61.64 .3999 36 11-15-6 .7778 46 79.25
Mass-Lowell 44 59.80 .3938 54T 8-21-7 .4694 23 127.4
Robert Morris 45 57.64 .3865 39 14-19-2 .7500 48 76.86
AL-Huntsville 46 54.56 .3756 42 13-19-3 .7073 47 77.13
Wayne State 47 52.21 .3670 49 12-21-2 .5909 44 88.35
RIT 48 48.46 .3526 8 21-11-2 1.833 54 26.43
Sacred Heart 49 46.30 .3440 10 21-11-4 1.769 55 26.17
Bowling Green 50 33.68 .2862 58 7-29-2 .2667 24 126.3
Air Force 51 33.06 .2830 22 19-15-5 1.229 51 26.91
Army 52 30.75 .2707 20 17-12-5 1.345 59 22.87
Connecticut 53 22.22 .2194 34 16-18-2 .8947 58 24.84
Merrimack 54 19.68 .2019 59 3-27-4 .1724 33 114.1
Bentley 55 15.06 .1667 51T 12-22-1 .5556 50 27.11
Mercyhurst 56 14.03 .1581 53 9-20-6 .5217 52 26.90
Holy Cross 57 14.02 .1580 51T 10-20-5 .5556 57 25.23
Canisius 58 11.34 .1341 56 9-23-3 .4286 53 26.47
American Intl 59 8.660 .1076 57 8-25-1 .3333 56 25.98

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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