KRACH Ratings for D1 College Hockey (2001-2002)

© 1999-2002, Joe Schlobotnik (archives)

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

See also

Current KRACH (including games of 2002 March 17)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Denver U 1 1872 .9080 1 32-7-1 4.333 4 432.0
Minnesota 2 1462 .8873 4 29-8-4 3.100 1 471.7
St Cloud 3 1043 .8540 6 29-10-2 2.727 8 382.3
New Hampshire 4 933.3 .8419 2 29-6-3 4.067 15 229.5
CO College 5 870.4 .8339 11 26-12-3 2.037 5 427.3
Mich State 6 633.0 .7945 5 27-8-5 2.810 16 225.3
Michigan 7 565.6 .7795 9 26-10-5 2.280 11 248.1
Boston Univ 8 561.8 .7785 8 25-9-3 2.524 20 222.6
Maine 9 484.5 .7578 12 23-10-7 1.963 12 246.8
AK-Fairbanks 10 411.4 .7339 13 22-12-3 1.741 13 236.4
Northern Mich 11 410.5 .7335 10 26-12-2 2.077 23 197.6
North Dakota 12 379.0 .7215 35 16-19-2 .8500 2 445.9
Wisconsin 13 355.9 .7118 34 16-19-4 .8571 6 415.3
Mass-Lowell 14 320.1 .6952 14 22-13-3 1.621 24 197.5
AK-Anchorage 15 292.4 .6808 43 12-19-5 .6744 3 433.6
NE-Omaha 16 284.5 .6764 18 21-16-4 1.278 18 222.7
Cornell 17 275.6 .6712 3 24-7-2 3.125 36 88.20
Ohio State 18 272.5 .6694 20 20-16-4 1.222 17 223.0
Western Mich 19 264.9 .6647 19 19-15-4 1.235 21 214.4
Boston Coll 20 233.6 .6440 27 18-18-2 1.000 14 233.6
MSU-Mankato 21 217.7 .6323 40 14-20-2 .7143 10 304.8
Northeastern 22 216.3 .6312 24 19-17-3 1.108 25 195.2
Minn-Duluth 23 206.9 .6237 47 13-24-3 .5686 9 363.8
Notre Dame 24 178.5 .5988 28 16-17-5 .9487 27 188.1
Ferris State 25 168.3 .5888 38 15-20-1 .7561 19 222.6
Providence 26 134.4 .5504 42 13-20-5 .6889 26 195.2
Michigan Tech 27 123.0 .5352 56 8-28-2 .3103 7 396.4
Miami 28 117.7 .5277 48 12-22-2 .5652 22 208.3
Wayne State 29 114.2 .5225 17 17-11-4 1.462 40 78.11
RPI 30 113.3 .5211 16 20-13-4 1.467 41 77.23
Clarkson 31 102.3 .5039 23 17-15-6 1.111 34 92.10
Harvard 32 101.9 .5032 26 15-14-4 1.062 32 95.90
Brown 33 83.96 .4707 30 14-15-2 .9375 35 89.56
Merrimack 34 80.32 .4634 51 11-23-2 .5000 31 160.6
Bowling Green 35 79.34 .4613 52 9-25-6 .4286 28 185.1
Dartmouth 36 77.20 .4569 25 14-13-5 1.065 43 72.52
Union 37 74.76 .4516 31 12-13-5 .9355 39 79.92
Mass-Amherst 38 63.18 .4244 53 8-24-2 .3600 29 175.5
Lake Superior 39 55.01 .4026 55 8-27-2 .3214 30 171.1
Yale 40 52.45 .3952 49 10-19-2 .5500 33 95.36
Bemidji State 41 50.98 .3909 44 10-17-4 .6316 38 80.72
St Lawrence 42 47.75 .3809 50 11-21-2 .5455 37 87.55
Princeton 43 47.54 .3803 45 11-18-2 .6316 42 75.27
Colgate 44 47.15 .3790 41 13-19-2 .7000 45 67.35
AL-Huntsville 45 44.26 .3696 37 14-17-1 .8286 46 53.41
Niagara 46 42.82 .3647 32 14-16-1 .8788 47 48.73
Air Force 47 31.79 .3226 33 12-14-2 .8667 48 36.68
Mercyhurst 48 21.50 .2724 7 24-8-3 2.684 49 8.009
Vermont 49 10.62 .1967 59 3-26-2 .1481 44 71.67
Quinnipiac 50 10.12 .1922 15 20-12-5 1.552 51 6.524
Sacred Heart 51 7.739 .1682 22 16-14-4 1.125 50 6.879
Holy Cross 52 7.053 .1604 21 15-12-5 1.207 52 5.844
Canisius 53 5.351 .1387 29 14-15-4 .9412 53 5.685
Connecticut 54 4.764 .1303 36 13-16-7 .8462 54 5.630
Iona 55 3.859 .1157 39 12-17-2 .7222 56 5.343
Army 56 3.263 .1050 46 9-17-6 .6000 55 5.439
American Intl 57 1.034 .0480 54 7-21 .3333 59 3.103
Fairfield 58 .9822 .0460 57 5-22-3 .2766 58 3.551
Bentley 59 .7357 .0360 58 4-26-2 .1852 57 3.973

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 KRACH-modified selection criteria which includes a list 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 59 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 an 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 10 parts opponents' winning percentage and 3 parts opponents' opponents' winning percentage. This means what while a team's RPI is only 7 parts winning percentage and 13 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.

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 35% of a team's winning percentage and 65% 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 .175 and .825, no matter what their strength of schedule was. Similarly, a team playing against an extremely weak or strong schedule only has .350 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.

Last Modified: 2020 February 1

Joe Schlobotnik /

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