KRACH Ratings for D1 College Hockey (2011-2012)

© 1999-2011, Joe Schlobotnik (archives)

URL for this frameset:

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

Today's KRACH (including games of 2012 March 17)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Boston Coll 1 486.8 .8028 1 29-10-1 2.810 10 173.3
Michigan 2 358.8 .7541 9 24-12-4 1.857 4 193.2
North Dakota 3 314.4 .7311 4T 25-12-3 1.963 18 160.2
Miami 4 305.9 .7262 13 24-14-2 1.667 6 183.5
Ferris State 5 286.8 .7145 6 23-11-5 1.889 22 151.8
Minn-Duluth 6 286.6 .7143 3 24-9-6 2.250 31 127.4
Western Mich 7 281.7 .7112 15 21-13-6 1.500 5 187.8
Maine 8 277.4 .7083 12 23-13-3 1.690 13 164.2
Boston Univ 9 273.4 .7056 14 23-14-1 1.621 12 168.7
Minnesota 10 267.2 .7014 4T 26-13-1 1.963 29 136.1
Denver U 11 260.7 .6967 11 25-13-4 1.800 25 144.8
Mass-Lowell 12 256.9 .6939 7 23-12-1 1.880 27 136.7
Mich State 13 252.4 .6906 21 19-15-4 1.235 2 204.3
Northern Mich 14 229.3 .6721 24 17-14-6 1.176 3 194.9
Union 15 226.3 .6695 2 24-7-7 2.619 36 86.40
Notre Dame 16 217.5 .6617 27T 19-18-3 1.051 1 206.9
Merrimack 17 205.2 .6501 17T 18-12-7 1.387 23 147.9
Ohio State 18 183.4 .6274 29T 15-15-5 1.000 7 183.4
Lake Superior 19 170.8 .6128 27T 18-17-5 1.051 14 162.5
St Cloud 20 161.1 .6008 29T 17-17-5 1.000 15 161.1
CO College 21 153.9 .5911 25 18-16-2 1.118 26 137.7
Cornell 22 153.3 .5904 8 18-8-7 1.870 40 82.02
Northeastern 23 148.9 .5843 39 13-16-5 .8378 8 177.8
Wisconsin 24 146.4 .5808 36 17-18-2 .9474 21 154.6
Bemidji State 25 137.7 .5679 35 17-18-3 .9487 24 145.2
New Hampshire 26 127.1 .5509 42 15-19-3 .8049 20 157.9
Mass-Amherst 27 121.4 .5412 43T 13-18-5 .7561 17 160.6
Harvard 28 117.0 .5333 23 13-10-11 1.194 35 98.02
Providence 29 115.2 .5300 45 14-20-4 .7273 19 158.3
Michigan Tech 30 111.5 .5231 37T 16-19-4 .8571 30 130.1
AK-Fairbanks 31 111.3 .5227 47 12-20-4 .6364 9 174.9
NE-Omaha 32 100.8 .5018 41 14-18-6 .8095 33 124.6
Bowling Green 33 96.56 .4926 48 14-25-5 .6000 16 160.9
Colgate 34 94.35 .4877 26 19-17-3 1.108 37 85.15
Quinnipiac 35 91.25 .4806 19 20-14-6 1.353 45 67.44
Yale 36 72.23 .4318 29T 16-16-3 1.000 42 72.23
Air Force 37 68.50 .4208 10 21-10-7 1.815 49 37.74
St Lawrence 38 62.96 .4037 43T 14-19-3 .7561 38 83.27
MSU-Mankato 39 61.05 .3975 51 12-24-2 .5200 34 117.4
Clarkson 40 61.00 .3973 34 16-17-6 .9500 46 64.22
RIT 41 59.71 .3930 16 20-13-6 1.438 48 41.53
Dartmouth 42 59.57 .3925 40 13-16-4 .8333 43 71.48
Niagara 43 58.58 .3892 17T 17-11-9 1.387 47 42.23
AK-Anchorage 44 52.36 .3670 53 9-25-2 .3846 28 136.1
Princeton 45 49.51 .3561 46 9-16-7 .6410 41 77.23
Mercyhurst 46 45.86 .3416 22 20-16-4 1.222 51 37.52
RPI 47 43.67 .3324 50 12-24-3 .5294 39 82.49
Holy Cross 48 41.76 .3241 20 20-15-4 1.294 54 32.27
Vermont 49 40.70 .3194 57 6-27-1 .2364 11 172.2
Brown 50 38.64 .3099 49 9-18-5 .5610 44 68.87
Robert Morris 51 36.99 .3021 29T 17-17-5 1.000 52 36.99
Bentley 52 31.80 .2759 29T 16-16-8 1.000 55 31.80
Connecticut 53 29.42 .2629 37T 16-19-4 .8571 53 34.32
Canisius 54 18.77 .1953 52 10-22-4 .5000 50 37.53
AL-Huntsville 55 11.13 .1327 58 2-28-1 .0877 32 126.9
American Intl 56 10.49 .1266 54 8-26-3 .3455 57 30.35
Army 57 8.354 .1052 55 4-23-7 .2830 58 29.52
Sacred Heart 58 7.825 .0996 56 6-28-3 .2542 56 30.78

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: 2012 March 26

Joe Schlobotnik /

HTML 4.0 compliant CSS2 compliant