KRACH Ratings for D1 College Hockey (2012-2013)

© 1999-2012, 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 2013 January 12)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
New Hampshire 1 575.4 .8270 3T 14-4-2 3.000 3 191.8
Boston Coll 2 569.3 .8256 5 13-4-2 2.800 2 203.3
Minnesota 3 471.0 .7986 2 16-3-3 3.889 26 121.1
Quinnipiac 4 453.5 .7929 1 17-3-3 4.111 33 110.3
Notre Dame 5 357.9 .7549 6 15-6 2.500 14 143.2
Boston Univ 6 355.6 .7539 12T 12-7 1.714 1 207.4
Denver U 7 312.5 .7315 11 13-6-4 1.875 5 166.7
North Dakota 8 280.1 .7118 8T 13-6-3 1.933 11 144.9
Dartmouth 9 239.9 .6826 14T 9-5-2 1.667 13 144.0
Yale 10 220.1 .6658 10 9-4-3 1.909 27 115.3
Miami 11 218.5 .6643 8T 12-5-5 1.933 29 113.0
Western Mich 12 213.0 .6593 7 13-5-4 2.143 39 99.38
NE-Omaha 13 210.5 .6570 19 13-9-2 1.400 9 150.4
Mass-Lowell 14 206.7 .6533 16 11-7-1 1.533 17 134.8
MSU-Mankato 15 202.0 .6487 14T 14-8-2 1.667 25 121.2
St Cloud 16 171.3 .6150 22 12-10 1.200 15 142.8
Colgate 17 150.0 .5872 17T 11-7-2 1.500 38 100.0
Cornell 18 142.5 .5762 23 7-6-2 1.143 21 124.7
Northern Mich 19 141.9 .5753 29 9-10-4 .9167 8 154.8
Wisconsin 20 140.4 .5731 24T 8-7-5 1.105 20 127.0
Niagara 21 132.9 .5614 3T 13-3-4 3.000 53 44.30
Providence 22 123.6 .5458 24T 9-8-3 1.105 30 111.9
CO College 23 123.1 .5448 41 9-13-2 .7143 4 172.3
Union 24 122.0 .5430 20 10-7-4 1.333 42 91.51
Lake Superior 25 121.5 .5420 27 12-11-1 1.087 32 111.8
Robert Morris 26 120.9 .5411 12T 11-6-2 1.714 47 70.54
Minn-Duluth 27 113.6 .5277 30T 9-10-3 .9130 22 124.5
Ohio State 28 113.6 .5276 30T 8-9-5 .9130 23 124.4
Ferris State 29 112.8 .5260 26 10-9-3 1.095 36 103.0
AK-Fairbanks 30 111.8 .5241 28 8-8-4 1.000 31 111.8
Mass-Amherst 31 103.2 .5069 44T 7-11-2 .6667 7 154.8
Michigan Tech 32 87.44 .4710 51 6-12-3 .5556 6 157.4
RPI 33 86.97 .4699 42T 6-10-5 .6800 19 127.9
Merrimack 34 86.32 .4683 33T 8-9-4 .9091 40 94.96
Princeton 35 85.38 .4659 38 6-8-4 .8000 35 106.7
Vermont 36 80.65 .4537 48 6-11-4 .6154 18 131.1
Harvard 37 79.97 .4519 46 5-8-1 .6471 24 123.6
Holy Cross 38 78.56 .4481 17T 11-7-2 1.500 50 52.37
Bemidji State 39 75.76 .4404 52 5-11-4 .5385 16 140.7
Mich State 40 74.71 .4374 53 6-13-3 .5172 12 144.4
Bowling Green 41 72.90 .4322 42T 6-10-5 .6800 34 107.2
St Lawrence 42 68.25 .4183 33T 9-10-2 .9091 46 75.07
Brown 43 64.29 .4058 39 5-7-4 .7778 44 82.65
Michigan 44 58.57 .3866 50 7-13-2 .5714 37 102.5
Northeastern 45 57.12 .3815 47 6-10-2 .6364 43 89.75
Mercyhurst 46 53.09 .3667 21 10-8-1 1.235 56 42.98
Maine 47 51.18 .3595 54T 5-13-3 .4483 28 114.2
AK-Anchorage 48 49.30 .3520 57 3-13-4 .3333 10 147.9
Connecticut 49 49.03 .3510 35 8-9-2 .9000 49 54.48
Air Force 50 45.01 .3344 36 7-9-5 .8261 48 54.49
Canisius 51 43.09 .3260 30T 8-9-5 .9130 51 47.20
Clarkson 52 42.43 .3231 54T 4-12-5 .4483 41 94.65
Bentley 53 36.48 .2953 37 8-10-1 .8095 52 45.07
Army 54 29.49 .2585 40 7-10-3 .7391 57 39.90
Penn State 55 26.17 .2391 49 6-10 .6000 55 43.62
RIT 56 25.66 .2360 44T 6-10-4 .6667 58 38.49
American Intl 57 14.88 .1610 56 4-12-3 .4074 59 36.53
AL-Huntsville 58 7.308 .0922 58 1-16-1 .0909 45 80.38
Sacred Heart 59 2.188 .0300 59 0-19-2 .0500 54 43.76

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: 2013 January 13

Joe Schlobotnik /

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