KRACH Ratings for D1 College Hockey (2018-2019)

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

Today's KRACH (including games of 2019 March 18)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
St Cloud 1 829.9 .8724 1 29-4-3 5.545 12 149.6
Mass-Amherst 2 408.6 .7775 3 28-8 3.500 22 116.7
Minn-Duluth 3 377.4 .7646 8 23-11-2 2.000 1 188.7
MSU-Mankato 4 368.7 .7607 2 31-7-2 4.000 41 92.17
Denver U 5 297.0 .7231 11 21-10-5 1.880 8 158.0
Quinnipiac 6 287.3 .7171 4 25-9-2 2.600 30 110.5
Ohio State 7 284.8 .7155 12 20-10-5 1.800 7 158.2
Northeastern 8 268.3 .7044 5 25-10-1 2.429 31 110.5
Clarkson 9 242.0 .6848 7 24-10-2 2.273 35 106.5
Cornell 10 236.7 .6806 10 19-9-4 1.909 19 124.0
Notre Dame 11 232.1 .6768 16 21-13-3 1.552 13 149.6
Penn State 12 226.1 .6717 17 22-14-2 1.533 15 147.5
Harvard 13 222.9 .6688 9 19-9-3 1.952 25 114.2
Western Mich 14 211.3 .6582 19T 21-15-1 1.387 11 152.3
Arizona State 15 204.3 .6515 14 21-12-1 1.720 21 118.8
Providence 16 203.5 .6507 13 22-11-6 1.786 26 114.0
Bowling Green 17 190.8 .6376 6 25-9-5 2.391 47 79.77
Minnesota 18 179.7 .6253 25 18-16-4 1.111 5 161.7
North Dakota 19 169.7 .6135 27T 18-17-2 1.056 6 160.8
Union 20 160.4 .6017 18 20-13-6 1.438 29 111.6
CO College 21 146.1 .5819 30 17-18-4 .9500 10 153.8
Mass-Lowell 22 145.0 .5803 19T 19-13-5 1.387 37 104.5
Lake Superior 23 144.4 .5795 15 23-13-2 1.714 45 84.26
Wisconsin 24 133.2 .5620 38 14-18-5 .8049 4 165.4
Boston Univ 25 131.1 .5587 31T 16-17-4 .9474 17 138.4
Northern Mich 26 126.4 .5508 21 21-16-2 1.294 38 97.64
Michigan 27 125.2 .5488 36T 13-16-7 .8462 14 148.0
Brown 28 122.9 .5447 23 15-13-5 1.129 32 108.8
Maine 29 114.3 .5291 35 15-17-4 .8947 18 127.8
Mich State 30 113.4 .5273 43 12-19-5 .6744 3 168.1
Yale 31 108.7 .5182 29 15-15-3 1.000 33 108.7
New Hampshire 32 97.35 .4942 36T 12-15-9 .8462 23 115.0
Boston Coll 33 91.05 .4796 46T 13-21-3 .6444 16 141.3
Miami 34 82.09 .4572 52 11-23-4 .5200 9 157.9
Bemidji State 35 78.14 .4466 34 15-17-6 .9000 44 86.82
Vermont 36 74.90 .4375 45 12-19-3 .6585 28 113.7
NE-Omaha 37 73.75 .4341 56 9-24-3 .4118 2 179.1
Dartmouth 38 73.49 .4334 39 13-17-4 .7895 40 93.09
Princeton 39 72.29 .4299 50 10-18-3 .5897 20 122.6
Connecticut 40 65.45 .4088 48 12-20-2 .6190 36 105.7
Michigan Tech 41 63.69 .4030 41 14-20-4 .7273 43 87.57
American Intl 42 62.57 .3993 22 20-16-1 1.242 50 50.36
AK-Fairbanks 43 53.67 .3676 49 12-21-3 .6000 42 89.44
RPI 44 53.43 .3667 53T 10-23-3 .4694 27 113.8
RIT 45 52.72 .3640 27T 17-16-4 1.056 51 49.95
Colgate 46 50.29 .3544 53T 10-23-3 .4694 34 107.1
Bentley 47 47.85 .3444 24 17-15-5 1.114 60 42.94
Sacred Heart 48 46.71 .3397 31T 16-17-4 .9474 53 49.31
Air Force 49 46.59 .3391 26 16-15-5 1.057 59 44.07
Niagara 50 41.61 .3172 33 16-18-5 .9024 56 46.11
Merrimack 51 38.17 .3008 57 7-24-3 .3333 24 114.5
Robert Morris 52 37.03 .2952 40 16-21-2 .7727 54 47.93
Ferris State 53 33.31 .2758 53T 10-23-3 .4694 49 70.96
Canisius 54 31.87 .2680 46T 12-20-5 .6444 52 49.45
Mercyhurst 55 30.42 .2598 42 13-20-5 .6889 58 44.15
Army 56 29.85 .2566 44 12-20-7 .6596 57 45.26
AL-Huntsville 57 25.55 .2307 58 8-28-2 .3103 46 82.34
Holy Cross 58 25.23 .2286 51 10-21-5 .5319 55 47.44
St Lawrence 59 22.53 .2109 59 6-29-2 .2333 39 96.57
AK-Anchorage 60 10.98 .1192 60 3-28-3 .1525 48 71.95

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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