KRACH Ratings for D1 College Hockey (2009-2010)

© 1999-2009, 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 2010 March 20)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Denver U 1 543.0 .8145 2 27-9-4 2.636 8 206.0
Miami 2 488.2 .7991 1 27-7-7 2.905 14 168.1
Wisconsin 3 481.3 .7970 7 25-10-4 2.250 5 213.9
North Dakota 4 434.2 .7812 9 25-12-5 1.897 3 229.0
Boston Coll 5 346.2 .7441 3T 25-10-3 2.304 24 150.2
St Cloud 6 345.3 .7437 10 23-13-5 1.645 6 209.9
Minn-Duluth 7 251.2 .6864 18 22-17-1 1.286 9 195.4
Northern Mich 8 234.2 .6730 13T 20-12-8 1.500 19 156.1
CO College 9 232.2 .6714 25T 19-17-3 1.108 7 209.5
Michigan 10 230.2 .6697 15 25-17-1 1.457 18 157.9
New Hampshire 11 224.4 .6648 19 17-13-7 1.242 12 180.6
Minnesota 12 222.5 .6632 34 18-19-2 .9500 2 234.2
Bemidji State 13 213.2 .6549 6 23-9-4 2.273 34 93.82
AK-Fairbanks 14 213.1 .6548 16 18-11-9 1.452 26 146.8
Ferris State 15 211.3 .6531 13T 21-13-6 1.500 30 140.9
Vermont 16 202.9 .6451 21 17-14-7 1.171 13 173.2
Cornell 17 201.7 .6439 5 21-8-4 2.300 37 87.69
Mich State 18 201.3 .6435 17 19-13-6 1.375 27 146.4
NE-Omaha 19 182.5 .6240 20 20-16-6 1.211 23 150.8
Maine 20 182.3 .6238 25T 19-17-3 1.108 15 164.5
Boston Univ 21 169.8 .6094 29 18-17-3 1.054 16 161.0
Mass-Lowell 22 165.0 .6036 22 19-16-4 1.167 29 141.4
Ohio State 23 159.9 .5972 35 15-18-6 .8571 11 186.5
Yale 24 159.4 .5965 8 20-9-3 2.048 40 77.83
MSU-Mankato 25 158.4 .5952 38 16-20-3 .8140 10 194.6
Northeastern 26 152.7 .5878 31T 16-16-2 1.000 21 152.7
Mass-Amherst 27 151.1 .5857 31T 18-18 1.000 22 151.1
Merrimack 28 134.6 .5618 37 16-19-2 .8500 17 158.4
Union 29 128.1 .5515 11 21-12-6 1.600 39 80.07
Lake Superior 30 118.2 .5348 36 15-18-5 .8537 31 138.5
Notre Dame 31 117.4 .5334 39 13-17-8 .8095 28 145.1
AK-Anchorage 32 107.7 .5153 51T 11-23-2 .5000 4 215.3
St Lawrence 33 82.37 .4599 23 19-16-7 1.154 48 71.39
Quinnipiac 34 80.97 .4564 27 20-18-2 1.105 46 73.26
RPI 35 79.14 .4517 30 18-17-4 1.053 42 75.18
Providence 36 75.10 .4411 50 10-20-4 .5455 32 137.7
Western Mich 37 73.65 .4371 51T 8-20-8 .5000 25 147.3
Colgate 38 72.48 .4339 31T 15-15-6 1.000 47 72.48
Robert Morris 39 59.31 .3939 48 10-19-6 .5909 33 100.4
Princeton 40 58.24 .3903 40 12-16-3 .7714 41 75.49
Niagara 41 56.96 .3860 47 12-20-4 .6364 36 89.52
AL-Huntsville 42 54.40 .3771 42 12-17-3 .7297 43 74.55
RIT 43 53.69 .3745 3T 26-11-1 2.304 53 23.30
Brown 44 50.40 .3625 43T 13-20-4 .6818 45 73.92
Michigan Tech 45 46.71 .3481 58 5-30-1 .1803 1 259.0
Bowling Green 46 43.92 .3368 56 5-25-6 .2857 20 153.7
Harvard 47 41.94 .3283 53 9-21-3 .4667 35 89.87
Dartmouth 48 41.64 .3270 49 10-19-3 .5610 44 74.23
Sacred Heart 49 37.23 .3071 12 21-13-4 1.533 50 24.28
Clarkson 50 35.48 .2987 54 9-24-4 .4231 38 83.87
Air Force 51 26.19 .2487 28 16-15-6 1.056 49 24.82
Canisius 52 23.48 .2320 24 17-15-5 1.114 58 21.07
Mercyhurst 53 17.07 .1874 41 15-20-3 .7674 55 22.25
Army 54 16.23 .1809 45 11-18-7 .6744 52 24.07
Holy Cross 55 15.79 .1774 43T 12-19-6 .6818 54 23.16
Bentley 56 14.08 .1634 46 12-19-4 .6667 57 21.12
Connecticut 57 7.234 .0970 55 7-27-3 .2982 51 24.26
American Intl 58 5.740 .0794 57 5-24-4 .2692 56 21.32

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 /

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