KRACH Ratings for D1 College Hockey (2008-2009)

© 1999-2008, 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 2009 March 21)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Boston Univ 1 711.8 .8507 2 31-6-4 4.125 13 172.5
Notre Dame 2 671.4 .8436 1 31-5-3 5.000 24 134.3
Michigan 3 407.5 .7725 4 29-11 2.636 18 154.6
Northeastern 4 360.9 .7527 7 25-11-4 2.077 10 173.8
Denver U 5 354.9 .7499 9 23-11-5 1.889 3 187.9
Vermont 6 289.1 .7140 11 20-11-5 1.667 11 173.4
Minn-Duluth 7 282.8 .7100 14T 21-12-8 1.562 6 181.0
North Dakota 8 267.6 .6999 12 24-14-4 1.625 16 164.7
New Hampshire 9 256.7 .6922 17 19-12-5 1.483 12 173.2
Yale 10 245.4 .6837 3 24-7-2 3.125 43 78.53
Wisconsin 11 240.6 .6799 23 20-16-4 1.222 1 196.8
Boston Coll 12 230.7 .6719 19T 18-14-5 1.242 4 185.7
Minnesota 13 226.7 .6686 19T 17-13-7 1.242 5 182.5
Mass-Lowell 14 217.7 .6606 21T 20-16-2 1.235 8 176.2
Miami 15 216.8 .6599 16 20-12-5 1.552 23 139.7
Ohio State 16 200.0 .6440 14T 23-14-4 1.562 27 128.0
Cornell 17 193.5 .6374 6 21-9-4 2.091 35 92.53
CO College 18 184.0 .6272 21T 16-12-10 1.235 20 148.9
St Cloud 19 180.3 .6232 29 18-17-3 1.054 15 171.0
Princeton 20 178.4 .6210 8 22-11-1 1.957 36 91.18
St Lawrence 21 177.7 .6202 13 21-12-5 1.621 32 109.6
Northern Mich 22 160.0 .5988 28 19-17-5 1.103 22 145.1
MSU-Mankato 23 157.2 .5951 35 15-17-6 .9000 9 174.6
AK-Anchorage 24 150.8 .5865 36 14-17-5 .8462 7 178.2
AK-Fairbanks 25 138.8 .5692 30 17-16-6 1.053 25 131.8
Mass-Amherst 26 123.4 .5444 37 16-20-3 .8140 19 151.5
NE-Omaha 27 112.7 .5253 34 15-17-8 .9048 28 124.5
Maine 28 102.4 .5050 44T 13-22-4 .6250 17 163.8
Air Force 29 98.61 .4970 5 27-10-2 2.545 51 38.74
Union 30 91.02 .4801 27 19-17-3 1.108 41 82.14
Western Mich 31 87.68 .4722 38 14-20-7 .7447 31 117.7
Mich State 32 84.37 .4641 49T 10-23-5 .4902 14 172.1
Dartmouth 33 81.15 .4559 31T 14-14-3 1.000 42 81.15
Quinnipiac 34 75.85 .4418 31T 18-18-3 1.000 44 75.85
Lake Superior 35 74.67 .4385 44T 11-20-8 .6250 30 119.5
Bemidji State 36 72.75 .4331 24 18-15-1 1.194 46 60.95
Ferris State 37 69.55 .4238 40T 12-19-7 .6889 33 101.0
Michigan Tech 38 65.59 .4117 55T 6-25-7 .3333 2 196.8
Colgate 39 64.94 .4096 39 12-18-7 .7209 37 90.07
Bowling Green 40 63.73 .4058 49T 11-24-3 .4902 26 130.0
RIT 41 60.95 .3967 10 23-13-2 1.714 55 35.56
Merrimack 42 58.37 .3880 51 9-21-4 .4783 29 122.1
Providence 43 56.44 .3812 53 7-22-5 .3878 21 145.6
Niagara 44 55.79 .3789 25 16-14-6 1.118 48 49.92
Harvard 45 55.06 .3763 43 9-16-6 .6316 38 87.18
Mercyhurst 46 54.00 .3724 18 22-15-3 1.424 52 37.91
Clarkson 47 51.94 .3647 46T 10-19-7 .6000 39 86.57
Bentley 48 43.79 .3317 26 19-17-2 1.111 50 39.42
RPI 49 37.04 .3007 52 10-27-2 .3929 34 94.29
Robert Morris 50 31.33 .2712 46T 10-19-7 .6000 47 52.22
Canisius 51 30.13 .2645 33 15-16-6 .9474 58 31.80
Holy Cross 52 25.05 .2344 40T 13-20-5 .6889 54 36.36
Brown 53 24.91 .2335 57 5-23-5 .2941 40 84.69
Army 54 22.39 .2171 42 11-19-6 .6364 56 35.18
AL-Huntsville 55 21.16 .2088 55T 5-20-5 .3333 45 63.48
Sacred Heart 56 19.19 .1947 48 11-23-4 .5200 53 36.91
Connecticut 57 14.93 .1617 54 9-26-2 .3704 49 40.32
American Intl 58 6.699 .0826 58 5-28-2 .2069 57 32.38

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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