KRACH Ratings for D1 College Hockey (2003-2004)

© 1999-2004, Joe Schlobotnik (archives)

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

Today's KRACH (including games of 2004 March 20)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
North Dakota 1 981.8 .8803 2 29-7-3 3.588 2 273.6
Maine 2 596.9 .8222 1 30-7-3 3.706 15 161.1
Boston Coll 3 575.6 .8173 3 27-8-4 2.900 12 198.5
Minnesota 4 554.2 .8121 7 26-13-3 1.897 1 292.2
Minn-Duluth 5 503.5 .7985 5 26-12-4 2.000 4 251.7
Denver U 6 397.7 .7627 8 23-12-5 1.759 8 226.1
Wisconsin 7 343.4 .7387 12 21-12-8 1.562 10 219.8
CO College 8 279.7 .7030 24 20-16-3 1.229 7 227.7
Michigan 9 274.9 .6998 6 26-13-2 1.929 23 142.5
Ohio State 10 269.2 .6960 9 26-15 1.733 17 155.3
St Cloud 11 260.8 .6903 28 18-16-4 1.111 6 234.7
Miami 12 222.3 .6605 11 23-13-4 1.667 27 133.4
New Hampshire 13 218.9 .6575 20 20-14-6 1.353 14 161.8
Mich State 14 211.7 .6511 17 23-16-2 1.412 20 149.9
Notre Dame 15 195.7 .6359 19 20-14-4 1.375 24 142.3
Mass-Amherst 16 181.2 .6209 26 17-14-6 1.176 18 154.0
Providence 17 179.3 .6188 27 16-14-7 1.114 16 160.9
AK-Anchorage 18 158.9 .5946 45 14-23-3 .6327 5 251.2
Boston Univ 19 157.7 .5931 37T 12-17-9 .7674 11 205.5
Colgate 20 149.7 .5825 10 22-12-5 1.690 40 88.57
Northern Mich 21 147.5 .5796 22 21-16-4 1.278 31 115.5
Cornell 22 140.2 .5693 14 16-10-6 1.462 37 95.91
Mass-Lowell 23 131.8 .5566 32 15-18-7 .8605 19 153.1
Harvard 24 129.5 .5531 23 18-14-3 1.258 34 103.0
Dartmouth 25 128.7 .5517 25 14-11-9 1.194 32 107.8
RPI 26 125.7 .5471 16 22-15-2 1.438 42 87.48
Northeastern 27 124.7 .5453 39 11-16-7 .7436 13 167.7
Western Mich 28 114.9 .5285 31 17-18-4 .9500 29 120.9
AK-Fairbanks 29 112.9 .5250 33T 16-19-1 .8462 26 133.5
Brown 30 110.8 .5210 21 15-11-5 1.296 43 85.44
MSU-Mankato 31 104.1 .5083 52 10-24-5 .4717 9 220.7
Michigan Tech 32 96.21 .4921 55 8-25-5 .3818 3 252.0
Clarkson 33 94.36 .4881 30 18-18-5 1.000 38 94.36
Bowling Green 34 93.32 .4858 43 11-18-9 .6889 25 135.5
Ferris State 35 92.78 .4846 37T 15-20-3 .7674 30 120.9
Bemidji State 36 91.79 .4824 13 20-13-3 1.483 44 61.90
Niagara 37 83.54 .4632 15 21-14-3 1.452 45 57.55
Merrimack 38 80.94 .4568 47 10-20-6 .5652 22 143.2
Union 39 74.91 .4411 33T 14-17-5 .8462 41 88.53
St Lawrence 40 71.40 .4315 41 14-21-6 .7083 36 100.8
Lake Superior 41 65.59 .4145 49 9-20-7 .5319 28 123.3
Yale 42 59.00 .3937 46 12-19 .6316 39 93.42
Holy Cross 43 56.13 .3840 4 22-9-4 2.182 55 25.73
NE-Omaha 44 53.52 .3749 56 8-26-5 .3684 21 145.3
Vermont 45 46.73 .3493 53 9-22-4 .4583 35 102.0
Mercyhurst 46 44.85 .3417 18 20-14-2 1.400 50 32.04
AL-Huntsville 47 35.83 .3015 40 11-16-4 .7222 47 49.61
Quinnipiac 48 30.71 .2755 29 15-14-6 1.059 51 29.00
Findlay 49 29.37 .2682 48 11-22-5 .5510 46 53.30
Princeton 50 25.23 .2441 58 5-24-2 .2400 33 105.1
Sacred Heart 51 23.00 .2301 33T 14-17-5 .8462 52 27.18
Air Force 52 22.81 .2289 44 14-21-2 .6818 49 33.46
Connecticut 53 20.80 .2155 36 12-16-7 .7949 53 26.17
Wayne State 54 20.01 .2100 54 9-24-3 .4118 48 48.59
Canisius 55 18.10 .1962 42 10-16-8 .7000 54 25.85
Bentley 56 10.84 .1353 50 9-19-4 .5238 56 20.70
Army 57 8.982 .1167 51 8-18-3 .4872 58 18.44
American Intl 58 5.169 .0728 57 5-25-4 .2593 57 19.94

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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