KRACH Ratings for D1 College Hockey (2010-2011)

© 1999-2010, 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 2011 March 19)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
North Dakota 1 597.0 .8313 3 30-8-3 3.316 7 180.0
Boston Coll 2 477.0 .7996 2 30-7-1 4.067 26 117.3
Yale 3 384.7 .7659 1 27-6-1 4.231 44 90.92
Denver U 4 355.6 .7528 8 24-11-5 1.963 5 181.2
Miami 5 334.9 .7426 7 23-9-6 2.167 14 154.6
Michigan 6 328.5 .7392 6 26-10-4 2.333 17 140.8
Minn-Duluth 7 314.1 .7313 9 22-10-6 1.923 10 163.3
Notre Dame 8 261.2 .6976 11 23-13-5 1.645 12 158.8
Merrimack 9 258.1 .6954 5 25-9-4 2.455 33 105.2
NE-Omaha 10 242.6 .6836 19 21-15-2 1.375 8 176.5
Union 11 231.9 .6748 4 26-9-4 2.545 42 91.08
New Hampshire 12 222.2 .6665 10 21-10-6 1.846 24 120.4
CO College 13 217.7 .6625 24 22-18-3 1.205 6 180.6
Western Mich 14 199.7 .6453 18 19-12-10 1.412 16 141.5
Minnesota 15 190.1 .6354 25 16-14-6 1.118 9 170.1
Wisconsin 16 186.0 .6310 23 21-16-4 1.278 15 145.6
AK-Anchorage 17 178.1 .6221 32 16-18-3 .8974 2 198.5
St Cloud 18 170.1 .6127 33T 15-18-5 .8537 1 199.3
Maine 19 168.4 .6105 21 17-12-7 1.323 21 127.3
Boston Univ 20 166.0 .6075 16 19-12-8 1.438 27 115.5
Bemidji State 21 160.8 .6009 33T 15-18-5 .8537 3 188.3
AK-Fairbanks 22 152.4 .5897 30 16-17-5 .9487 11 160.7
Dartmouth 23 142.5 .5756 15 19-12-3 1.519 40 93.86
Ferris State 24 139.8 .5715 26 18-16-5 1.108 22 126.1
RPI 25 136.5 .5665 13 20-12-5 1.552 45 87.99
MSU-Mankato 26 127.0 .5511 40T 14-18-6 .8095 13 156.8
Lake Superior 27 112.7 .5257 38T 13-17-9 .8140 18 138.5
Mich State 28 111.5 .5233 40T 15-19-4 .8095 19 137.7
Princeton 29 107.6 .5157 22 17-13-2 1.286 46 83.69
Cornell 30 107.1 .5148 27 16-15-3 1.061 35 101.0
Northern Mich 31 104.1 .5086 38T 15-19-5 .8140 20 127.9
Ohio State 32 103.1 .5066 35T 15-18-4 .8500 23 121.3
Quinnipiac 33 99.41 .4987 29 16-15-8 1.053 38 94.44
Northeastern 34 96.08 .4914 31 14-16-8 .9000 32 106.8
Air Force 35 78.13 .4474 12 20-11-6 1.643 48 47.55
Brown 36 75.45 .4400 44 10-16-5 .6757 29 111.7
Clarkson 37 72.84 .4326 42 15-19-2 .8000 43 91.05
RIT 38 70.82 .4267 14 19-11-8 1.533 49 46.19
St Lawrence 39 67.63 .4171 45 13-22-5 .6327 31 106.9
Robert Morris 40 59.95 .3923 17 18-12-5 1.414 52 42.40
Harvard 41 59.89 .3921 48 12-21-1 .5814 34 103.0
Niagara 42 58.49 .3873 20 18-13-4 1.333 51 43.87
Vermont 43 57.06 .3822 50 8-20-8 .5000 28 114.1
Providence 44 51.01 .3598 49 8-18-8 .5455 41 93.52
Bowling Green 45 49.69 .3546 52 10-27-4 .4138 25 120.1
Holy Cross 46 42.61 .3249 28 17-16-5 1.054 53 40.42
Colgate 47 41.64 .3205 51 11-28-3 .4237 36 98.26
Mercyhurst 48 38.00 .3036 35T 15-18-4 .8500 50 44.71
Mass-Amherst 49 37.75 .3023 54 6-23-6 .3462 30 109.1
Canisius 50 35.44 .2909 43 13-19-6 .7273 47 48.73
Michigan Tech 51 34.61 .2866 57 4-30-4 .1875 4 184.6
Connecticut 52 34.05 .2837 35T 15-18-4 .8500 54 40.06
Mass-Lowell 53 24.93 .2314 56 5-25-4 .2593 37 96.14
Bentley 54 21.70 .2101 46 10-18-6 .6190 58 35.06
Army 55 21.10 .2060 47 11-20-4 .5909 57 35.71
AL-Huntsville 56 17.47 .1794 58 4-26-2 .1852 39 94.32
American Intl 57 12.77 .1405 53 8-24-1 .3469 56 36.80
Sacred Heart 58 12.73 .1402 55 6-25-6 .3214 55 39.61

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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