KRACH Ratings for D1 College Hockey (2017-2018)

© 1999-2017, 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 2018 March 17)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
St Cloud 1 486.2 .8092 4 25-8-6 2.545 3 191.0
Notre Dame 2 468.0 .8036 3 25-9-2 2.600 9 180.0
Cornell 3 389.0 .7747 1 25-5-2 4.333 36 89.78
Ohio State 4 370.2 .7665 5 24-9-5 2.304 14 160.7
Denver U 5 369.3 .7660 7T 22-9-8 2.000 7 184.6
MSU-Mankato 6 300.1 .7298 2 29-9-1 3.105 30 96.63
Michigan 7 234.3 .6829 16T 20-14-3 1.387 13 168.9
Minn-Duluth 8 234.2 .6828 20 21-16-3 1.286 8 182.2
Minnesota 9 230.6 .6797 23T 19-17-2 1.111 1 207.5
Penn State 10 213.2 .6640 21 18-14-5 1.242 11 171.6
North Dakota 11 209.4 .6603 22 17-13-10 1.222 12 171.3
Clarkson 12 209.0 .6599 7T 23-10-6 2.000 20 104.5
Northeastern 13 199.1 .6500 6 23-9-5 2.217 35 89.79
Providence 14 195.9 .6466 9 23-11-4 1.923 23 101.9
NE-Omaha 15 185.2 .6350 28T 17-17-2 1.000 5 185.2
Boston Univ 16 181.0 .6301 13 21-13-4 1.533 17 118.0
Boston Coll 17 166.8 .6128 16T 20-14-3 1.387 16 120.2
Princeton 18 150.1 .5901 14 19-12-4 1.500 26 100.1
Bowling Green 19 148.6 .5878 10 23-12-6 1.733 41 85.71
Wisconsin 20 140.9 .5762 42T 14-19-4 .7619 6 184.9
Western Mich 21 140.6 .5758 39T 15-19-2 .8000 10 175.8
Union 22 133.9 .5650 18 21-15-2 1.375 29 97.37
Northern Mich 23 133.8 .5648 12 25-15-3 1.606 45 83.29
Miami 24 132.8 .5633 45 12-20-5 .6444 2 206.1
CO College 25 132.3 .5624 35 15-17-5 .8974 15 147.4
Michigan Tech 26 117.0 .5352 19 22-16-5 1.324 38 88.38
Maine 27 108.8 .5188 23T 18-16-4 1.111 28 97.88
Harvard 28 107.5 .5161 27 15-14-4 1.062 25 101.1
Mich State 29 107.1 .5153 50T 12-22-2 .5652 4 189.4
Mercyhurst 30 95.45 .4896 11 21-12-4 1.643 58 58.10
Bemidji State 31 93.96 .4860 23T 16-14-8 1.111 42 84.57
Quinnipiac 32 89.16 .4743 34 16-18-4 .9000 27 99.07
Mass-Amherst 33 88.24 .4720 37 17-20-2 .8571 21 102.9
Connecticut 34 84.23 .4616 39T 15-19-2 .8000 19 105.3
Yale 35 83.50 .4597 28T 15-15-1 1.000 44 83.50
Air Force 36 82.73 .4576 15 22-14-5 1.485 60 55.71
Mass-Lowell 37 82.47 .4569 36 17-19 .8947 34 92.18
Dartmouth 38 81.30 .4537 32 16-17-2 .9444 40 86.09
Colgate 39 79.65 .4491 28T 17-17-6 1.000 46 79.65
Merrimack 40 69.59 .4194 48 12-21-4 .6087 18 114.3
Canisius 41 64.85 .4040 23T 19-17-2 1.111 57 58.37
Army 42 58.43 .3815 28T 15-15-6 1.000 56 58.43
Vermont 43 58.38 .3813 49 10-20-7 .5745 24 101.6
Robert Morris 44 57.66 .3787 33 18-20-3 .9070 51 63.58
New Hampshire 45 53.91 .3645 50T 10-20-6 .5652 33 95.38
Ferris State 46 51.54 .3551 47 14-23-1 .6170 43 83.53
AL-Huntsville 47 48.30 .3416 52 12-23-2 .5417 37 89.16
Holy Cross 48 48.08 .3407 38 13-16-7 .8462 59 56.82
American Intl 49 47.66 .3389 41 15-20-4 .7727 54 61.67
RIT 50 47.10 .3365 42T 15-20-2 .7619 53 61.82
Arizona State 51 45.96 .3316 57 8-21-5 .4468 22 102.9
Bentley 52 45.65 .3302 42T 13-18-6 .7619 55 59.91
Brown 53 41.36 .3105 56 8-19-4 .4762 39 86.85
Sacred Heart 54 40.16 .3048 46 13-22-4 .6250 50 64.25
AK-Fairbanks 55 38.08 .2945 53T 11-22-3 .5319 49 71.60
Lake Superior 56 37.83 .2933 55 10-22-4 .5000 47 75.65
Niagara 57 33.51 .2706 53T 11-22-3 .5319 52 63.01
St Lawrence 58 30.78 .2554 58 8-27-2 .3214 32 95.77
RPI 59 26.49 .2296 59 6-27-4 .2759 31 96.02
AK-Anchorage 60 15.57 .1520 60 4-26-4 .2143 48 72.66

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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