KRACH Ratings for D1 College Hockey (2016-2017)

© 1999-2016, Joe Schlobotnik (archives)

URL for this frameset:

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

Today's KRACH (including games of 2017 March 18)

Team KRACH Record Sched Strength
Rk Rating RRWP Rk W-L-T PF/PA Rk SOS
Denver U 1 680.9 .8440 2 29-7-4 3.444 8 197.7
Minn-Duluth 2 657.6 .8397 3 25-6-7 3.000 3 219.2
Harvard 3 580.1 .8233 1 26-5-2 4.500 21 128.9
Western Mich 4 352.2 .7476 15T 22-12-5 1.690 5 208.4
Mass-Lowell 5 349.4 .7463 6 26-10-3 2.391 13 146.1
Boston Univ 6 338.3 .7408 9T 23-11-3 1.960 9 172.6
North Dakota 7 293.7 .7161 21 21-15-3 1.364 4 215.3
Union 8 281.7 .7086 4 25-9-3 2.524 31 111.6
Cornell 9 280.9 .7080 7 21-8-5 2.238 24 125.5
Minnesota 10 279.9 .7074 9T 23-11-3 1.960 15 142.8
Notre Dame 11 265.4 .6977 12 21-11-5 1.741 11 152.4
Penn State 12 251.0 .6874 8 24-11-2 2.083 25 120.5
Providence 13 242.2 .6806 11 22-11-5 1.815 19 133.4
Boston Coll 14 228.6 .6696 22 21-15-4 1.353 10 169.0
Vermont 15 206.5 .6498 20 20-13-5 1.452 16 142.2
NE-Omaha 16 203.2 .6467 30 17-17-5 1.000 7 203.2
Ohio State 17 196.6 .6402 13T 21-11-6 1.714 30 114.7
Wisconsin 18 192.4 .6359 24 20-15-1 1.323 14 145.5
St Cloud 19 173.9 .6155 35 16-19-1 .8462 6 205.5
Quinnipiac 20 167.0 .6074 19 23-15-2 1.500 32 111.4
Northeastern 21 164.6 .6044 27 18-15-5 1.171 17 140.5
Air Force 22 162.8 .6021 5 26-9-5 2.478 43 65.70
St Lawrence 23 161.2 .6001 25T 17-13-7 1.242 20 129.7
Clarkson 24 140.9 .5722 29 18-16-5 1.108 22 127.1
Miami 25 123.5 .5447 49 9-20-7 .5319 2 232.3
Merrimack 26 119.0 .5369 31 15-16-6 .9474 23 125.6
Michigan Tech 27 104.0 .5084 18 23-14-7 1.514 42 68.70
MSU-Mankato 28 103.7 .5077 17 22-13-4 1.600 45 64.82
Yale 29 103.3 .5068 34 13-15-5 .8857 28 116.6
Michigan 30 97.31 .4942 42 13-19-3 .7073 18 137.6
Princeton 31 95.41 .4901 33 15-16-3 .9429 36 101.2
Robert Morris 32 95.34 .4899 13T 22-12-4 1.714 51 55.61
Bemidji State 33 95.14 .4895 23 22-16-3 1.343 40 70.85
New Hampshire 34 92.53 .4836 37 15-20-5 .7778 26 119.0
Canisius 35 90.07 .4779 15T 21-11-7 1.690 56 53.31
CO College 36 89.70 .4770 55 8-24-4 .3846 1 233.2
Connecticut 37 88.28 .4736 36 12-16-8 .8000 33 110.4
Bowling Green 38 69.34 .4230 28 21-18-2 1.158 48 59.89
Dartmouth 39 62.75 .4024 47 10-18-3 .5897 34 106.4
Army 40 62.63 .4020 25T 18-14-5 1.242 58 50.41
Maine 41 58.49 .3881 48 11-21-4 .5652 35 103.5
Arizona State 42 53.34 .3695 52 8-19-3 .4634 29 115.1
Mich State 43 51.45 .3623 56 7-24-4 .3462 12 148.6
Holy Cross 44 49.41 .3544 32 14-15-7 .9459 57 52.24
Bentley 45 47.64 .3472 39 13-19-7 .7333 44 64.96
Colgate 46 47.40 .3462 51 9-22-6 .4800 37 98.75
Sacred Heart 47 43.83 .3311 40T 13-19-5 .7209 46 60.80
Northern Mich 48 43.04 .3276 46 13-22-4 .6250 41 68.86
Ferris State 49 41.02 .3186 40T 13-19-5 .7209 50 56.90
Mercyhurst 50 38.62 .3073 38 15-20-4 .7727 59 49.97
Lake Superior 51 37.36 .3012 43 11-18-7 .6744 52 55.40
AK-Fairbanks 52 36.44 .2966 45 12-20-4 .6364 49 57.26
RIT 53 34.79 .2882 44 14-22-1 .6444 54 53.98
RPI 54 27.75 .2492 57 8-28-1 .2982 39 93.05
AK-Anchorage 55 25.18 .2334 54 7-21-6 .4167 47 60.44
AL-Huntsville 56 24.22 .2272 53 9-22-3 .4468 53 54.20
American Intl 57 23.66 .2235 50 8-20-8 .5000 60 47.31
Mass-Amherst 58 23.39 .2218 58T 5-29-2 .2000 27 117.0
Brown 59 18.50 .1872 60 4-25-2 .1923 38 96.20
Niagara 60 10.67 .1208 58T 5-31-3 .2000 55 53.34

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 /

HTML 4.0 compliant CSS2 compliant