# Danish Rating System

__Keywords__: Tournament

### Danish Rating System

This is a description of the Danish Rating System.

The system was developed at some time back in the 1980's.

### Rank and rating

The rating is calculated with 2 decimals. In general: A negitive number is a kyu player and a positive number is a dan player, but Shodan's (1 dan) has negative numbers from -0,50 and positive number up to +0,49. (Scroll down to the end of this page to see all rank and corresponding rating interval.)

### Rated Games

Not all games are to be rated, or more precise, not all games are to be rated for both players. When the Rank difference between 2 players is very large, then only some of the games are rated.

- If a players rating is below 5 kyu = -5,50 (not including), and he is playing WHITE, then games are only rated when he didn't give more than 5 stones in handicap. BLACK's games are always rated.
- If a players rating is above 5 kyu = -5,50 (including), then games are only rated when there are not more than 4 stones in handicap. This is for both colors.

+----------+--------+-------+ | Rating | BLACK | WHITE | +----------+--------+-------+ | < -5,50 | Always | Max 5 | +----------+--------+-------+ | >= -5,50 | Max 4 | Max 4 | +----------+--------+-------+

### Winning percent

Prior to a games in a tournament the winning percentage for both players is calculated using an accumulated normal distribution. The distribution uses the parameters x_=0 ; and s=3,568. (Scroll down and see the table for the accumulated normal distrution at the end of this page.)

From 2 players with ranking -2,13 and -2,63 the winning percent is calculated to be:

- Player 1 (-2,13) : N( -2,13 - -2,63 ; 0 ; 3,568 ) = 56 %
- Player 2 (-2,63) : N( -2,63 - -2,13 ; 0 ; 3,568 ) = 44 %

### Handicap stones

The system also works using handicap stones. When a handicap is given in a game the rating difference is reduced by the number of stones given. A player with rating -2,13 and a player with rating -5,87 has a difference of -2,13 - -5,87 = 3,74. In the accumulated normal distribution this gives a winning percent of 85 % for the player with rating -2,13, and 15 % for the other. If they played a handicap game with 3 stones this would be -2,13 - -5,87 - 3 = 0,74, and a winning percent of 58 % for the stronger player and 42 % for the weaker.

### Compare with actual score

The winning percent is calculated for all games in the turnament. All winning percent are accumulated and compared with the actual score in the turnament. Example:

- A player has an accumulated winning percent of 2,37 from 6 games, and he won 3. So he had +0,63 better score than expected.

### Bonus

If the difference between expected score and actual score is larger than 1, then a bonus is given by doubling the part over 1. Example: If the expected score is 1.88 and actual score is 3, the difference is 3 - 1,88 = 1,12. Bonus 1 + 0,12 x 2 = 1,24.

### Coefficient

Before the new rating for a player is found, the score difference is multiplied by a coefficient to reduce fluxtuation. The coefficient is selected using 2 parameters; Main thinking time and the strength of the player. If the time is between [ 45 min ; 75 min [ the coefficient is:

- Rating: [ -18,50 ; -9,50 [ => 0,40
- Rating: [ -9,50 ; -4,50 [ => 0,30
- Rating: [ -4,50 ; -0,50 [ => 0,20
- Rating: [ -0,50 ; +8,50 [ => 0,13

If the time is between [ 75 min ; inf [ the coefficient is:

- Rating: [ -18,50 ; -9,50 [ => 0,60 = -9,88 ( =0.80
- Rating: [ -9,50 ; -4,50 [ => 0,45
- Rating: [ -4,50 ; -0,50 [ => 0,30
- Rating: [ -0,50 ; +8,50 [ => 0,20

So with a score differencece of +0,63 and a thinking time of 45 min, the player with rating -2,13 would become -2,13 + 0,63 x 0,20 = -2,00

A player with rating +1,06, a score difference of -0,17 and a thinking time of 75 min would become: +1,06 + -0,17 x 0,20 = +1,03.

A player with rating -10,00, a score difference of +1,34 and a thinking time of 90 min would become: -10,00 + ( 1,00 + 0,34 x 2 ) x 0,60 = -8,99.

### Accumulated Normal Distribution

Here is a list of the an accumulated normal distribution with x_=0 and s=3,568. When the rating diffence is negative, then 1 - accumulated value must be used.

- [ 0,00 ; 0,05 [ => 0,50
- [ 0,05 ; 0,14 [ => 0,51
- [ 0,14 ; 0,23 [ => 0,52
- [ 0,23 ; 0,32 [ => 0,53
- [ 0,32 ; 0,41 [ => 0,54
- [ 0,41 ; 0,50 [ => 0,55
- [ 0,50 ; 0,59 [ => 0,56
- [ 0,59 ; 0,68 [ => 0,57
- [ 0,68 ; 0,77 [ => 0,58
- [ 0,77 ; 0,86 [ => 0,59
- [ 0,86 ; 0,96 [ => 0,60
- [ 0,96 ; 1,05 [ => 0,61
- [ 1,05 ; 1,14 [ => 0,62
- [ 1,14 ; 1,24 [ => 0,63
- [ 1,24 ; 1,33 [ => 0,64
- [ 1,33 ; 1,43 [ => 0,65
- [ 1,43 ; 1,53 [ => 0,66
- [ 1,53 ; 1,62 [ => 0,67
- [ 1,62 ; 1,72 [ => 0,68
- [ 1,72 ; 1,82 [ => 0,69
- [ 1,82 ; 1,93 [ => 0,70
- [ 1,93 ; 2,03 [ => 0,71
- [ 2,03 ; 2,14 [ => 0,72
- [ 2,14 ; 2,25 [ => 0,73
- [ 2,25 ; 2,36 [ => 0,74
- [ 2,36 ; 2,47 [ => 0,75
- [ 2,47 ; 2,58 [ => 0,76
- [ 2,58 ; 2,70 [ => 0,77
- [ 2,70 ; 2,82 [ => 0,78
- [ 2,82 ; 2,94 [ => 0,79
- [ 2,94 ; 3,07 [ => 0,80
- [ 3,07 ; 3,20 [ => 0,81
- [ 3,20 ; 3,34 [ => 0,82
- [ 3,34 ; 3,48 [ => 0,83
- [ 3,48 ; 3,63 [ => 0,84
- [ 3,63 ; 3,78 [ => 0,85
- [ 3,78 ; 3,94 [ => 0,86
- [ 3,94 ; 4,11 [ => 0,87
- [ 4,11 ; 4,29 [ => 0,88
- [ 4,29 ; 4,48 [ => 0,89
- [ 4,48 ; 4,68 [ => 0,90
- [ 4,68 ; 4,90 [ => 0,91
- [ 4,90 ; 5,14 [ => 0,92
- [ 5,14 ; 5,41 [ => 0,93
- [ 5,41 ; 5,71 [ => 0,94
- [ 5,71 ; 6,05 [ => 0,95
- [ 6,05 ; 6,47 [ => 0,96
- [ 6,47 ; 7,00 [ => 0,97
- [ 7,00 ; 7,75 [ => 0,98
- [ 7,75 ; 9,20 [ => 0,99
- [ 9,20 ; inf [ => 1,00

### Ranking and rating interval

Here is a list of the rank and the intervals

- 6 dan = [ 4,50 ; 5,50 [
- 5 dan = [ 3,50 ; 4,50 [
- 4 dan = [ 2,50 ; 3,50 [
- 3 dan = [ 1,50 ; 2,50 [
- 2 dan = [ 0,50 ; 1,50 [
- 1 dan = [ -0,50 ; 0,50 [
- 1 kyu = [ -1,50 ; -0,50 [
- 2 kyu = [ -2,50 ; -1,50 [
- 3 kyu = [ -3,50 ; -2,50 [
- 4 kyu = [ -4,50 ; -3,50 [
- 5 kyu = [ -5,50 ; -4,50 [
- 6 kyu = [ -6,50 ; -5,50 [
- 7 kyu = [ -7,50 ; -6,50 [
- 8 kyu = [ -8,50 ; -7,50 [
- 9 kyu = [ -9,50 ; -8,50 [
- 10 kyu = [ -10,50 ; -9,50 [
- 11 kyu = [ -11,50 ; -10,50 [
- 12 kyu = [ -12,50 ; -11,50 [
- 13 kyu = [ -13,50 ; -12,50 [
- 14 kyu = [ -14,50 ; -13,50 [
- 15 kyu = [ -15,50 ; -14,50 [
- 16 kyu = [ -16,50 ; -15,50 [
- 17 kyu = [ -17,50 ; -16,50 [
- 18 kyu = [ -18,50 ; -17,50 [
- 19 kyu = [ -19,50 ; -18,50 [
- 20 kyu = [ -20,50 ; -19,50 [
- 21 kyu = [ -21,50 ; -20,50 [
- 22 kyu = [ -22,50 ; -21,50 [
- 23 kyu = [ -23,50 ; -22,50 [
- 24 kyu = [ -24,50 ; -23,50 [
- 25 kyu = [ -25,50 ; -24,50 [