College Football Performance Rating
Calculation Example

The definition of Wilson's College Football Performance Rating is:
   A team's rating is the average of its opponents' ratings plus 100
   for a win or minus 100 for a loss.  Wins that lower the rating
   and losses that raise the rating count one twentieth as much
   as the other games.  Post-season games count double.

The ratings are calculated by setting everyone's rating to a default value based on division and then calculating the ratings using the above definition over and over again until they settle down. The default values used actually has no effect on the final ratings except for teams in those conferences that play no non-conference games.

For this example, we'll use the record of one such isolated conference--the New England Small College Athletic Conference (NESCAC) as of 10/23/99. Here are the game results. "Win" or "Loss" refers to how the team at the left did. Each game appears in the table twice.

At the time I worked out this example, the weight for wins that lower a rating and losses that raise a rating was one hundreth (0.01) rather than the current one twentieth (0.05).

Amherst Bates Bowdoin Colby Hamilton Middlebury Trinity Tufts Wesleyan Williams
AmherstLossWinWin Loss  Win 
BatesWin   Loss LossLossLoss
BowdoinLoss  Win LossLoss Loss
ColbyLoss  WinLossLoss Win 
Hamilton  LossLoss LossLossLoss 
MiddleburyWinWin Win   LossLoss
Trinity  WinWinWin Win Loss
Tufts WinWin Win Loss  Loss
WesleyanLossWin LossWinWin    
Williams WinWin  WinWinWin 

This is all the information needed to calculate the ratings. Other information, such as margin of victory, home field advantage or order of the games is not used in this system.

All the teams start with an initial rating of 360, the average rating for other Division III teams. Here's the first step rating calculation for Trinity:

Trinity, Conn.            Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product
--------------   ------   ------   ------   ------   -------
Colby             WIN      360.0    460.0    1.00      460.0
Williams          LOSS     360.0    260.0    1.00      260.0
Hamilton          WIN      360.0    460.0    1.00      460.0
Tufts             WIN      360.0    460.0    1.00      460.0
Bowdoin           WIN      360.0    460.0    1.00      460.0
                                            =====    =======
                                             5.00     2100.0
                              2100.0
   Trinity, Conn.'s rating =  ------ =  420.0
                               5.00
The new estimate for Trinity's rating is not 420, however. The new estimate is set at 390, half way between the old estimate (360) and this newly calculated rating (420). This is done because the new rating calculation tends to overestimate the change that should be made.

step Amherst Bates Bowdoin Colby Hamilton Middlebury Trinity Tufts Wesleyan Williams
0360.0360.0360.0360.0360.0360.0360.0360.0360.0360.0
1370.0330.0330.0350.0310.0370.0390.0370.0370.0410.0
2370.0324.0320.0346.0286.0378.0402.0372.0368.0434.0

Now, let's look at the calculation of Trinity's rating for the third step.

Trinity, Conn.            Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product
--------------   ------   ------   ------   ------   -------
Colby             WIN      346.0    446.0    1.00      446.0
Williams          LOSS     434.0    334.0    1.00      334.0
Hamilton          WIN      286.0    386.0    0.01        3.9
Tufts             WIN      372.0    472.0    1.00      472.0
Bowdoin           WIN      320.0    420.0    1.00      420.0
                                            =====    =======
                                             4.01     1675.9
                              1675.9
   Trinity, Conn.'s rating =  ------ =  417.9
                               4.01
Even after adding the 100 points to Hamilton's rating for the win, the game rating of 386 is below Trinity's rating of 402 at the end of step 2. This means that averaging this game rating in with the others would lower Trinity's rating. Games such as this where a win would lower the rating (and the loss would raise the rating when Hamilton's rating is calculated) are given a wieght of 0.01 in the average. Thus, there is a small lowering of Trinity's rating for playing such a weak opponent.

The new estimate for Trinity's rating is set at 410, half way between the old estimate (402) and this newly calculated rating (417.9). After this step, the average rating for all ten teams was 360.7, so all ratings were reduced by 0.7 to keep the average rating at 360.

step Amherst Bates Bowdoin Colby Hamilton Middlebury Trinity Tufts Wesleyan Williams
3367.9322.4313.1342.7268.2382.5409.3371.9363.7458.1
4366.3321.9308.7340.4254.1386.7415.2373.7360.2472.7
5365.4322.3304.0338.9240.2389.9422.4377.1357.4482.5
6363.7322.2299.0336.7231.0391.4427.2379.7355.5493.5
7362.2322.3295.3335.4216.4392.8431.5381.0354.3508.7
8361.0322.4291.1335.2207.3393.1436.2381.8353.6518.4
9359.7322.2287.2335.1200.5392.8442.3382.1352.9525.2
10358.5321.8284.0334.7195.2392.3447.0382.6352.2531.6
12356.5321.1278.2333.6187.9391.2454.6384.2351.1541.7
14354.6320.6273.3332.3182.0390.1460.8387.1349.9549.4
16352.8320.3269.4331.0177.3389.0466.0389.8348.8555.5
18351.3320.0266.4329.8173.6388.0470.4392.2347.7560.6
20350.0319.8263.9328.6170.7387.0474.1394.3346.7564.8
25347.5319.2259.5326.2165.6385.0481.1398.2344.7572.9
30345.7318.7256.8324.5162.6383.6485.8400.7343.2578.3
35344.5318.3255.0323.4160.7382.6488.9402.4342.2581.9
40343.7318.1253.9322.6159.4381.9491.0403.5341.5584.3
45343.2317.9253.2322.0158.7381.4492.4404.3341.0585.9
50342.9317.8252.7321.6158.1381.1493.3404.8340.7587.0
60342.5317.6252.2321.2157.6380.7494.4405.3340.3588.2
70342.3317.6252.0321.0157.3380.5494.8405.6340.2588.7
80342.2317.5251.9321.0157.2380.5495.0405.7340.1589.0
90342.2317.5251.8320.9157.2380.4495.1405.7340.1589.1
100342.1317.5251.8320.9157.2380.4495.1405.7340.0589.1
120342.1317.5251.8320.9157.1380.4495.2405.7340.0589.2

At this point the ratings have stabilized and we have the final answer. The final rating calculations are:

Amherst                   Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Bates             LOSS     317.5    217.5    1.00      217.5   -124.6
Bowdoin           WIN      251.8    351.8    1.00      351.8     +9.7
Middlebury        LOSS     380.4    280.4    1.00      280.4    -61.7
Colby             WIN      320.9    420.9    1.00      420.9    +78.8
Wesleyan          WIN      340.0    440.0    1.00      440.0    +97.9
                                            =====    =======
                                             5.00     1710.6
                              1710.6
          Amherst's rating =  ------ =  342.1
                               5.00


Bates                     Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Amherst           WIN      342.1    442.1    1.00      442.1   +124.6
Tufts             LOSS     405.7    305.7    1.00      305.7    -11.8
Williams          LOSS     589.2    489.2    0.01        4.9     +1.7
Wesleyan          LOSS     340.0    240.0    1.00      240.0    -77.5
Middlebury        LOSS     380.4    280.4    1.00      280.4    -37.1
                                            =====    =======
                                             4.01     1273.2
                              1273.2
            Bates's rating =  ------ =  317.5
                               4.01


Bowdoin                   Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Williams          LOSS     589.2    489.2    0.01        4.9     +2.4
Amherst           LOSS     342.1    242.1    1.00      242.1     -9.7
Tufts             LOSS     405.7    305.7    0.01        3.1     +0.5
Hamilton          WIN      157.1    257.1    1.00      257.1     +5.3
Trinity, Conn.    LOSS     495.2    395.2    0.01        4.0     +1.4
                                            =====    =======
                                             2.03      511.2
                               511.2
          Bowdoin's rating =  ------ =  251.8
                               2.03


Colby                     Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Trinity, Conn.    LOSS     495.2    395.2    0.01        4.0     +0.7
Middlebury        LOSS     380.4    280.4    1.00      280.4    -40.5
Wesleyan          WIN      340.0    440.0    1.00      440.0   +119.1
Amherst           LOSS     342.1    242.1    1.00      242.1    -78.8
Hamilton          WIN      157.1    257.1    0.01        2.6     -0.6
                                            =====    =======
                                             3.02      969.1
                               969.1
            Colby's rating =  ------ =  320.9
                               3.02


Hamilton                  Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Tufts             LOSS     405.7    305.7    0.01        3.1     +1.5
Wesleyan          LOSS     340.0    240.0    0.01        2.4     +0.8
Trinity, Conn.    LOSS     495.2    395.2    0.01        4.0     +2.4
Bowdoin           LOSS     251.8    151.8    1.00      151.8     -5.3
Colby             LOSS     320.9    220.9    0.01        2.2     +0.6
                                            =====    =======
                                             1.04      163.4
                               163.4
         Hamilton's rating =  ------ =  157.1
                               1.04


Middlebury                Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Wesleyan          LOSS     340.0    240.0    1.00      240.0   -140.4
Colby             WIN      320.9    420.9    1.00      420.9    +40.5
Amherst           WIN      342.1    442.1    1.00      442.1    +61.7
Williams          LOSS     589.2    489.2    0.01        4.9     +1.1
Bates             WIN      317.5    417.5    1.00      417.5    +37.1
                                            =====    =======
                                             4.01     1525.4
                              1525.4
       Middlebury's rating =  ------ =  380.4
                               4.01


Trinity, Conn.            Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Colby             WIN      320.9    420.9    0.01        4.2     -0.7
Williams          LOSS     589.2    489.2    1.00      489.2     -6.0
Hamilton          WIN      157.1    257.1    0.01        2.6     -2.4
Tufts             WIN      405.7    505.7    1.00      505.7    +10.6
Bowdoin           WIN      251.8    351.8    0.01        3.5     -1.4
                                            =====    =======
                                             2.03     1005.2
                              1005.2
   Trinity, Conn.'s rating =  ------ =  495.2
                               2.03


Tufts                     Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Hamilton          WIN      157.1    257.1    0.01        2.6     -1.5
Bates             WIN      317.5    417.5    1.00      417.5    +11.8
Bowdoin           WIN      251.8    351.8    0.01        3.5     -0.5
Trinity, Conn.    LOSS     495.2    395.2    1.00      395.2    -10.6
Williams          LOSS     589.2    489.2    0.01        4.9     +0.8
                                            =====    =======
                                             2.03      823.7
                               823.7
            Tufts's rating =  ------ =  405.7
                               2.03


Wesleyan                  Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Middlebury        WIN      380.4    480.4    1.00      480.4   +140.4
Hamilton          WIN      157.1    257.1    0.01        2.6     -0.8
Colby             LOSS     320.9    220.9    1.00      220.9   -119.1
Bates             WIN      317.5    417.5    1.00      417.5    +77.5
Amherst           LOSS     342.1    242.1    1.00      242.1    -97.9
                                            =====    =======
                                             4.01     1363.5
                              1363.5
         Wesleyan's rating =  ------ =  340.0
                               4.01


Williams                  Oppon.    Game
   Opponent      Result   Rating   Rating   Weight   Product   Effect
--------------   ------   ------   ------   ------   -------   ------
Bowdoin           WIN      251.8    351.8    0.01        3.5     -2.4
Trinity, Conn.    WIN      495.2    595.2    1.00      595.2     +6.0
Bates             WIN      317.5    417.5    0.01        4.2     -1.7
Middlebury        WIN      380.4    480.4    0.01        4.8     -1.1
Tufts             WIN      405.7    505.7    0.01        5.1     -0.8
                                            =====    =======
                                             1.04      612.7
                               612.7
         Williams's rating =  ------ =  589.2
                               1.04


                    New England Smalls

 360 Ave. Rating  /--Conference---\  /---Over-all----\ Perf
                   W  L   Pts  Opp    W  L   Pts  Opp  Rate
                   -  -   ---  ---    -  -   ---  ---  ----
Williams           5  0   146   31    5  0   146   31   589
Trinity, Conn.     4  1   105   44    4  1   105   44   495
Tufts              3  2   107   89    3  2   107   89   406
Middlebury         3  2    88   94    3  2    88   94   380
Amherst            3  2    82   71    3  2    82   71   342
Wesleyan           3  2   119   97    3  2   119   97   340
Colby              2  3   100  121    2  3   100  121   321
Bates              1  4    77  102    1  4    77  102   318
Bowdoin            1  4    42  128    1  4    42  128   252
Hamilton           0  5    74  163    0  5    74  163   157

<- Parent Directory

David Wilson / dwilson@cae.wisc.edu