Strength of Schedule 

Strength of Schedule (SOS) is a rating which applies to a teams schedule
where the stronger the opponents, the higher the SOS rating, or conversely,
the weaker the opponents, the lower the SOS rating. In order to tabulate
an SOS rating, the opponent's strength has to be evaluated. A measure of
an opponent's strength is its power rating. The higher the rating, the
stronger the team.

There are three methods that come to mind when doing an SOS rating.
The first method uses an average of all opponents PR's to arrive at the SOS.
A second method uses a weighted average of the PR's so that 'tougher games'
are not "diluted" by games against weaker games. A third method uses the
Bowl Championship Series (BCS) SOS rating as applied to college football:
it employs the W-L record of the opponents and the W-L record of the
opponent's opponent. This method employs the Ratings Percentage Index (PRI).

All three methods are demonstrated below with the advantages and disadvan-
tages for each illustrated. The results of all three methods are shown for:
(1) average, (2) weighted average and (3) BCS. We use the college
football schedule for the year 2001 through Oct 31,2001 and use the game results
of Oklahoma to derive the three calculations. Note that the results vary
based on the method.

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Oklahoma has played 8 games (Power Ratings as of October 31 shown in
parentheses):

 North Carolina  (71.61)
 Air Force       (43.94)
 North Texas     (46.95)
 Kansas State    (70.70)
 Texas           (82.47)
 Kansas          (52.75)
 Baylor          (44.53)
 Nebraska        (84.16)

The three methods are:

(1) Average of the Opponent's Power Ratings

The average power rating would simply be the sum of all opponent ratings
divided by the number of games played. If the game was against a team
outside division I, then the game would not be included in the SOS rating.
All games for Oklahoma were in Division I. Averaging the opponent's power
ratings we get.

   71.61
   43.94
   46.95
   70.70
   82.47
   52.75
   44.53
   84.16
   -----
  497.11 / 8    =  62.19

Advantages: Simple to understand and explain. Disadvantages: Does not
emphasize playing the best teams and penalizes for adding weaker teams or
requires computer ratings.

(2) Weighted Average of the Opponent's Power Ratings

If we want to give added weight to tougher games so that a team
that plays a strong team (PR = 75.) and a weak team (PR = 25.)
will have a higher rating than a team that plays two average
teams (PR = 50.), we average the square root of the ratings and
then normalize the results.

  (71.61/100.) **0.5 * 100.
  (43.94/100.) **0.5 * 100.
  (46.95/100.) **0.5 * 100.
  (70.70/100.) **0.5 * 100.
  (82.47/100.) **0.5 * 100.
  (52.75/100.) **0.5 * 100.
  (44.53/100.) **0.5 * 100.
  (84.16/100.) **0.5 * 100.
  ------------
  Rating =   78.17

which changes every teams SOS rating. Some teams benefited by
playing stronger teams where the weaker foes did not reduce their
ratings as much as the average method.

Advantages: Emphasizes playing the best team while not penalizing
for adding weaker teams. Disadvantages: Not simple to understand or
explain; requires computer ratings.

(3) The Bowl Championship Series Method

Oklahoma Opponent record = 34/26

North Carolina   5   3
Air Force        4   3
North Texas      2   5
Kansas State     3   4
Texas            7   1
Kansas           2   5
Baylor           2   5
Nebraska         9   0
                ------
                34  26    (total games = 60)

Opponent's  W/L Record  =   34/26   % = 34/60 =  0.57
and each opponent's record was:
North Carolina  42  18
Air Force       22  24
North Texas     18  18
Kansas State    32  22
Texas           30  31
Kansas          29  15
Baylor          37  16
Nebraska        39  27
               -------
               249 171 (total games = 420)


W/L Record =  249/171 % = 249/420 = 0.59

Oklahoma's opponent's records can be found here.

Now the BCS SOS takes 2/3 of the opponent's record and 1/3 of the opponent's
opponent's record yielding an SOS of 0.575. it then ranks them (1 to 117)
and divides by 25 to generate the quartile (as shown on the SOS pages).
SOS rating = 0.667*0.57 + 0.333*0.59 = 0.575

Advantages: Independent of computer ratings. Disadvantages: Difficult to
compute; data is somewhat abstract.

In summary, there is not a lot of difference between average and weighted
average (for football) but there are noticeable differences between the BCS
and the average/weighted average approaches. There are additional issues
such as: (1) Should only games played count in the sos or should all games
scheduled count? (2) Should games outside a division count?