I went back and calculated something very similar to Andrew's suggestion. It adds the standard deviations for raw bonus conversion and opponent-adjusted tossup conversion, converts the result to a whole number, and corrects for order-of-finish deviations. Steps as follows:
1. Find the tossup points per tossup heard and the bonus conversion.
2. Compute the Opponent-Averaged Opponent Average Performance (OAOAP, I'm searching for a better and less cumbersome descriptor). This is a strength-of-schedule factor that is unique to each team, and is computed as follows:
A. Set up an n
matrix, where n
is the number of teams in the sectional.
B. For each matrix entry (R, C), compute the tossup points per tossup heard of team R in all games that did not involve team C. Obviously, (R, R) is not applicable and is not used in subsequent steps. Note that forfeit games are not currently counted as games involving other teams. Any future workaround solutions for counting forfeits (e.g. a game against empty chairs) would be included in this step.
C. Take the average of each column weighted by number of games played against each team. So if you had 8 teams, played a full round robin and then split into 4-team brackets with a double RR in each bracket, your column's average would be (1*(each of the teams not in your bracket)+3*(each of the teams in your bracket))/(13 total games played).
D. The resulting number is a rough measure of how hard you had to work to get tossups. Higher numbers mean that your opponents scored tossups more easily against other teams, and therefore you would have had to work harder to get the tossups you did earn. Lower numbers mean that your opponents scored fewer tossups against other teams, and therefore you would not have had to work as hard to get your tossups.
3. Multiply each team's TPTH by its OAOAP and divide by the average OAOAP. This is a team's raw tossup score (RTSC), and represents the TPTH you are likely to have gotten playing an exactly average schedule.
4. Convert from the RTSC to the normalized tossup score (NTSC), which is the number of standard deviations away from the mean RTSC.
5. Convert bonus conversion to the normalized bonus score (NBSC), which is the number of standard deviations away from the mean BC.
6. Compute the cumulative standard deviations (CSD) = NTSC + NBSC.
7. Convert the CSD to a 3 or 4 digit Raw S Value (RSV) = 1000+100*CSD.
8. Convert the RSV to the Adjusted S Value (ASV) as follows:
A. Rank each team within sectional by order of finish and by RSV.
B. For each sectional, if a team's RSV rank is k
spots below its order of finish rank (note that this does not apply to a team that is, e.g., 5th in RSV and in a 3 way tie for 3rd in order of finish, only a team whose RSV rank could not possibly equal its order of finish rank):
For that team and the k
-1 teams above it, the ASV is equal to (1/(k
), where RSV_i
is the i
th place RSV out of the k
So, for a team that is 1 spot lower by RSV than its order of finish, its score is (2/3)*higher RSV +(1/3)*lower RSV. For a team that is 2 spots lower, its score is (1/2)*(highest of 3 RSVs)+(1/4)*(middle of 3 RSVs)+(1/4)*(lower of 3 RSVs). And so on.
C. For all other teams, the ASV is simply the RSV.
D. Continue this process until for each sectional, the order of finish and RSV within-sectional rankings match.
9. Round all ASVs to the nearest integer and sort from highest to lowest. If there is a tie in ASV, higher CSD breaks the tie. Note that steps 7 and 8 can be switched and it won't affect the ASV.
The ASV will always
correct for order-of-finish such that a team that finishes higher at the SCT itself will always be invited first. Additionally, the OAOAP is a better strength-of-schedule factor because it takes into account that every team at the sectional plays a different schedule, and because it never takes into account games that are played between two teams, it is impervious to within-game gaming (it is still theoretically susceptible to forfeit manipulation, though note that if the lower-TPTH team forfeits, it artificially increases its opponent's OAOAP and decreases its own; it remains to be seen whether this artificial increase/decrease is enough to offset the potential TPTH gain/loss, but I am skeptical that it will). The ASV is given as an easy-to-read 3 or 4 digit number and calculations can be done using two Excel Sheets (one for OAOAP and one for everything else). This method will also work if raw D2/D1 numbers are converted to numbers in the other division, though a way to do that has not yet been established.
EDIT: People wishing to see the relevant calculations should see the OAOAP and ASV sheets here
EDIT2: I'd just like to point out that independent of any other advantages of this system, the raw-to-adjusted S value conversion completely solves part 1 of NAQT's "three most difficult parts of assembling this system." "Team A" will always be invited ahead of "Team B" and the invitation order for "Team C" can be found directly from the ASV rankings.