Math lesson. In the context of football rankings.
The math ranking uses a method of minimization of the sum of squares of errors. This example will try to make sense of that.
There are currently 131 FBS (football bowl series?) teams. Every team plays 11 or 12 other teams in the regular season; due to championships and playoffs it’s possible to play up to 15 games per year.
If you look at my final rankings, one of the most glaring weirdnesses is that Texas, who lost 5 games, is ranked (6th or 7th? I don’t have it in front of me and don’t remember offhand.) But clearly, by almost everyone’s estimation, it’s too high (everyone’s but mine and math’s, I mean.) Here is how that happens. Here is Texas’ actual schedule and results for the season that just ended;
Texas vs LA-Monroe 52-10, actual margin 42
Texas vs Alabama 19-20, actual margin -1
Texas vs UTSA 41-20, actual margin 21
Texas vs Texas Tech 34-37, actual margin -3
Texas vs West Virginia 38-20, actual margin 18
Texas vs Oklahoma 49-0, actual margin 49
Texas vs Iowa State 24-21, actual margin 3
Texas vs Oklahoma State 34-41, actual margin -7
Texas vs Kansas State 34-27, actual margin 7
Texas vs TCU 10-17, actual margin -7
Texas vs Kansas 55-14, actual margin 41
Texas vs Baylor 38-27, actual margin 11
Texas vs Washington 20-27, actual margin -7
Texas is always listed first here, unlike convention that gives the visiting team’s score first. A positive actual margin is a win for Texas, negative is a loss. Now, here’s another copy of that list, but this time assigning each team’s “power rating” rather than the actual game score. Texas has a power rating of 57, so that will show up for them every time; the margin then becomes a “projected margin”; that is, if the system were foolproof, these would be the margins you could expect to play out in real life. You might guess that it is NOT a foolproof system, which is impossible due to something called “circular triads.” So the system tries to assign ranks that will smooth the most extreme mistakes for all 131 teams simultaneously.
Texas vs LA-Monroe 57-15, projected margin 42
Texas vs Alabama 57-64, projected margin -7
Texas vs UTSA 57-36, projected margin 21
Texas vs Texas Tech 57-45, projected margin 12
Texas vs West Virginia 57-36, projected margin 21
Texas vs Oklahoma 57-44, projected margin 13
Texas vs Iowa State 57-41, projected margin 16
Texas vs Oklahoma State 57-39, projected margin 18
Texas vs Kansas State 57-55, projected margin 2
Texas vs TCU 57-56, projected margin 1
Texas vs Kansas 57-40, projected margin 17
Texas vs Baylor 57-42, projected margin 15
Texas vs Washington 57-46, projected margin 11
So, if the power ratings were flawless, you would expect Texas to be 12-1 and they were a far cry from that. Here’s what the system is measuring to try to give the best results it can. Consider the Alabama game. Texas was expected to lose by 7 points, but they only lost by 1. This is a difference of 6 points between the real score and the projected score. Now, it doesn’t matter whether you have it as “6” or “-6”, because we immediately square these “errors,” so either way, the contribution to the sum of squares due to this individual game is 36. Here’s a list of the differences and their squares for the entire schedule.
LA-Monroe diff 0, square of diff 0
Alabama diff 6, square of diff 36
UTSA diff 0, square of diff 0
Texas Tech diff -15, square of diff 225
West Virginia diff -3, square of diff 9
Oklahoma diff 36, square of diff 1296
Iowa State diff -13, square of diff 169
Oklahoma State diff -25, square of diff 625
Kansas State diff 5, square of diff 25
TCU diff -8, square of diff 64
Kansas diff 24, square of diff 576
Baylor diff -4, square of diff 16
Washington diff -18, square of diff 324
Finally, Texas’ contribution to the total sum of squares is the sum of the last numbers from each line, SS = 3365. A rationale for the sum of squares method is that it sees the most extreme “mistakes” (for example, Texas was projected to beat Oklahoma by 13, and beat them by 49. So even though Texas’ entire SS is 3365, almost 1300 of that total comes from the single game, where the system was off by 36 points.
Now, clearly, Texas is already ranked too highly, and so we wouldn’t make the mistake of increasing their power rating. But if we DID increase it to 58
Texas vs LA-Monroe 52-10, actual margin 42
Texas vs Alabama 19-20, actual margin -1
Texas vs UTSA 41-20, actual margin 21
Texas vs Texas Tech 34-37, actual margin -3
Texas vs West Virginia 38-20, actual margin 18
Texas vs Oklahoma 49-0, actual margin 49
Texas vs Iowa State 24-21, actual margin 3
Texas vs Oklahoma State 34-41, actual margin -7
Texas vs Kansas State 34-27, actual margin 7
Texas vs TCU 10-17, actual margin -7
Texas vs Kansas 55-14, actual margin 41
Texas vs Baylor 38-27, actual margin 11
Texas vs Washington 20-27, actual margin -7
Texas vs LA-Monroe 58-15, projected margin 43
Texas vs Alabama 58-64, projected margin -6
Texas vs UTSA 58-36, projected margin 22
Texas vs Texas Tech 58-45, projected margin 13
Texas vs West Virginia 58-36, projected margin 22
Texas vs Oklahoma 58-44, projected margin 14
Texas vs Iowa State 58-41, projected margin 17
Texas vs Oklahoma State 58-39, projected margin 19
Texas vs Kansas State 58-55, projected margin 3
Texas vs TCU 58-56, projected margin 2
Texas vs Kansas 58-40, projected margin 18
Texas vs Baylor 58-42, projected margin 16
Texas vs Washington 58-46, projected margin 12
LA-Monroe diff -1, square of diff 1
Alabama diff 5, square of diff 25
UTSA diff -1, square of diff 1
Texas Tech diff -16, square of diff 256
West Virginia diff -4, square of diff 16
Oklahoma diff 35, square of diff 1225
Iowa State diff -14, square of diff 196
Oklahoma State diff -26, square of diff 676
Kansas State diff 4, square of diff 16
TCU diff -9, square of diff 81
Kansas diff 23, square of diff 529
Baylor diff -5, square of diff 25
Washington diff -19, square of diff 361
sum of squared diffs is 3408
If we increase Texas’ power rating by one point, the sum of squares goes up from 3365 to 3408. But the thing is, the same thing happens if we try to drop Texas’ rating to 56.
Texas vs LA-Monroe 52-10, actual margin 42
Texas vs Alabama 19-20, actual margin -1
Texas vs UTSA 41-20, actual margin 21
Texas vs Texas Tech 34-37, actual margin -3
Texas vs West Virginia 38-20, actual margin 18
Texas vs Oklahoma 49-0, actual margin 49
Texas vs Iowa State 24-21, actual margin 3
Texas vs Oklahoma State 34-41, actual margin -7
Texas vs Kansas State 34-27, actual margin 7
Texas vs TCU 10-17, actual margin -7
Texas vs Kansas 55-14, actual margin 41
Texas vs Baylor 38-27, actual margin 11
Texas vs Washington 20-27, actual margin -7
Texas vs LA-Monroe 56-15, projected margin 41
Texas vs Alabama 56-64, projected margin -8
Texas vs UTSA 56-36, projected margin 20
Texas vs Texas Tech 56-45, projected margin 11
Texas vs West Virginia 56-36, projected margin 20
Texas vs Oklahoma 56-44, projected margin 12
Texas vs Iowa State 56-41, projected margin 15
Texas vs Oklahoma State 56-39, projected margin 17
Texas vs Kansas State 56-55, projected margin 1
Texas vs TCU 56-56, projected margin 0
Texas vs Kansas 56-40, projected margin 16
Texas vs Baylor 56-42, projected margin 14
Texas vs Washington 56-46, projected margin 10
LA-Monroe diff 1, square of diff 1
Alabama diff 7, square of diff 49
UTSA diff 1, square of diff 1
Texas Tech diff -14, square of diff 196
West Virginia diff -2, square of diff 4
Oklahoma diff 37, square of diff 1369
Iowa State diff -12, square of diff 144
Oklahoma State diff -24, square of diff 576
Kansas State diff 6, square of diff 36
TCU diff -7, square of diff 49
Kansas diff 25, square of diff 625
Baylor diff -3, square of diff 9
Washington diff -17, square of diff 289
sum of squared diffs is 3348
Now, it looks like it drops a bit, but it actually rises a tad; this is the TV version; I have rounded all of the ratings to nice whole numbers, which doesn’t actually happen if you include the decimal forms of the ratings (Texas’s rating is not really 57; it’s 57.16058.)
So the system assigns a rating to each of the 131 teams in such a way that it minimizes the total sum square error in the system simultaneously. It’s a cute trick, and while not foolproof, it’s one way to make it as mathematically sound as possible.
As for circular triads, this season Vandy beat Kentucky, Kentucky beat Missouri, and Missouri beat Vandy, so there’s no linear ranking of those teams that could get all three outcomes correct; there are dozens and dozens of these triads in this year’s results.