Updated March 7, 2018
As posted on a wiki talk page, the popularly used quarterback rating formula is so far from adequate measure that I put upon myself to make one that is more sensible. My aim is to make reasonable models of the major categories involved in assessing a quarterback's PASSING performance. Certainly, until which time the NFL decides to parse certain statistics like no-fault passer incompletions and not counting spiked balls used to stop the clock, my proposed model below is working off of traditional statistics. Overall quarterback performance would be far more involved and would have much obfuscating aspects involved such as the play caller, coach calls, errors by the offense (such as delay of game not caused by the quarterback, off-side penalties as well as other play action penalties), running, dodging sacks, etc.
My objections five years ago of the NFL adopted passer rating index include:
1) the bizarre numerical rating of 158.3 to be the highest figure possible,
2) the 10 pass minimum to qualify for a perfect rating, as if snowy day and most all plays are on the ground, and with a weak defense, the offense is able to only have say, 30 total plays, it would not be the quarterback’s failure to not pass extensively.
3) averaging 12.5 yards per attempt as “perfect” which means throwing for more yardage would not be any better
4) the 11.875% touchdown passes percentage which could be highly dependent on what the quarterback is told to do.
I opted to make 100 the number considered to be perfect. In my first formula, in using exponential functions, the 100 number would be impossible to be obtained but could get to practically that number. In this newer model, I am allowing the 100 number to be achieved.
I originally gave greater points to quarterbacks doing extensive passing but opted to avoid this. Besides the scenario of a snowy day as mentioned prior, suppose a first-string quarterback gets injured and the replacing quarterback makes only 9 passes and all were completions of 20+ yards - would that make the second quarterback not be considered perfect? And so, any number of passes would qualify for perfection and this would not hurt anything as besides trivial short-lasting presence on the field, the yearly or career statistics could have basic requirements to be included.
The NFL's adopted formula using 12.5 yards per attempt qualifying for perfect, I find this to be about right and thus am incorporating this figure in the newer formula.
I am fine with the 77.5% completion percentage but I am adopting herein a 75% completion percentage to account for spiked balls and receiver dropped balls. I’d up the number to 90% if basic statistics honed in more to the quarterback’s passer performance that would not be unduly affected by confounding factors.
The newer formula I am proposing deals with three items only:
1) Percentage completions per attempt
2) Yards per attempt
3) Interceptions
Proposal for Quarterback Rating:
Passer Rating = 100*F1*F2*F3
Where:
F1=(27/32)*(C/A)^3/(729/4096+(C/A)^6) for C/A<=0.75, F1=1 for C/A>0.75
F2=5^(3/2)*2^(¼)*(Y/A)^(¾)/(125/4*2^(½) + (Y/A)^(3/2)) for Y/A<=12.5, F2=1 for Y/A>12.5
F3=1-(200*I/Y)^3/(7+(200*I/Y)^3)
where A=attempts, C=completed passes I= interceptions, Y=total yards passing.
The numbers in the formula may appear to be complicated but it’s really not so. The numbers have been chosen to enable curves of the individual passer rating components to have a zero slope at certain critical points or other defining properties to maintain flow without any cusps.
Here are the break-down graphs of the components used in the formula (use of wolfram alpha):
Pass completion percentage (function of complete passes per attempt):
Production (function of yards per attempt):
Interceptions (function of interceptions relative to counterpart of passing yards - shown here using 200 passing yards as a basis):
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
COMPARE THIS NEW FORMULA WITH THE CURRENTLY USED FORMULA. COMPARE THE STATS AND YOU DETERMINE THE BETTER FORMULA! LOOK CLOSELY AT THE SEVERAL PLAYERS WITH MUCH DISPARATE RANKINGS AND INSPECT THE STATS TO SEE WHICH FORMULA YOU JUDGE THE PLAYERS TO BE MORE ACCURATELY PLACED.
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| Preferred Rating Formula | Currently Used Rating Formula | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Rank | Rating | Rating | Rank | PLAYER | TEAM | COMP | ATT | PCT | YDS | YDS/A | TD | INT | SACK | G | YDS/G |
| 1 | 89.99 | 105.8 | 2 | Peyton Manning, QB | DEN | 400 | 583 | 68.6 | 4,659 | 7.99 | 37 | 11 | 21 | 16 | 291 |
| 2 | 88.48 | 108 | 1 | Aaron Rodgers, QB | GB | 371 | 552 | 67.2 | 4,295 | 7.78 | 39 | 8 | 51 | 16 | 268 |
| 3 | 87.8 | 99.1 | 5 | Matt Ryan, QB | ATL | 422 | 615 | 68.6 | 4,719 | 7.67 | 32 | 14 | 28 | 16 | 295 |
| 4 | 87.55 | 102.4 | 3 | Robert Griffin III, QB | WSH | 258 | 393 | 65.6 | 3,200 | 8.14 | 20 | 5 | 30 | 15 | 213 |
| 5 | 81.82 | 100 | 4 | Russell Wilson, QB | SEA | 252 | 393 | 64.1 | 3,118 | 7.93 | 26 | 10 | 33 | 16 | 195 |
| 6 | 81.37 | 98.7 | 6 | Tom Brady, QB | NE | 401 | 637 | 63 | 4,827 | 7.58 | 34 | 8 | 27 | 16 | 302 |
| 7 | 81.13 | 90.7 | 9 | Matt Schaub, QB | HOU | 350 | 544 | 64.3 | 4,008 | 7.37 | 22 | 12 | 27 | 16 | 251 |
| 8 | 80.86 | 90.5 | 10 | Tony Romo, QB | DAL | 425 | 648 | 65.6 | 4,903 | 7.57 | 28 | 19 | 36 | 16 | 306 |
| 9 | 80.07 | 97 | 7 | Ben Roethlisberger, QB | PIT | 284 | 449 | 63.3 | 3,265 | 7.27 | 26 | 8 | 30 | 13 | 251 |
| 10 | 77.88 | 96.3 | 8 | Drew Brees, QB | NO | 422 | 670 | 63 | 5,177 | 7.73 | 43 | 19 | 26 | 16 | 324 |
| 11 | 75.26 | 88.6 | 11 | Philip Rivers, QB | SD | 338 | 527 | 64.1 | 3,606 | 6.84 | 26 | 15 | 49 | 16 | 225 |
| 12 | 73.1 | 85.3 | 16 | Carson Palmer, QB | OAK | 345 | 565 | 61.1 | 4,018 | 7.11 | 22 | 14 | 26 | 15 | 268 |
| 13 | 72.46 | 87.7 | 12 | Joe Flacco, QB | BAL | 317 | 531 | 59.7 | 3,817 | 7.19 | 22 | 10 | 35 | 16 | 239 |
| 14 | 71.77 | 87.4 | 13 | Andy Dalton, QB | CIN | 329 | 528 | 62.3 | 3,669 | 6.95 | 27 | 16 | 46 | 16 | 229 |
| 15 | 71.4 | 79.1 | 23 | Nick Foles, QB | PHI | 161 | 265 | 60.8 | 1,699 | 6.41 | 6 | 5 | 20 | 7 | 243 |
| 16 | 70.48 | 87.2 | 14 | Eli Manning, QB | NYG | 321 | 536 | 59.9 | 3,948 | 7.37 | 26 | 15 | 19 | 16 | 247 |
| 17 | 69.9 | 79.8 | 22 | Matthew Stafford, QB | DET | 435 | 727 | 59.8 | 4,967 | 6.83 | 20 | 17 | 29 | 16 | 310 |
| 18 | 69.28 | 81.2 | 21 | Christian Ponder, QB | MIN | 300 | 483 | 62.1 | 2,935 | 6.08 | 18 | 12 | 32 | 16 | 183 |
| 19 | 69.07 | 86.2 | 15 | Cam Newton, QB | CAR | 280 | 485 | 57.7 | 3,869 | 7.98 | 19 | 12 | 36 | 16 | 242 |
| 20 | 68.65 | 82.6 | 18 | Sam Bradford, QB | STL | 328 | 551 | 59.5 | 3,702 | 6.72 | 21 | 13 | 35 | 16 | 231 |
| 21 | 66.45 | 83.3 | 17 | Ryan Fitzpatrick, QB | BUF | 306 | 505 | 60.6 | 3,400 | 6.73 | 24 | 16 | 30 | 16 | 213 |
| 22 | 64.96 | 76.1 | 27 | Ryan Tannehill, QB | MIA | 282 | 484 | 58.3 | 3,294 | 6.81 | 12 | 13 | 35 | 16 | 206 |
| 23 | 64.03 | 81.3 | 20 | Jay Cutler, QB | CHI | 255 | 434 | 58.8 | 3,033 | 6.99 | 19 | 14 | 38 | 15 | 202 |
| 24 | 63.43 | 78.1 | 24 | Michael Vick, QB | PHI | 204 | 351 | 58.1 | 2,362 | 6.73 | 12 | 10 | 28 | 10 | 236 |
| 25 | 63.07 | 77.4 | 25 | Blaine Gabbert, QB | JAC | 162 | 278 | 58.3 | 1,662 | 5.98 | 9 | 6 | 22 | 10 | 166 |
| 26 | 58.32 | 72.6 | 29 | Brandon Weeden, QB | CLE | 297 | 517 | 57.4 | 3,385 | 6.55 | 14 | 17 | 28 | 15 | 226 |
| 27 | 57.78 | 81.6 | 19 | Josh Freeman, QB | TB | 306 | 558 | 54.8 | 4,065 | 7.29 | 27 | 17 | 26 | 16 | 254 |
| 28 | 57.02 | 74 | 28 | Jake Locker, QB | TEN | 177 | 314 | 56.4 | 2,176 | 6.93 | 10 | 11 | 25 | 11 | 198 |
| 29 | 55.47 | 76.5 | 26 | Andrew Luck, QB | IND | 339 | 627 | 54.1 | 4,374 | 6.98 | 23 | 18 | 41 | 16 | 273 |
| 30 | 50.78 | 66.7 | 32 | Matt Cassel, QB | KC | 161 | 277 | 58.1 | 1,796 | 6.48 | 6 | 12 | 19 | 9 | 200 |
| 31 | 50.41 | 72.2 | 30 | Chad Henne, QB | JAC | 166 | 308 | 53.9 | 2,084 | 6.77 | 11 | 11 | 28 | 10 | 208 |
| 32 | 45.91 | 66.9 | 31 | Mark Sanchez, QB | NYJ | 246 | 453 | 54.3 | 2,883 | 6.36 | 13 | 18 | 34 | 15 | 192 |
