Passer rating

Passer rating (known as passing efficiency or pass efficiency in NCAA football) is a measure of the performance of quarterbacks or any other passers in American football and Canadian football. There are at least two formulae currently in use: one officially used by the National Football League and the Canadian Football League, and one used in college football. Passer rating is calculated using each quarterback's completion percentage, passing yardage, touchdowns and interceptions.

A perfect passer rating in the NFL is 158.3. A perfect passing efficiency in college football is 1261.6.

History
The NFL's complex passer rating was devised in 1971 by a special committee headed by Don R. Smith, an executive at the Pro Football Hall of Fame. Don Weiss created the passer rating. Prior to that, the NFL had struggled with how to crown a passing king using multiple statistics.

NFL
The calculation of the NFL passer rating involves more steps than the NCAA formula. In order to establish a maximum value for an NFL player's passer rating, a separate calculation needs to be completed involving each of the following four categories: Completion Percentage, Average Yards Per Attempt, Percentage of Touchdown Passes, and Percentage of Interceptions. If the result in any category is less than 0, the given result should be 0. If the result in any category is greater than 2.375, the given result should be 2.375. This makes the maximum possible quarterback rating for the NFL 158.3. A perfect rating requires at least a 77.5% completion rate, at least 12.5 yards per attempt, a touchdown on at least 11.875% of attempts, and no interceptions.

The four separate calculations can be expressed in the following equations:

$$a = \left ({\text{COMP} \over \text{ATT}} - .3 \right ) \times 5$$

$$b = \left ({\text{YARDS} \over \text{ATT}} - 3 \right ) \times .25$$

$$c = \left ({\text{TD} \over \text{ATT}} \right ) \times 20$$

$$d = 2.375 - \left ({\text{INT} \over \text{ATT}} \times 25 \right )$$

where
 * ATT = Number of passing attempts
 * COMP = Number of completions
 * YARDS = Passing yards
 * TD = Touchdown passes
 * INT = Interceptions

Then, the above calculations are used to complete the passer rating:

$$Passer Rating_{NFL} = \left ({mm(a) + mm(b) + mm(c) + mm(d) \over 6} \right ) \times 100$$ where $$mm(x) = \text{max}(0, \text{min}(x, 2.375))$$

NCAA
Passer rating, known formally in college football as passing efficiency or pass efficiency, is based on player performances. The NCAA passing efficiency formula is far simpler than the NFL formula, as it lacks limits on the four components:

$$Passer Rating_{NCAA} = {(8.4 \times YDS) + (330 \times TD) + (100 \times COMP) - (200 \times INT) \over ATT}$$

The NCAA passer rating has an upper limit of 1,261.6 (every attempt is a 99-yard completion for touchdown), and a lower limit of -731.6 (every attempt is completed, but results in a 99-yard loss). A passer who throws only interceptions will have a -200 rating, as would a passer who only throws completed passes losing an average of 35.714 yards.

Limitations
As mentioned, both the NFL and NCAA passer rating formulae are only based on completion percentage, passing yardage, touchdowns, and interceptions. As the NFL notes, "It is important to remember that the system is used to rate passers, not quarterbacks. Statistics do not reflect leadership, play-calling, and other intangible factors that go into making a successful professional quarterback." For example, it does not factor in rushing yards gained by such quarterback as Randall Cunningham, Steve Young, Michael Vick, and John Elway, who were also known for their running ability. Nor does it measure a quarterback's win-loss record, how many times he has been sacked, or evaluate his leadership and performance during different situations.

In 2011, the American sports network ESPN developed an alternative statistic called the total quarterback rating which purports to more accurately measure quarterbacks' contribution to their teams' winning.

NFL
Aaron Rodgers holds the NFL record for the highest career passer rating for any player with at least 1,500 attempts, with a mark of 100.5. The highest passer rating for a complete season is 121.1 set by Peyton Manning in 2004 (Manning also has completed four games with a perfect passer rating, another NFL record). Also in 2004, Ben Roethlisberger posted a mark of 98.1, setting a record for rookies. Wide receiver Antwaan Randle El, with a passer rating of 157.5 from 21 completed passes of a possible 26, has the highest career rating of any player with more than twenty attempts. As of 2010, 60 NFL quarterbacks have completed a game with a perfect passer rating of 158.3, and seven have done so multiple times.

NCAA
In Division I FBS (formerly Division I-A), the career record for passing efficiency is held by Sam Bradford of Oklahoma, who had a career mark of 175.6 between 2007 and 2009. The single-season record belongs to Colt Brennan of Hawaii, who amassed a passer rating of 186.0 over the 2006 season, while the freshman record belongs to Michael Vick of Virginia Tech, whose rating during the 1999 season was 180.4. Current Division I FBS passing efficiency ratings can be found here.