Quarterback Rankings

Introduction

Passer Rating is a commonly used tool to judge QB performance, but it is notable for having some serious limitations. True to its name, it only considers passing stats, which unfairly penalizes quarterbacks who produce for their team by rushing (or receiving for that matter, but that's substantially less common - unless you'd like to think of Mohammed Sanu as a QB for the purposes of this experiment), and fails to account for fumbles. Attempts have been made in the past to account for this, such as ESPN's Total QBR, but given the proprietary nature of that statistic, the legitimacy of it has come into question. Here I have come up with a formula of my own that I hope gives a better picture of what a QB brings the table. Before you say it, yes I am a Panthers fan, yes I think Cam Newton is unfairly represented by his passer rating, no I did not design this specifically to give Cam a higher score.

Methodology

I used stats taken from pro-football-reference and included a number of quarterbacks, most of whom are still active. I included (almost - I forgot AJ McCarron) every projected starter for next season (which was kind of a b*tch because Patrick Mahomes has so few snaps to work with), as well as some recently retired names like Peyton Manning, Tony Romo, and Jay Cutler, as well as Colin Kaepernick, as his situation is the source of a lot of debate.

Passer rating is a stat comprised of components, named a through d.

a= (comp/att-0.3)*5
b= (yds/att-3)*0.25
c= (TD/att)*20
d= 2.375-(INT/att)*25

With the final passer rating defined by: Passer Rating = (a+b+c+d)*100/6

According to wikipedia each component has a maximum of 2.375 and a minimum of 0 and the total rating ranges from 0 to 158.3. Initially the statistic was designed to reflect an average quarterback's score at 66.7, with 100 being considered very good, but over time, with the emergence of the NFL's new passing era, we have seen inflation of passer rating, with the average now falling in the high 80s.

To design a rating system that reflects rushing statistics, I chose to use comparable statistics as those used for passing, so I included rushing attempts, rushing yards, rushing touchdowns, and fumbles. Immediately I was faced with one major issue. In the "b" component, yards per attempt of each QB is reduced by 3, which is a seemingly arbitrary number. Because rushing yards per attempt is held to a different standard than passing yards per attempt, they need to be adjusted for in some way. To address this, I took an average of the Y/C and compared it to the average Y/A of all the quarterbacks in the sample. The difference was approximately 3.56 yards/attempt. Given that the passing Y/A are already docked 3.0 y/a, to equalize this, I added 0.56 to the Y/C number to equalize the value of a yard for rushing and passing. Keep in mind that this addition favors QBs who rush less frequently over those who rush more frequently, as the additional 0.56 yards/carry loses weight with every additional rushing attempt (see formula for b*). This is how I created a new component for the formula: e.

e= (rush yds/attempt +0.56)*0.25

As e is analogous to b in the original passer rating formula, I chose to average the two weighting by number of attempts for each type of play, to create b*.

b*=(b*passing att + e*rushing att)/(passing att + rushing att)

This b* would then go on to replace b in the passer rating formula.

I did the same thing with touchdowns, but this is easier because a touchdown is a touchdown is a touchdown and no adjustment was needed. I created an f variable that is the rushing analogue to the c component.

f= (Rushing TD/att)*20

Using an attempt weighted average gave me c*, to replace c in the passer rating formula:

c*=(c*passing att + f*rushing att)/(passing att + rushing att)

Because pro-football-reference doesn't sort out fumbles by rushing or passing (or at least, I did not bother to check) and fumbles can occur on both QB rushes and passing downs, I created a d* variable to account for total turnovers combining fumbles and interceptions.

d*=(Int+fumbles)/(passing att + rushing att)

This gives us a new set of 4 variables to use in a Overall QB Rating formula:

OVR QB Rating = (a+b*+c*+d*)*100*M/6

You're probably wondering what "M" is supposed to be. When I calculated the formula, I noticed that every single QB's rating dropped and none improved or stayed the same. This is because of the inclusion of fumbles, the lack of which have been artificially inflating passer rating because for some reason, despite being equivalent (or worse) than an interception, they aren't included in the formula. To adjust for that, I took the average of the QB ratings and the average of the passer ratings and added a multiplier to normalize the two to the same mean. This came out to be a factor of about 1.0584=M.

Having reworked the formula to factor in rushing stats, let's look at the results...