Nevertheless, in spite of passer rating's success as an indicator, it has no inherent "meaning" or theoretical value. Below, I examine the current passer rating formula, methods for simplifying the formula, and finally, a more explanatory alternative.
For future reference, I will use the top ten quarterbacks in passing yards from 2017. According to ESPN:
I. The Current Formula
The current formula for passer rating, Q0, is as follows:
Let:
w = the median of {0.3, 0.775, comp/att}
x = the median of {3, 12.5, yds/att}
y = the median of {0, 0.11875, td/att}
z = the median of {0, 0.095, int/att}
Q0 = (250/3)w + (25/6)x + (1000/3)y - (1250/3)z + (25/12)
The "median" rules serve to place upper and lower bounds on each category. The upper bounds, lower bounds, coefficients, and constant are all arbitrary. Kinda complicated on the surface, but any spreadsheet should be able to handle it just fine. Here are the 2017 rankings:
Alex Smith was 2017's best passer according to Q0, largely due to his high completion percentage (68%) and yards per attempt (8).
II. Simplification
One strategy for improving passer rating is simplifying it. If passer rating has no theoretical backing, one might say, it should, at least, be easy to calculate. Q1 is essentially the same formula, but with the bounds removed (they are generally unattainable over the course of a full season anyway).
Q1 = (250/3)comp/att + (25/6)td/att + (1000/3)yds/att - (1250/3)int/att + (25/12)
So, Q1 looks exactly like Q0 for the 2017 season, but it is important to note that, for extreme individual games, the values may differ.
But, enough is enough. For simplicity, it would be ideal to not have to plug in five separate statistics (comp, att, yds, td, int). Luckily enough, for the 2017 season, yds/att and int/att strongly correlated with passer rating. Here are charts based upon all of the 2017 starters:
The reward for passing yards and the penalty for interceptions must be increased to compensate for the removal of completions and touchdowns. The regression of 2017 starters yielded the following formula, Q2, which is normalized to approximately the same scale:
The reward for passing yards and the penalty for interceptions must be increased to compensate for the removal of completions and touchdowns. The regression of 2017 starters yielded the following formula, Q2, which is normalized to approximately the same scale:
Q2 = 13.5*yds/att-340*int/att
Easy enough. Once again, no theoretical value, but only three statistics are needed to compute Q2. Here is how Q2 correlates with Q0 for 2017's starting QBs:
The two values correlate extremely well. The r-squared value indicates that over 80% of the varience in passer rating is predictable by exclusively yds/att and int/att. Here is the updated chart for 2017's best quarterbacks:
The two values correlate extremely well. The r-squared value indicates that over 80% of the varience in passer rating is predictable by exclusively yds/att and int/att. Here is the updated chart for 2017's best quarterbacks:
III. A Theoretically-Backed System
First, one must decide how to convert between yards and points. Since a touchdown is worth 6 points, being at the opponent's 0 yard line is worth 6 points. And, since a safety is worth 2 points for the defending team, being at one's own 0 yard line is worth -2 points. With these numbers in mind, one can calculate that traveling the entire 100 yards is worth 8 points. Or, 1 yds = 0.08 pts. Since a touchdown is worth 6 points, 1 td = 75 yds.
Second, one must convert turnovers to yards. A turnover, without any advancement by the opposing team, is worth -4 points. To illustrate this:
If John Doe is at the opposing team's goal-line (let's say at the opponent's 0 yard line) and throws an interception, he surrenders the 6 points of field position he previously had. However, if the opposing team does not return the ball at all, they get the ball at their own 0 yard line, worth -2 points of field position to the opposition (or 2 points for Doe's team). Thus, the overall play is worth 2 - 6 points, or -4 points.
Since -4 pts = 1 turnover and 1 yds = 0.08 pts, 1 yds = -0.02 turnovers.
Thus, 1 int = 1 fumb = -50 yds.
Third, completions are, theoretically, unimportant. It may be an important predictive measure (an "accurate" quarterback may be more likely to throw more completions for more yards and touchdowns), but it is unimportant in evaluating quarterback performance. This is because completions themselves are essentially included in yard totals. An incompletion is a 0 yard gain, whereas a completion already has its yard value attached.
Fourth, quarterbacks with high rushing production are typically hurt by passer rating (Remember Cam Newton's MVP Season?), and quarterbacks who lose many yards on sacks typically see their passer ratings inflated (Kirk Cousins in 2017). To compensate, rushing yards and rushing touchdowns should be added to a quarterback's total, and sack yards should be deducted from a quarterback's total.
Putting it all together, Q3 is equal to modified yards per passing attempt. This gives it "meaning," in addition to its theoretical value. The formula for Q3 is below:
Q3 = (passYds + rushYds - sackYds + 75*passTd + 75*rushTd - 50*int - 50*fumb) / passAtt
Here are the values for 2017's top passers (Q4 is just 10*Q3 because it scales to approximately the old Q0 scale):
IV. Final Recap of Alternate Formulae
Q1 = (83.333*comp + 4.167*td + 333.333*yds - 416.667*int)/att + 2.083
Q2 = 13.5*yds/att-340*int/att
Q3 = (netYds + 75*allTd - 50*turnover) / passAttQ2 = 13.5*yds/att-340*int/att
Q4 = (10*netYds + 750*allTd - 500*turnover) / passAtt