Passer rating is a dumb stat. Let’s get that out of the way. As I’ve written before, passer rating is stupid because it gives a 20-yard bonus for each completion, a 100-yard penalty for each interception, and an 80-yard bonus for each touchdown. In reality, there should be no (or a very small) weight on completions (or, better yet, a bonus for completions that go for a first down), a 45-yard weight on interceptions, and a 20-yard weight on touchdowns. But given how ubiquitous passer rating is in analysis of passing, let’s at least try to understand it more.

Let’s begin with the formula one needs to calculate passer rating in Excel:

=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

To make this formula work, you need to put the following categories in these cells:

C2 = Attempts

D2 = Completions

E2= Passing Yards

F2 = Passing Touchdowns

G2 = Interceptions

That formula probably seems like gibberish to you, so let’s unpack it a little bit.

=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

This part is simple enough: if a quarterback doesn’t have at least 224 pass attempts (during a 16-game season), they fail to qualify for the passer rating crown. You can lower this number for non-16-game seasons as necessary.

Passer Rating – Four Components

Passer rating comprises four components: completion percentage, yards per attempt, touchdowns per attempt, and interceptions per attempt. Let’s see how the above formula addresses these concerns:

Completion Percentage

=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

Take a look at the bolded blue text — What are we doing? Taking completions and dividing them by attempts is how we come up with completion percentage, of course. You take that result and subtract 0.3, or 30%. Savvy readers will pick up on the fact that if your completion percentage is 29% or 0%, you get the same credit in passer rating: there is a floor of 30%.

But there’s also a ceiling, which isn’t obvious at first glance. Okay, so we take completion percentage, and remove 30%. Let’s say our hypothetical passer completed 65% of his passes. The formula above would result in 0.35 being multiplied by 5; that result is 1.75. But if you look at the beginning of the bolded blue part of the formula, you see that we take the median of three numbers: the result (here, 1.75), 0, and 2.375. Why 2.375? We’ll get to that later.

But this means we have lower and upper bounds of 0 and 2.375 for our result after we multiply by 5. So that means before we multiply by 5, our lower and upper bounds are 0 and 0.475. And that, of course, is after we subtract 30% from our hypothetical passer’s completion percentage.

This means, in passer rating, a quarterback won’t get penalized for a completion percentage below 30% or credited for a completion percentage above 77.5%.

We are now done with step 1 of the passer rating formula, which is completion percentage: it has a floor of 30% and a ceiling of 77.5%. To get the completion percentage variable, you need to take a quarterback’s completion percentage, remove 30%, and multiply the result by 5 (why 5? We’ll get to that later). Our hypothetical passer with a 65% completion percentage has a completion percentage variable of 1.75.

Yards per Attempt

=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

Here, the bolded E2 value represents Passing Yards; we simply divide passing yards by attempts — to create yards per attempt — and then subtract 3.0. That means, of course, that there is a built in floor of 3.0 yards per attempt. We then multiply that result by 0.25 (or divide by 4), so if a quarterback averages 7.0 yards per attempt, they have a Yards per Attempt variable of 1.0.

Once again, though, we have a ceiling. The yards per attempt variable is bounded by 0 and 2.375, as are all of the variables in passer rating. Work backwards, and you will see that 12.5 is the upper limit: 12.5 minus 3 is 9.5, and 9.5 divided by 4 is 2.375. Therefore, a quarterback can’t get credit for any yards per attempt produced above 12.5, or penalized for anything below 3.0.

We are now done with step 2 of the passer rating formula, which is yards per attempt. The calculation is relatively simple: take yards per attempt, subtract 3, and multiply the result by 0.25 (why 0.25? We’ll get to that later). Our hypothetical passer with a 7.0 yards per attempt average has a yards per attempt variable of 1.0.

Passing Touchdowns per Attempt



=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

Next up: touchdown rate. By now, the formula shouldn’t look so scary. F2 is the cell for touchdown passes; C2 the one for pass attempts. After some simple division, we multiply the result by 20 (why 20? Later!). So if a quarterback throws 30 touchdowns on 500 passes, he has a 6% touchdown rate, which turns into 1.20 once we multiply it by 20. That’s the touchdown rate variable.

What are the lower and upper bounds? The lower bound is still zero; the upper bound? Well, the touchdown rate variable is capped at 2.375, which means an 11.875% touchdown rate is the best rate this formula will reward. That’s a touchdown every 8.42 pass attempts, which is pretty rare. The best example, in terms of recency and number of attempts, would be Matt Flynn in 2011, who had 6 touchdowns on 49 attempts, mostly in one game.

So that’s the touchdown rate variable, perhaps the simplest one of them all. Take a quarterback’s touchdown passes, divide by his number of attempts, and multiply by 20. That’s the touchdown rate variable, although there is an upper cap at an 11.875% touchdown rate.

Interception Rate

=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

Here is our final one, and the bolded blue part of the formula looks a little different. But don’t be alarmed: that’s only because interceptions are a bad thing. It’s still pretty straightforward: take interceptions (here, in cell G2) and divide by pass attempts. So if a quarterback throws 10 interceptions on 500 passes, that’s a 2% rate. Multiply that result by 25, to get 0.50.

But since we want a low number here — 0.50 is better than 1.50 — we need to flip things around. Think of the TD rate: without saying it, we start with the lower limit of 0, and then add the TD rate number after multiplying by 20. Now, we start with the upper limit of 2.375, and subtract the INT rate number after multiplying by 25. So the 0.50 turns into 1.875. That is the interception variable, which like all others, is bounded by 0 and 2.375.

Just like a quarterback can’t do any worse than 0 touchdowns, a quarterback can’t do any better than 0 interceptions: that just operates as a cap here instead of a floor. The floor for interception rate? That’s at a whopping 9.5%, or nearly 1 in every 10 passes. That was a meaningful floor when the passer rating formula was created, but it’s essentially meaningless now (although Drew Stanton did throw 6 interceptions on 51 pass attempts back in ’09).

Putting it all together

We had a hypothetical passer complete 65% of his passes, while averaging 7.0 yards per attempt, with a 6.0% touchdown rate and a 2% interception rate. That gave him a 1.75 for the completion percentage variable, 1.0 for the yards per attempt variable, 1.20 for the touchdown rate variable, and 1.875 for the interception rate variable.

Now what?

=IF(C2>223,SUM(MEDIAN(0,2.375,(D2/C2-0.3)*5),MEDIAN(0,2.375,((E2)/C2-3)*0.25),MEDIAN(0,2.375,F2/C2*20),MEDIAN(0,2.375,2.375-(G2/C2*25)))/6*100,0)

We sum those four numbers, divide by 6, and then multiply by 100. The last part is because the formula is designed to give a result between 0 and 1, and for added accessibility for fans , we multiply it by 100 to give a score of between 1 and 100 (you know, back when passers weren’t expected to produce ratings in excess of 100). If we add our four variables — 1.75, 1.0, 1.2, 1.875 — we get 5.825. Divide that by 6, and we get 0.971; multiply that by 100, and we get a 97.1 passer rating.

So a quarterback who completes 325 of 500 passes for 3500 yards, with 30 TDs and 10 INTs will have a 97.1 passer rating. It took us a while to get there, but hopefully that made some sense.

What’s Up With 2.375?

Now, what about those things I said we would get too later? Why is the upper limit 2.375 for all of these variables? That’s the result one gets from dividing 19 by 8, which maybe doesn’t make any more sense than saying 2.375. But remember that passer rating was created by the NFL after the 1972 season, when Archie Manning led the NFL in pass attempts and the great Johnny Unitas just finished his final season with the Colts. Manning, of course, wore number 8, while Unitas famously wore 19. So 19 divided by 8 serves as a nice limit.

Okay, that may not be the correct answer. When we looked at the three “good” variables, the upper limit didn’t seem intuitive, but you need to think back to what the creators of passer rating were trying to do. If you have a 1.0 average on the four variables, you get a passer rating of 66.7 — that would sum to 4.0, which you then divide by 6, and multiply by 100. That was the intended baseline, and the average passer rating for the 10 years prior to the introduction of passer rating was 63.4 (which was deflated a bit by backup and third string passers). So 66.7 was suppose to be a nice “round” number representing average, with 100 being exceptional.

To get to 66.7, you need to work backwards. All four variables need to average 1.0. You can do that however you want with TD rate, completion percentage, and yards per attempt, but it gets tricky with interception rate. How do you get an interception rate variable of 1.0? Well remember, this variable is derived by taking interception rate, multiplying by 25, and then subtracting from that yet-to-be-understood 2.375 number. So to be at 1.0, you need to have an interception rate of 5.5%. Do you see why? 0.055 times 25 is 1.375, which is 1.0 better than 2.375.

The interception rate for the 10 years of pro football prior to the introduction of the passer rating formula? Well, that was 5.5%. So it naturally follows that if you want to have an “average” interception rate of 1.0, you need to start with 2.375, and subtract from that a player’s interception rate (multiplied by 25). That doesn’t quite answer the question of why are we multiplying by 25 (more on this in a future post), but whatever number they picked, they would have to work backwards to get the upper limit. So since the average interception rate was 5.5%, and to get the interception variable to equal 1.0 you need to start with 2.375, and to make all four variables consistent, that’s how we live in a world where 2.375 works as the upper limit throughout the formula.

Obtaining An Average Result

This strive for 1.0 is why you see all of the weird numbers being used as multipliers throughout the table. With completion percentage, you subtract 30%, and multiply by 5. Why? Well, the average completion percentage for the 10 prior years was 50.4%. So if you want to get an average value of 1.0, and your average is 50%, that’s one good way to do that.

What about touchdown rate? You take TD rate and multiply by 20. Why? The average touchdown rate was 4.84% — and higher historically — so it’s reasonable to assume that the designers figured 5.0% was an average touchdown rate; multiplying by 20 is, then, the simplest way to get at an average grade of 1.0. In addition, by doing it this way, you have the same multiplier (of 5) for both TD rate and Completion Percentage, if that is appealing to folks.

How about yards per attempt? Recall that we subtract 3.0 from a player’s Y/A average, and then multiply by 0.25 or divide by 4. That means 7.0 is average, and in fact, the average Y/A from 1963 to 1972 was 6.94. Again, you can quibble with the methodology — you could subtract 4.0, and divide by 3, or subtract 5.0 and multiply by 0.5 — but while that would reward the top passers, it would also raise the floor for the bad ones.

As you can see, all four variables were designed with 1.0 in mind as the average result, and the “work backwards” method for interception rate produced a cap of 2.375. While there are many dumb things about passer rating, at least you can understand some of the decisions they made on the math side, given the variables they chose.

Conclusion

One thing we could do is try to adjust for era. This has been done many times before, including by Rupert Patrick in a 2011 Pro Football Researchers Association article in the Coffin Corner. But that was 6 years ago, and numbers have only skyrocketed since then. In addition, Bryan Frye wrote a great guest post here. There are many ways to adjust for era. But over the next few articles, I’m going to provide some additional thoughts passer rating, now that we all fully understand the formula. And I’ll discuss another way we can adjust passer rating for era.