What is Passer Rating v. QBR?

Passer rating and QBR are metrics which attempt to tell us how well a quarterback played in a particular game or over the course of a season. Passer rating is a very old way of doing this which mostly just combines box scores in a (pretty weird) way. Total quarterback rating looks at how much better or worse than average a quarterback does in a context dependent way. Even though the statistics are completely different, they try to answer the same question: how good of a game did a quarterback have.

Passer Rating	QBR
Average Changes by Year	Normalized by Year
Bad Accuracy	Good Accuracy
Easy Formula	Complicated Formula
Hard to Interpret	Interpretable

In this article, we’ll build up some intuitive description of these stats to help us understand them better. Along the way, there will be a bit of discussion about the relative merit of these two statistics.

Understanding passer rating v. QBR is a great way to get a feel for the practice of sports analytics as a whole. I’ve found this book to be a great example of practical sports analytics and how to use them in your own work.

What is Passer Rating?

Passer rating is a combination of four parameters: touchdowns, completions, yards, and interceptions all normalized on a per-attempt basis. As someone who has studied a lot of different mathematical and statistical models, passer rating is really weird. It is almost a linear function in the four variables TD/ATT, Comp/ATT, YDS/ATT, and INT/ATT. If it were, that wouldn’t be weird at all. In fact, most (almost all?) sports statistics are linear functions of things showing up in a box score. Rather, the weird thing about passer rating is that the contributions of each of these four statistics is ‘capped’.

A 100% completion percentage doesn’t contribute any more to your passer rating than a 77.5% completion percentage. Similarly, the effect of yards and touchdowns per attempt is capped. This is what makes the stat weird. Because of the capping, a perfect game doesn’t require perfect performance. Two players with vastly different stats – one could even be noticeably better than the other! – can be labelled as both having perfect games by passer rating.

But how is passer rating actually computed?

How Passer Rating is Computed

The traditional description of passer rating involves a process centered around four components or variables. I will label these four variables Y, C, T, and I for the yards component, completion percentage component, touchdown component, and interception component, respectively. Step 1 in computing Y, C, T, and I is to compute the following four statistics:

• Y= 0.25\cdot\frac{Yards}{Attempt} - 0.75
• C = 5\cdot \frac{Completions}{Attempt}- 1.5
• T = 20 \cdot \frac{Touchdowns}{Attempt}
• I = 2.375 - 25\cdot \frac{Interceptions}{Attempt}

These four expressions were originally designed so that the league average value of Y, C, T, and I was exactly one. Since the league has greatly changed over the past decades, that is no longer true, but that was the original motivation. After this first step in calculation, each of the values Y, C, T, and I is capped so that they are in the interval [0, 2.375]. If one of these values is negative, it is set to 0. If one of the values is larger than 2.375, it is set to 2.375. This is the weird part of the calculation.

Finally, these four variables are combined with equal weight and rescaled to give the formula

\text{Passer Rating} = \left( \frac{Y + C + T + I}{6}\right) \times 100

In this way, passer rating combines yards, completions, touchdowns and yards in order to determine how good of a game a quarterback had. The graphic below shows a flow chart for how passer rating is computed.

What is Total Quarterback Rating (QBR)?

Total quarterback rating – sometimes just quarterback rating and colloquially “QBR” – is ESPN’s answer to many of the deficiencies that passer rating has. Motivating the calculation for this stat is easier if we appropriately appreciate a few key deficiencies of the more traditional passer rating.

• Passer Rating is context independent. A 10 yard pass on first and 10 is much more impressive than a 10 yard pass on 3rd and 30. Similarly, a QB in garbage time – their team being down big at the end of the game – will tend to accrue significant stats as the defense just tries to run out the clock. QBR will adjust for the context surrounding a play.
• Passer Rating is opponent independent. Throwing for 200 yards against the best defense in the league might be more impressive than racking up 400 yards against the league’s worst defense. Total Quarterback Rating will adjust for the quality of the opponent against which the stats are accrued.
• Passer Rating can award quarterbacks for excellent plays by their receivers. A few broken tackles can turn a 10 yard throw-and-catch into a 90 yard touchdown. Those broken tackles, though, should be credited to the skill of the receiver, not the quarterback. Total QBR differentiates between value contributed by the quarterback and value contributed by the receiver.
• Passer rating only looks at passing statistics while in the modern NFL a quarterback’s role can generally be much broader. For example, a running quarterback like Lamar Jackson will be significantly undervalued if we only look at his passer rating. ESPN’s QBR accounts for all of a quarterback’s actions, not just his passing ability, in order to better measure their total impact on the game.
• Passer rating is on a meaningless, hard-to-interpret scale. Passer rating is on a scale from 0 to 158.3. This arbitrary scale which is non-normal in distribution makes it difficult to rationalize what a specific passer rating means. Total quarterback rating is normalized on a 0-100 scale where 50 is average and the difference between 50 and 60 is larger than the difference between 60 and 70.

Because of all of these points, comparing passer rating v. QBR should make it clear that QBR is the better statistic. Passer rating is dated, has too many flaws, fails to take too many things into consideration, and simply wasn’t a very well designed statistic in the first place.

But, the question still remains: how is QBR calculated? Like most advanced modern statistics, QBR is the result of an algorithm, not just a formula that pops out of the box score stats. So, in the next section I’ll describe how this algorithm works and why it was designed the way it was.

How is Total Quarterback Rating Computed?

Total Quarterback Rating is built on two ideas: measuring the expected points added by a quarterback and translating these expected points added to winning probabilities. Let’s go through each of these components separately.

Expected Points Added

First, let’s try to understand expected points added. Really, football and any other game is simple: score more points than your opponent. Then, if one could measure ‘how many points’ a particular player contributes to your team’s total, then you could figure out definitively how much a player helps you win.

Each down and distance from each spot on the field in the NFL has an expected points scored on the drive. For example, if it is first and 1 from the opponents’ 1 yard line, the expected points on the drive is almost certainly around 7. If it is 4th and 10 from your own 1 yard line, the expected points should be nearly 0. Computing the change in expected points from one play to the next lets us determine the ‘value’ of a play. Let’s do an example.

It is 3rd and 10 at our own one yard line, the expected points on this drive are nearly 0. However, the running back rips off a 98 yard run to the opponents 1 yard line. Now the expected points on the drive is about 7. The previous play added 7 expected points to my team’s total. If we determined that the running back was solely responsible for the outcome of that play, then the running back contributed 7 expected points to his team’s total from that one play. We can compute a player’s expected points added over all plays in the game this way to figure out how much a player contributes to their team’s success. This is a lot like RE24 in baseball.

The first step in determining total QBR is determining the quarterback’s expected points added over the course of the game. Because there are 11 players on each side of the ball, it can be hard to assign credit for the outcome of a play. In fact, entire articles have been written trying to assign credit for specific outcomes to one player or another.

However, for quarterbacks, the problem is often a lot simpler. For example, the quarterback gets credited for completing the pass and a few yards after contact, the rest is the receiver. A similar judgement and division of credit can be made for quarterback runs and sacks though that information is still proprietary to ESPN. In this way, we can calculate the expected points added by a quarterback. Now we need to convert this to a meaningful scale.

Logistic Regression for Winning Probability

After computing a quarterback’s expected points added, we want to convert this number to something easy-to-digest. In particular, converting from expected points added to estimated winning probability is about as valuable as a conversion one could make. Why? Because then the numbers are easy to interpret.

A 50% chance at winning is a perfectly average performance; you’ll win half the time you’ll lose half the time. A QBR of 75 is significantly better. A QB with a season-long QBR of 75 should be expected to win 75% of their games if they play against league-average opponents. All else held equal, a quarterback with a 75 rating should lead your team to a 12-4 record. Most importantly, a perfect QBR of 100 is actually impossible to attain – as it should be! No matter how well your quarterback performs, you cannot ever guarantee a win because the other quarterback could always do the exact same thing.

How is the conversion from expected points to winning probability done? This is the perfect opportunity to use logistic regression. Logistic regression transforms ‘continuous’ predictive variables (like expected points added) into probabilities of an event happening. In particular, the logistic regression model fits a curve (the logistic curve) to existing data to translate from expected points added to winning probability. It is kind of like a least squares regression line or line of best fit in this way.

(As an aside, a good tool to learning logistic regression is this “hands on, practical” book that I’ve used before. Logistic regression is one of the top most important tools that data scientists and statisticians should learn. I consider it a “must know”.)

While not entirely correct, if the data is sufficiently nicely distributed, one could actually think of total quarterback rating as a percentile of expected points added over all time. Then, a 99.9 QBR would correspond to the 99.9th percentile of quarterback performances. Again, this isn’t exactly correct, but it is a nice interpretation that is close enough to the truth to be valuable.

After that, that’s really it. Total quarterback rating is calculated by finding out a quarterback’s expected points added and converting it to an estimated winning probability.

Why No QBR Calculator

It would be really nice if I could build a little widget and embed it on this page that would let the user enter a quarterback’s stats and it would spit out that quarterback’s total QBR. However, that isn’t really possible in such a setting. Because the statistic relies on knowing the down, distance, time of game, and opponent for every single event, you can’t build a QBR calculator just by entering in a few box score stats. In order to do that, you would need to basically enter by hand the play-by-play of the game. This is why I can’t provide – and you won’t find anywhere on the internet really – a total QBR calculator.

Conclusions

I tend to like most of the statistics that ESPN comes up with. They are very much like things that I would do. Get to the core of what ‘value’ is, take into account teammates, opponents, and other contextual factors, and try to extract how much value a certain player contributes to the overall. The only way I could necessarily see to improve total QBR is to somehow take into account quality of one’s own teammates.

In the current QBR, I don’t believe any adjustments are made for the quality of your own teammates. If you have Derrick Henry lining up at running back, it gets easier to pass because the defense has to account for the possibility of him running. If you have an elite offensive line, then you will probably complete more passes because you have more time to throw. This division of credit problem is hard to do, if I wanted to improve upon QBR at all I think I would have to use tracking data to determine grades for lineman, running backs, etc. to help actually determine what my quarterback contributed.

One thing that I don’t understand is the claim that computing total QBR spans some 10,000 lines of code. Quarterback rating can be calculated by looping over every play the quarterback was on the field for, querying a lookup table to compute expected points added, and applying the logistic regression to convert from the 0-100 scale. This is not 10,000 lines of code. So, either ESPN is inflating this number to make themselves seem smarter, they are creatively counting how many lines of code are involved, or there are extra proprietary steps that go into the calculation of total QBR.

Passer rating, on the other hand, is a dumpster fire. It should be done away with. Passer rating doesn’t tell us much of anything. If passer rating was never brought up again, I think the NFL would be better for it.

To read similar articles explaining advanced statistics, see some of our previous posts below:

• What is usage rate?
• What is real plus minus?
• RE24 and the run expectancy matrix

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