Group Study Session: The Perfect QB Rating

Qb_rating_equation__with_limits__mediumThis is the equation for the infamous "quarterback rating". It's a mystery, right? Well, it may seem that way, at the moment, but we're going to figure this thing out. Knowledge is not given, it is earned, doggonit. 

Now, before you start to have a nervous breakdown because there are letters and numbers jumbled together, rest assured that this isn't an algebra exam. There won't be a quiz, at the end, (just a simple multiple choice in a poll) and you won't have to retake this class in summer school with Jamarcus Russell and Al Davis. 

Don't you dare worry, MHR. Wipe those blank stares off your faces. You in the back! Stop crying. I am here to help. We're going to get through this together. Think of it like a group study session. The teacher gave us a problem and it is our duty to figure out what it all means.

With that said, are there any volunteers to start the session?....

Alright, since nobody else in the study group decided to step up, I'll do it. If you have questions or concerns, please, wait until I am finished. My train of thought is often broken by interruptions, loud noises and ....... awww look at the butterfly....

See what I mean?

As you probably noticed, I didn't put up the full QB Equation, at the top, in one big piece. It's much easier to take small bites than to shove the whole equation in your mouth. You might choke.

Since, we've been a study group for 2 or 3 minutes, and we're all basically best friends, I'm going to tell you a secret. We are friends, right? ... Right?....Maybe?... Your silence worries me, but I'll still tell you.

The truth is, when I first saw this equation, I thought, "Yikes, I might hurl...", but once you get over the feeling of awe, stop cursing quietly to yourself and finally get rid of the vomit stains on your shirt, it's not so bad. Essentially, we are looking at one big equation that can be broken down into 4 subunits, which we I have done, already. 

Well, enough fooling around. Let's get down to it and finish this, so we can go outside and play some flag football or something...

 

What is QB Rating?

In a nutshell, QB Rating attempts to quantify a quarterback's production. It takes four different ratios into account (completion percentage, touchdowns per attempt, interceptions per attempt, and yards per attempt). That's all swell, but the tricky part comes when you try to equalize the affects of each ratio. How do you make 2 interceptions as important, statistically, as having 500 yards passing? Hopefully, we can figure that out, in this article. Figuring out the relationship between the four ratios is going to be the key to understanding the logic behind the QB rating equation.

 

Limits, We Don't Need No Stinkin' Limits

To begin, let's look at the limits that were shown. The equation says that each variable (a, b, c, and d) must be between 0 and 2.375. For now, we'll just focus on the upper limit, to determine what it takes to get a perfect QB Rating. Did the stat makers just pull a number out of the sky? Where did 2.375 come from? Well, to find out the answer to that, let's see what happens if we set a, b, c, and d equal to 2.375.

Note: I'm using a wicked awesome mathematics program (Maple 13) to save time, make more use of my page space, and ensure that I won't make any errors in my calculations. So, if you're wondering where my steps are.... well, they're in my computer somewhere.

Perfect_qb_rating_medium 
Here we have a perfect score. The upper limit, 2.375, is what it takes to get a perfect score. Now that we know the significance of 2.375, let's take a more detailed look at what it takes to get a 2.375 in each subunit. Maybe we can even figure out how the equation maker arrived at the number, in the first place. Let's find out, shall we? I'll go ahead and input 2.375 for each variable (a, b, c, and d) and we'll see what we see...

Qb_rating__upper_limit_of_2

What jumps out right away is "Interceptions = 0". What does that mean? As far as I can tell, the "interceptions per attempt" is the limiting factor of this entire equation. Obviously, you cannot throw fewer than 0 interceptions. If you know of a way to throw a negative interception, call Jay Cutler, immediately. He'll pay you substantially for your knowledge, I'm sure.

What makes the interception subunit (D) different than the other subunits? You'll notice that in the D subunit, the equation begins with a "0.095 minus... That is because a high QB rating is inversely proportional to the number of interceptions thrown. In other words, the more picks you throw, the lower your QB Rating is.

Another interesting note about interceptions... It doesn't matter if every other subunit is maxed out at 2.375, if you throw 1 interception, it is numerically impossible to have a perfect QB Rating. Don't believe me? I wrote a quick excel program and put in the following stats.

Qb_rating_with_1_int_medium 
It's pretty darn close to a perfect 158.3, but it is not close enough...

Here are a few other things I deduced from the subunit calculations:

       Any completion percentage greater than 77.5% will result in a perfect subunit score.
       Throwing more than 12.5 Yards per Attempt will result in a perfect subunit score.
       Throwing a TD pass more than once for every 8.42 passes  will result in a perfect subunit score.
              (That comes from 1 divided by 0.11875) See below for proof.
       You must be have a perfect score in every subunit to receive a perfect QB Rating.

Qb_rating__perfect_8

Qb_rating__not_so_perfect__medium

Why is 158.3 a perfect score?

We have all seen the mystical 158.3 QB Rating score and have been told that it's perfect. What makes that so?

I've done some calculations and... I have no idea. 158.3 is an arbitrary number that doesn't really hold any significance. The basic formula for QB Rating, as shown earlier, divides by a constant (0.06). Why? Beats me...

Fortunately, Brian Shrout has a great idea that you can find in the comments. For the sake of argument, let's talk about making QB Rating into a 100 point scale.

I present to you.... the 100 point scale QB Rating Equation. It's actually not that different from the original QB Rating equation. Instead of dividing by the constant, 0.06, let's divide by 0.095, instead....

100_point_scale_qb_rating_medium 

And here is what QB Rating looks like, when converted to a 100 point scale.

Qb_rating_and_100_point_scale_medium

Well, that's all, for now. There's more to the QB Rating statistic than was covered in this post, but we'll take a break so we can all have a moment to digest this information. I hope that our little group study session was beneficial.

I declare the group study session adjourned. My mom is outside  in her car and I don't want to keep her waiting....

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