Simplifying the QB Rating System

This graph illustrates a comparison between the QB Rating and only the components for Completions and Interceptions.  The components for Yards per Attempt and Touchdowns are ignored.  The data shows a correlation of 0.970.  Interestingly, the Yards per Attempt and Touchdowns components, when added together, have a slight negative (-0.17) correlation with the QB Rating calculation.  In a previous post, I had shown that correlations of just the Completions component with the QB Rating was 0.956, and the Interceptions component was 0.965, respectively.  What this shows, at least to me anyway, is that we could simplify the QB Rating to only include 2 components - Completion Rate and Interception Rate.

The QB Rating formula is as follows:

Q = (J+K+L+M)*100/6

where,

Q = QB Rating

J = max(min(C, 2.375), 0)

K = max(min(Y, 2.375), 0)

L = max(min(T, 2.375), 0)

M = max(min(I, 2.375), 0)

and where,

C = ([Completions/Attempts]*100 - 30)/20

Y = ([Yards/Attempts] - 3)/4

T = [Touchdowns/Attempts]*20

I = (2.375 - [Interceptions/Attempts]*25)

Here's how we will go about simplifying the QB Rating calculation:

As discussed above, we can eliminate the K and L terms, since the J and M terms have an almost perfect positive correlation with the QB Rating.

We get, 

Q ≈ k*(J + M)*100/6

where,

k = non-zero scalar

Let's further simplify the equation by doing the following:

Let's remove the restrictions on a max and a min for each term (in my opinion, these restrictions are arbitrary, and when looking at annual league average data, the restrictions haven't come into play since 1943).

And, by removing the approximation and by substituting back for the J and M terms, we get

Q*100/6 = ([Completions/Attempts]*100 - 30)/20 + 2.375 - [Interceptions/Attempts]*25

Re-arranging terms, we get

(Q*100/6 - 2.375 + 1.5)/5 = [Completions/Attempts] - [Interceptions/Attempts]*5

or,

Q(new) = C/A - b*I/A

where,

Q(new) = New QB Rating

C/A = Completions/Attempts

I/A = Interceptions/Attempts

b = factor to be determined (in my preliminary research, it appears that b would be a number close to 3)

So, there we have it, a (much) simplified new QB Rating System.  It only looks at completion rate and interception rate, and no fancy mathematical gyrations.  It is easy to understand, easy to explain, not arbitrary, simple to calculate and it also has the amazing beauty of never exceeding 100.0%!  It still does have the issue of not being comparable over time (i.e. players in the 1940's versus players from the 1990's).  I will address this issue in an upcoming post.

For simplicity, I'd like to call this new rating system CMI, or, short for Completions Minus Interceptions (sort of like the OPS statistic in baseball - on-base plus slugging).