I knew that I wasn't happy with the N.F.L's Passer rating
system. I knew that it didn't take the evolution of the game into account,and
that the improvement in a Passer's rating would depend on WHICH area he had
improved in (even though each category was apparently meant to be equally
important). I also knew I wanted to start with a clean sheet of paper,rather
than attempting to modify what there already was (if you have a Mini,and want
a Porsche,you start from scratch. You don't try to modify the Mini to BECOME a
Porsche :-) ). I KNEW these things,but where did I start??
Before trying to launch straight into writing a new
formula,I first needed to establish what RESULTS I was aiming for. i.e. from
a statistical point-of-view,who should end up at No. 1? I needed something
simpler than calculating all the stat's (since that is what the formula is
essentially FOR - making things simpler),but the stat's would give me an idea
of what results I SHOULD be getting - the results that I should be aiming
for,in order for them to be statistically sound.
I started with a selection of Quarter-backs to test things
out,and for each of them,calculated how they compared to average in the 4
categories; completion percentage; yards/completion; touch-down percentage;
and intercept percentage. i.e. if the average completion percentage (for a
particular year) was 50%,and the Passer in question had 55%,that would be
1.10 (10% above average).
Now comes one of the MOST important things to understand
about the P.P.R. system. Instead of "averaging" (adding,then dividing) these
4 figures,I MULTIPLIED them. This is one of the 2 biggest differences between
the P.P.R. and N.F.L. systems. Unlike addition (where you are averaging
averages),multiplying them gives you statistically-sound results.
Imagine the figures of 2 QB's - one who has 1.1; 1.1; 1.1;
1.1; and the other who has 0.7; 0.7; 1.0; and 2.0 (be it each one in
what-ever category you choose).
Both of these "average" to 1.1 (4.4/4),and any system that
USES this approach would declare them to be "equal". Problem is,they're not.
Some quick statistics,in laymen's terms (as far as that is
possible). Naturally,these
numbers - for a particular Passer - vary from year to year,and even category
to category. In amongst all that,there is a "true mean",which represents his
inherent ability (and is what we're trying to find). When the
figures don't exactly reflect this true mean,then that is because of random
variation. So the figures vary above and below the true mean.
We create a hypothesis that says this true mean is 1.0
(average),and TEST if the variation is consistent with that hypothesis.
What we find is that the first Q.B. (with all 1.1's),has
statistically-significant results (we would reject the idea that he is
"average"),whilst the 2nd Q.B. does NOT have statistically-significant results
(chances are,he IS average).
Getting back to those laymen's terms, :-) we can see that
this is the case because all 4 of the first Q.B's figures are above
average,whereas the 2nd Q.B. has 2 figures below average (the 0.7's),one
average,and one above average. Even though his above average figure is a LOT
above average,there are still those below-average figures to consider. The
first Q.B. doesn't have ANY below-average figures. The difference is
consistency.
We therefore need to do something with the figures that will
REFLECT this.....
When we multiply those 2 sets of figures,we get 1.1^4=1.4641
and 0.7*0.7*1.0*2.0=0.98. These numbers DO accurately reflect what we found
with the statistical tests. The first Passer is significantly above
average,whereas the second one is "average" (close to).
So now I had established I'd be using multiplication (and
division) exclusively - no addition or subtraction. Where to next?
The OTHER important thing to take care of,was to actually
compare these Passers TO an average - not some arbitrary,static figures. How
to do this without involving having to calculate all those averages?
Well,having done away with any idea of using addition,it
actually became quite SIMPLE to arrive at an easy formula - it was already
staring me in the face! :-)
Once you only have multiply and divide in an equation,you
can change the order of the operations in any way you want,and still get the
same answer. More to the point,you can also end up with common factors
cancelling out. If you can bear with me for a moment with the maths
involved,then you will see EXACTLY what it is the P.P.R. system is doing. Even
if you know the formula the N.F.L. uses,can you say that you know what the
formula is DOING? :-)
Here is how the P.P.R. came to be what it is....
Starting out with what I wanted to achieve ideally (4
figures which each relate how much above/below average the Passer is - e.g.
1.10 represents 10% above average - and then put them all together),I firstly
have the QB's completion percentage,divided by the average completion
percentage,all squared,and then multiply by the QB's yards per completion
divided by the average yards per completion.
I squared the percentage figure as I wanted it to be the
most important (completion percentage is the figure which is a mark of
consistency,and consistency is the most important thing). The N.F.L.
effectively adopts the same approach by using yards per attempt,rather than
yards per completion (yards per attempt is just yards per completion
multiplied by the completion percentage,and the completion percentage is thus
counted twice).
Finally you multiply by QB touchdown percentage on average
touchdown percentage,and divide by QB intercept percentage on average
intercept percentage. You divide by the intercept figure,rather than
multiply,because the SMALLER it is,the better it is (as opposed to as with
the other figures).
This all gives us.......
(QBcom%/AVcom%)^2 * (QBydscom/AVydscom) * (QBtd%/AVtd%) / (QBint%/AVint%)
Expanding this into it's factors,we get....
((QBcom/QBatt)/(TOTcom/TOTatt))^2 * ((QByds/QBcom)/(TOTyds/TOTcom))
* ((QBtd/QBatt)/(TOTtd/TOTatt)) / ((QBint/QBatt)/(TOTint/TOTatt))
Now,because this is all multiply and divide (and the brackets
are shown ONLY for clarity),we can re-arrange the factors into a more
convenient order...
((QBcom/QBatt)^2 * (QByds/QBcom) * (QBtd/QBatt) / (QBint/QBatt))
/ ((TOTcom/TOTatt)^2 * (TOTyds/TOTcom) * (TOTtd/TOTatt) / (TOTint/TOTatt))
In THIS form,we can see that there is repetition,and in fact
we are just using the same formula twice! We use the formula with the Passer's
figures,and then divide by the results of using the same formula with the
"average" figures (in fact,we only need the totals,and don't need to
calculate the actual averages).
Finally,there is some cancelling out of common factors that
we can do,which leaves us with....
(QBcom / QBatt^2 * QByds * QBtd / QBint)
/ (TOTcom / TOTatt^2 * TOTyds * TOTtd / TOTint)
....which is the P.P.R. formula! :-) We took something that
was very messy at first appearances,and turned it into something quite simple
and elegant. Isn't maths wonderful?! (Oh?! You don't agree?? ;-) )
So,there you have the P.P.R. system in a nut-shell.
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