Another way to explain OPR.
Let's take the St Joseph High School District event as an example.
http://www2.usfirst.org/2013comp/Eve...chresults.html
Here is the first row of the Qual Match Data. It shows the first qual match of the event:
11:00 AM 1 3767 3452 4568 453 3572 4835 20 34
In Qual Match 1 shown above, Red Alliance (teams 3767, 3452, & 4568) played Blue Alliance (teams 453, 3572, & 4835), and the Blue Alliance won 34 to 20.
From this
one match, there are
2 equations:
OPR3767 + OPR3452 + OPR4568 = 20
and
OPR453 + OPR3572 + OPR4835 = 34
... where OPR3767, OPR3452, etc represent the (as yet unknown) OPR values for those teams
So, for each qual match at the competition, there are 2 equations. And since there were 78 qual matches at this event, there are 2*78 = 156 equations for the entire event.
But there were only 39 teams at the event.
So we have
156 equations and 39 unknown OPR values.
There are two important things about this system of equations:
1) Each equation is linear (a simple linear combination of the variables), and
2) There are too many equations for the number of unknowns. This is called an "overdetermined" system of equations. In general, there is no set of values for the 39 variables which will satisfy all 156 equations.
So... instead of trying to solve for *the* solution (which does not exist), we instead say OK let's try to find "the best" solution, in some sense of the word "best".
One of the ways to define "best" is to proceed as follows:
First re-write the 156 equations as
OPR3767 + OPR3452 + OPR4568 - 20 = 0
OPR453 + OPR3572 + OPR4835 - 34 = 0
etc
Notice that all I did was to put everything on the left-hand side. It's still the same set of equations.
Now plug the proposed "best" solution into the left-hand side of each of the 156 equations and do the arithmetic to get a numerical value of the left side for each equation. Square each of those 156 values, and add all the 156 squared values together.
The solution which minimizes this sum of the squared values is called the "Least Squares" solution, and it is one of the ways to define "best". It is used extensively in problems like this. There are theoretical reasons for saying that the Least Squares solution is indeed the best solution, under certain circumstances. But the Least Squares definition of "best" is often used even when it is really not the "best", because the math for computing it is elegant and simpler than for other definitions of "best".
For computing OPR, using the Least Squares definition of "best" is probably as good as any other method, and since the math is relatively straightforward, it makes sense to use that definition.
If any students are still with me at this point and want "the rest of the story" let me know and I will continue with an explanation of how that system of 156 equations in 39 unknowns becomes the system of 39 equations in 39 unknowns that Ian showed, whose solution will be the Least Squares solution to the 156 equations.