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Friday, October 12, 2007

The Cost of Kicking Field Goals

A famous article (in the right circles) by Virgil Carter and Robert E. Machol (1978) established empirically what every fan seems to know intuitively: football coaches kick too many field goals.1 They looked at how many points a possession was worth, on average, and how many points teams earned, on average, by kicking a field goal. Logically, if it is fourth and short and the field goal is relatively long, the chances of getting the first down and, subsequently, an easier field goal attempt or a touchdown make it worthwhile to go for it.

But they were speaking in generalities--what the average team should do--but individual teams are not the average teams. So, I've made an attempt here to build on that work thirty years later.

I have used four variables--yards to go, yard line, quality of kicker (yard line from which he will make 10% of his fieldgoals), and offensive yards per play. If if it fourth and long (high yards to go) that will favor a field goal attempt, as will a high quality of kicker. A high offensive yards per play increase the likelihood of converting on fourth down as well as that of scoring a touchdown, and should be correlated with fewer field goal attempts.

Obviously, I am making several simplifying assumptions. For example, passing teams will do better on fourth and long and running teams will have the advantage on fourth and short--I remember BYU taking delay of game penalties on the goal line so they would have more room for the receivers.

The model I will be using after a fourth down conversion is very simple, and it doesn't allow a team to go for it on fourth down again; they must kick field goal if they get to fourth down again.

Finally, I ignore the subsequent possession. An advantage of going for it on the opponent's 5 is that they are then forced to 95 yards if you don't make it. I hope to add in this factor as well as add some variable options (passing/rushing yards per play and standard deviations, etc.) in a future model, but that will take some time.

I have a second problem. I have five variables as it is. To graph the results we need one dimension per variable and, if you haven't noticed, we experience reality in three dimensions, not five. (Technically I could use time as the fourth dimension, but good luck interpreting that on a with blogarific graphics.) Instead, I use a disaggregated form.

First, I modeled data they had on field goal accuracy, adjusting a little for college football (where the average kicker has less distance and is less accurate). To my utter disgust, the line of best fit was straight= P(fg) = (45-yard line)/.475 . This would suggest that a kicker has a zer0 percent chance of hitting a 52 yard field goal, a 42% chance from 42 yards, and an 84% chance from 22 yards. To adjust the model for accuracy and kicker leg strength I have added a few things:

P(fg) = (38+ LS -yard line)/(47.5-AC)

where LS is a LS rating (0=average, +=better than average) and AC is kicker accuracy (0=average, +=better than average).

The probability of getting a first down on first down is:

P(fd) = sqrt((x/4.5)/(YTG^1.75+.6))

where x is the offensive yards per play and YTG is the yards needed. This equation has some real problems (i.e. if a team averages 8 YPP then they have a greater than 100% chance of getting the first down), but with moderate values it seems to fit the data relatively well.

Next, I use the probability that the team will get "y" yards after they have converted the first down. I use the following formula:

P(yards=>y) =1/((y/10)^1.5*(4.5/x)+1)

And the probability that a team will get exactly "y" yards is:

P(yards==y) =1/((y/10)^1.5*(4.5/x)+1) - 1/(((y-1)/10)^1.5*(4.5/x)+1)

To avoid a rather complicated calculus maneuver I have just decided on the discrete values of each yard line, so that the probability of making a field goal later in the drive after the fourth down conversion is the sum of probabilities from goal line to (yard line - ytg).

The probability of a touchdown is simply

P(y=>yard line - ytg) = 1/((y/10)^1.5*(4.5/x)+1)

Then, to calculate the expected points, I just multiply the P(pfg) (probability of a post-conversion field goal) by 3 and of a touchdown by 7.

So, for an average team with an average kicker getting 4.5 yards per play facing a 4th and 5, we get distributions like this:

If the image were a better quality, you would see quite clearly that a team is better off kicking a field goal on 4th and 5 if they are inside the 28 or so (a 45 yard field). The main factor here is the probability of getting that initial first down.

If the situation is 4th and 3, the average team should never kick a field goal.

The effects of increasing the quality of the kicker or the offense have the expected results--they shift the curves up and down relative to one another. A team with a better kicker is not much more likely to kick the field goal, because he would also be kicking any subsequent field goals.

Because my actual numbers are a little suspect this little study is not too conclusive, but we can draw some basic conclusions. Most importantly, on 4th and short (and the definition of short varies on the offensive prowess of the team) teams kick too many field goals, but kicking the field goal is not a bad option on 4th and 5 or more--so don't fire the kicker.

In some future blog, I will try to clean up some problems (for example, allowing teams to punt or to turn the ball over). I think it is also necessary to consider the field position you will be handing over to the opponent.

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