There are two well known adjusted yards per attempt formulas, which easily reduce to simple scoring models. The first is the equation introduced by Carroll et al. in “The Hidden Game of Football“, which they called the New Passer Rating.

(1) AYA = (YDs + 10*TDs- 45*INTs)/ ATTEMPTS

And the Pro Football Reference formula currently in use.

(2) AYA = (YDs +20*TDs – 45*INTs)/ATTEMPTS.

Formula (1) fits well to a scoring model with the following attributes:

- The value at the 0 yard line is -2 points, corresponding to scoring a safety.
- The slope of the line is 0.08 points per yard.
- At 100 yards, the value of the curve is 6 points.
- The value of a touchdown in this model is 6.8 points.

The difference, 0.8 points, translated by the slope of the line, (i.e 0.8/0.08) is equivalent to 10 yards. 4 points, the value of a turnover, is equal to 50 yards. 45 was selected to approximate a 5 yard runback, presumably.

Formula (2) fits well to a scoring model with the following attributes:

- The value at the 0 yard line is -2 points, corresponding to scoring a safety.
- The slope of the line is 0.075 points per yard.
- At 100 yards, the value of the curve is 5.5 points.
- The value of a touchdown in this model is 7.0 points.

The difference, 1.5 points, translated by the slope of the line, (i.e 1.5/0.075) is equivalent to 20 yards. 3.5 points, the value of a turnover, is equal to 46.67 yards. 45 remains in the INT term for reasons of tradition, and the simple fact this kind of interpretation of the formulas wasn’t available when Pro Football Reference introduced their new formula. Otherwise, they might have preferred 40.

**Adjusted yards per attempt or adjusted expected points per attempt?**

Because these models show a clearly evident relationship between yards and points, you can calculate expected points from these kinds of formulas. The conversion factor is the slope of the line. If, for example, I wanted to find out how many expected point Robert Griffin III would generate in 30 passes, that’s pretty easy, using the Pro Football Reference values of AYA. RG3’s AYA is 8.6, and 0.075 x 30 = 2.25. So, if the Skins can get RG3 to pass 30 times, against a league average defense, he should generate 19.35 points of offense. Matt Ryan, with his 7.7 AYA, would be expected to generate 17.33 points of offense in 30 passes. Tony Romo? His 7.6 AYA corresponds to 17.1 expected points per 30 passes.

Peyton Manning, in his best year, 2004, with a 10.2 AYA, could have been expected to generate 22.95 points per 30 passes.

This simple relationship is one reason why, even if you’re happy with the correlation between the NFL passer rating and winning (which is real but isn’t all that great), that you should sometimes consider thinking in terms of AYA.

**A Probabilistic Rule of Thumb.**

If you think about these scoring models in a simplified way, where there are only two results, either a TD or a non-scoring result, an interesting rule of thumb emerges. The TD term in equation (1) is equal to 10 yards, or 0.8 points. 0.8/6.8 x 100 = 11.76%, suggesting that the odds of *not* scoring, in formula (1), is about 10%. Likewise, for equation (2) whose TD term is 20, 1.5/7 x 100 = 21.43%, suggesting the odds of *not* scoring, in formula (2), is about 20%.

February 1, 2014 at 9:08 pm

[…] league as a whole. We’ll use the Pro Football Reference formula. Later, we’ll use the known conversion factors for AYA to turn that contribution to points, and the subtract the league average from that […]