- you try to get the ball in the hole;
- scoring works by counting the number of times you hit the ball with the club; lower scores are better;
- "par" is really good, not average.
Anyway, Tiger Woods won the U. S. Open, and Ian Ayres, guest-blogging at Freakonomics asks why golf commentators don't give the probability that a putt from distance x will go in. Commentators in just about every other sport do; Ayres' example is free-throw percentage in basketball, mine would have been batting average in baseball.
Ayres then gives a plot showing the success rate of golf putts as a function of difference from the hole, taken from this paper. Not all that surprisingly, the probability of making a putt from distance x scales like 1/x; essentially you can assume that the angular error in putting doesn't change as the distance of the putt increases. But the apparent size of the target is smaller at longer distances. Basically, from twice as far away the hole looks half as big.
It turns out that's what Andrew Gelman and Deborah Nolan thought, too. (Andrew Gelman and Deborah Nolan, "A Probability Model For Golf Putting", Teaching Statistics, Volume 24, Number 3, Autumn 2002. (This is the source for Ayres' figure.)) Their actual model is a bit more complicated, because they actually do the trigonometry correctly, and they assume that errors in putting are normally distributed while I'm assuming they're uniform. Read it, it's two and a half pages.) This fixes the problem that my crude model would have. At five feet, pros make 59 percent of their putts; thus it would predict that at two and a half feet, they make 118 percent!
The result of Gelman and Nolan is that the probability of the success of a putt from distance x is
where R, r are the radii of the ball and the hole; 2R = 4.25 inches and 2r = 1.68 inches. σ is the standard deviation of the angular error of a shot (in radians), which can be found from empirical data to be 0.026 (about 1.5 degrees). Φ is the standard normal distribution.
If you assume that x is large enough that (R-r)/x is small enough that we can make the small angle approximation arcsin x ≈ x, then this becomes 2Φ((R-r)/(σ x)) - 1. But Φ(z) is linear near z = 0, with Φ(0) = 1/2 and Φ'(0) = 1/(2π)1/2. So the probability of succeeding from distance x, for large x, is approximately
R-r is 1.285 inches, or .10708 feet.
So we get that the probability of making a putt from distance x, in the limit of large x, is about (3.29 feet)/x, although this is really only a good approximation above x = 6 feet or so. This has the advantage of being easy to remember -- well, somewhat easy, because you still have to remember the constant. But if you measure in meters, there are 3.28 feet in a meter, so the constant is basically 1; clearly golf should be done in metric.
Incidentally, I think it's a good idea to put citations that at least include the author and title in blog posts, even though strictly speaking they're not necessary as pointers if a link is provided. Why? Because that makes it more likely that people who Google the paper's title or authors find the people talking about it. (Occasionally I've recieved comments from self-Googling authors.)