What other people may find in poetry or in art museums, I find in the flight of a good drive.
—Arnold Palmer
A trip to Oakhurst Links in White Sulphur Springs, West Virginia, is a journey back in time to the 1880s. The setting is pastoral, both in the sense of a beautiful rural setting and in the sense of the herd of sheep that helps keep the links trimmed.1 The 9-hole course at Oakhurst was restored to its original layout in 1994 by owner Lewis Keller and architect Bob Cupp. Dating back to 1884, Oakhurst was the first golf club in the United States. To honor that historic distinction, you play Oakhurst with hickory sticks and gutta-percha balls, imported from St. Andrews, Scotland. You form your own sand tees on the tee box, and the “stymie rule” is in effect.
I had the pleasure of playing Oakhurst in July 2008. Mr. Keller, at 85 years young, was a wonderfully upbeat, inquisitive, and gracious host. He was pleased to hear that this book would be about the mathematics of golf, as he had recently discussed similar ideas with Luke List, a Vanderbilt golfer who was runner-up in the 2004 U.S. Amateur championship at Winged Foot, Keller’s old club. Mr. Keller has an enviably rich collection of friends and stories. As told in Oakhurst: The Birth and Rebirth of America’s First Golf Course (coauthored by Vikki Keller, Lewis Keller’s daughter), Sam Snead played an important role in the decision to restore Oakhurst Links, and he hit the first shot at the re-opening of the course. The special club built for the occasion by Karsten Solheim did not perform well. The clubhead broke away from the shaft at impact and flew off along with the ball. Snead, never at a loss for words, immediately asked the shocked gallery, “Did either of ’em get on the green?”2
Newcomers to Oakhurst must adjust to the four-club “set” consisting of a driver, a niblick, a putter, and your choice of a cleek or a mashie. On the tee, there is the visual distraction of holding a “needlenose” driver with a head shaped more like a wooden banana than the modern metallic grapefruit. In spite of its relatively small size, the wooden driver head is very heavy, so that finding the right swing rhythm is a challenge.3 On the green, you use a putter that looks disconcertingly similar to the driver. The putter is only slightly shorter but is much lighter and less lofted. The length and angle of the putter head are reminiscent of a hockey stick, which probably explains why a wristy putting stroke is effective.
Interestingly, the hickory shafts do not feel especially different from the shafts of today’s clubs. The flexibility of hickory shafts is similar to the flexibility of modern graphite and soft steel shafts.4 The gutta-percha balls are small and light and do not fly very far. In spite of the lack of dimples on the gutties, I can testify that monstrous slices are possible. Although the hickory sticks of the 1880s are recognizable as tools for golfing, the modern golf club represents a complete and extreme makeover from grip to clubhead. In this chapter, we explore how adept modern golfers are with their customized drivers.
The technological explosion has forced golf’s governing bodies to take action. Drivers are now limited to a clubhead volume of 460 cc, and the coefficient of restitution cannot exceed 0.83. This has put club manufacturers in the awkward position of not being able to significantly change the most important characteristics of drivers, while needing to market a “revolutionary” new design.
The clubhead volume most directly affects the moment of inertia (MOI) of the club, although this is an imprecise statement, as the clubhead has different MOIs for different directions of rotation. For a given axis of rotation, the larger the MOI is, the more resistant the club is to rotation about that axis. The rotations that we want to minimize are the club twists caused by off-center hits. In effect, an increased MOI can enlarge the sweet spot of the clubhead. Due to the reduced twisting, even off-center hits can produce solid drives.
The coefficient of restitution (COR) is a measure of the “liveliness” of the driver. When clubheads were made of solid wood, the COR was determined by the type of wood used. With hollow metal clubheads, the driver face can be made thin and flexible, so that the golf ball receives a trampoline effect and springs off the clubface. The cap on the COR levels the playing field, in that most modern drivers are equally lively. The main question now is how well the characteristics of the club match a particular swing.5
One surprising aspect of driver construction is that small changes in the center of gravity of the clubhead can have large effects on launch angle and spin rate. This is one of many reasons to test-drive a new club and have a professional custom-fit you with an appropriate club. Many discussions about drivers focus on loft: Is it better to use a 9° driver or a 10°? As Tom Wishon argues in The Search for the Perfect Club, for most golfers this is the wrong question. A better choice for the typical golfer is between a 12° driver and a 14° driver. Each golfer has a typical swing speed and angle of attack, and for the ball speed and spin rate produced there is a precise angle that gives maximum distance. The important fact is that the ideal angle is different for different swing speeds. Wishon has shown that the slower your swing speed is, the more loft you need on your driver. An average golfer with a 90-mph swing speed is best served by a 13° or 14° driver.6
Since professional drivers are legislated to be similar in performance, it should not be surprising that driving statistics from the Tour changed little from 2004 to 2008. Figure 10.1a shows average driving distance, with measurements taken only on holes designated as “driving holes”—that is, holes on which most pros would actually hit driver. Figure 10.1b shows percentages of fairways hit over time. The vertical scales are very narrow, which exaggerates the differences in values, but the average driving distance has varied less than 1 yard from its 6-year average, and the percentage of fairways hit has varied less than 1 percentage point from its 6-year average.
Another driving statistic of interest is the longest drive of the year. Of the six years of my data sets, 2004 was the big year for long drives, including a whopping 476-yard launch by Davis Love III. The longest drives for the succeeding years were 442 (2005), 427 (2006), 437 (2007), 435 (2008), and 467 (2009) yards. In 2009, there were 32 recorded drives of over 400 yards. There are some big hitters out there.7
Figure 10.1 Driving stats: (a) average driving distance on driving holes, 2004–2009; (b) percentage of fairways hit, 2004–2009
The driving statistics vary with the length of the hole. The ShotLink data sets do not record which club is used, so we must infer whether a driver was used by the length of the shot hit. There is not a typical length for a par 4. Table 10.1 shows how frequently par 4s of different lengths were played in 2007. Most par 4s are between 380 and 490 yards in length. On the shorter holes, many of the pros will back off of the driver and hit an iron or hybrid off the tee. The length of the average drive depends on the length of the hole, as shown in figure 10.2. Once the length of the hole clears 350 yards, the length of the average drive climbs steadily from about 250 yards to 310 yards.
For some of the shorter hole lengths, the average drive distance is 10–20 yards longer for drives that ended up in the rough. This would seem to be backwards: if the ball gets into the rough, it should not roll as far. The apparent paradox resolves itself if you think of the causal relationship in the opposite direction. That is, the act of hitting the ball farther on short holes (for example, by hitting driver instead of a safer iron) increases the likelihood that the ball will find the rough. The percentage of fairways hit (see figure 10.3) also varies with distance, although not in a perfectly precise way. The fairway on a short hole can be hard to hit if it is very narrow, and the fairway on a long hole can be easy to hit if it is the size of Montana.
Table 10.1 Percentages of par-4 holes of different lengths, 2007
Figure 10.2 Average drive distance on par 4s of different lengths, 2007
Figure 10.3 Percentage of fairways hit on par 4s of different lengths, 2007
This is the background necessary to compute a “par” for holes of different lengths. For example, from 420 yards the average drive is 274 yards, leaving 146 yards to the pin. The average approach shot from 146 yards in the fairway is 25.3 feet from the hole (see chapter 9), producing an average of 1.93 putts (see chapter 7). Therefore, the average score on the hole when the fairway is hit is 3.93. From the rough, the average approach shot from 146 yards is 63.5 feet from the hole, producing an average of 2.2 putts and a score of 4.20. If 66% of the drives hit the fairway, then the average score on the hole is 66% of 3.93 and 34% of 4.20, or 0.66(3.93) + 0.34(4.20) ≈ 4.02. So the 420-yard hole really has a par of 4.02.
In the above example, missing the fairway cost our average golfer 0.27 strokes, slightly over a quarter of a stroke. The difference between the expected score from the fairway and the expected score from the rough is about 0.25 for most distances. To be sure, missing a fairway does hurt golfers’ scores. However, it rarely pays to give up distance to increase accuracy, as you’ll see in what follows.
The calculation of average score illustrated above can be done for other distances, as shown in figure 10.4. The calculated average score gets consistently higher as the length of the hole increases. If my only interest here was in showing that the pros’ performances on par 4s depend on the length of the hole, I would simply compute their average scores on par 4s of different lengths. As shown in table 10.2, the match is quite close for most distances. There are consistent underestimates, starting at 420 yards and escalating when the hole length exceeds 480 yards, that are somewhat problematic. My predicted scores do not anticipate penalty strokes or major problems in the trees, and unanticipated disasters may explain the discrepancies in scores.
Figure 10.4 Estimated average score of par 4s of different lengths, 2007
The reason for computing the average scores is to set up the rating system that will be introduced in chapter 11. This rating system depends on our ability to predict an average score from different positions on the golf course.
To complete our analysis of tee shots, we examine par-3 tee shots next. One of the opportunities that the par-3 environment gives us is the chance to estimate how much landing in a bunker costs the pros. Not surprisingly, the distance to the hole after the tee shot depends on the length of the hole. Figure 10.5 illustrates the dependence. Figure 10.5 should remind you of figure 9.3 for approach shot distances from the fairway. Apparently, whatever sidehill, uphill, and scruffy lies the pros find in the fairway do not degrade their accuracy much from that enjoyed from the par-3 tee box.
Distance from the hole is only part of the story. As in real estate, location is also key. Of the 64,324 par 3s played in 2008, the pros hit the green 39,561 times (61.5%). The average score when they hit the green was 2.87. They landed in bunkers 6,411 times (10%), with an average score of 3.52. From tee shots that finished in the rough (6,841 of them, or 10.6%), the average score was 3.46. The pros actually scored better from lies classified as rough than from bunkers. From this evidence, the difference between hitting the green and dumping their tee shots into a bunker was more than half a stroke. The pros found water 983 times (1.5%), averaging 4.76 strokes. (This works out to a penalty stroke plus about a quarter of a stroke more than if they found the rough or a bunker.) From the fringe, the pros averaged 3.06 strokes, while from the fairway, they averaged 3.24 strokes. Corresponding statistics for par 4s show average scores of 3.01 from the green, 3.94 from the fairway, 4.01 from the intermediate rough, 4.27 from the primary rough, and 4.35 from bunkers. The average score for drives into fairway bunkers is worse than the average score from the primary rough. The average score for driving into greenside bunkers is 3.73, which is again more than half a stroke worse than being on the green. Bunkers cause more of a problem than I had realized.
Table 10.2 Predicted average scores and actual average scores for par-4 holes of various lengths, 2007
Figure 10.5 Average distance to hole after par-3 tee shots, 2004–2008
Turning to player statistics, we start with par-3 tee shots. Many of the issues that apply to approach shot statistics apply here. In particular, compiling average distances to the hole for all par 3s played can unfairly penalize players who play tournaments with longer par 3s. A more accurate measure is to use average distances better or worse than Tour average. Table 10.3 shows the top tens for 2007, 2008, and 2009.
Table 10.3 Top ten average par-3 tee-shot distances, compared to Tour averages, 2007–2009
On average, in 2008 Tiger’s par-3 tee shots finished 6 feet closer to the hole than the Tour average from the same distance. In 2007, Tiger distanced himself from the Tour by an even wider margin. As with irons from the fairway, however, Tiger’s performance slipped in 2009. If you’re wondering, Tiger’s average par-3 tee shot in 2008 finished 32.2 feet from the hole, ranking 22nd on tour. While pure distance averages over all par 3s give little useful information, an interesting pattern emerges when the averages are broken down by distance. Table 10.4 shows top ten averages for par-3 tee shots over the 5-year span of 2004–2008 for the most common distances. From 180–210 yards, Tiger ranked 15th at 34.5 feet. From less than 150 yards, Tiger ranked 201st at 30.1 feet. Statistically, then, Tiger averaged about the same distance to the hole from all distances of 210 yards or less. Very odd.
Golf fans have always had a special fascination with long hitters. The special aura that goes with the long ball has been enhanced over the years by the characters who let it rip. The best example of the breed is John Daly.
John Daly became an instant hero when he dominated the 1991 PGA Championship. His victory was a true Cinderella story, and his good-old-boy charm magnifies his popularity.8 But John Daly’s legend is built on power—a huge swing that produces outrageous height and distance. Commenting on Daly’s drives, Ian Baker-Finch said, “I don’t go that far on my holidays,” and Gay Brewer exclaimed, “Man, I can’t even point that far.”9 Age and declining fitness have helped displace Daly as king of the long drives.
Table 10.4 Top ten average par-3 tee-shot distances, 2004–2008
The driving distances given in table 10.5 are averages for the special driving holes on which the PGA predicts most players will use a driver. Five-year driving averages for both distance and accuracy are given in table 10.6.
For driving distance, there is little mystery in the fact that Watson and Holmes dominate. Quite elementary. In driving accuracy, Phil Mickelson was 180th at 58.5%, Tiger Woods was 191st at 57.5%, and the only two under 50% were David Duval (48.8%) and John Daly (49.0%). Top tens in driving accuracy for 2008 and 2007 are given in Appendix table A10.1. The top tens for distance and accuracy provide some perspective on the relative importance of these two measures. While accuracy is clearly important (we have seen that a missed fairway can raise the average score by a quarter of a stroke), you will find more major tournament winners on the long-driving lists than on the most-accurate lists.
Table 10.5 Top ten driving distances, 2007–2009
Table 10.6 Top ten driving distances and driving accuracy (fairways hit), 2004–2008
Figure 10.6 Accuracy (percentage of fairways hit) versus distance, 2008
The scatter plot in figure 10.6 shows that most players excel at distance or accuracy but not at both. The correlation between distance and accuracy for 2008 averages is −0.62.
We will return to the issue of distance versus accuracy when we focus on rating players off the tee.
When players miss the fairway, do they consistently miss in particular places? The following quirky study was prompted by one of those odd coincidences of life. Shortly after reading about Ben Hogan reconstructing his swing to eliminate the left side of the golf course, I was told that Tiger Woods never misses to the left.10 This struck me as an interesting thread to pursue. ShotLink records whether a drive has gone into the left rough or the right rough. It is then a simple matter to find the percentage of drives that miss on the left compared to the drives that miss on the right. There is no “good” value here—just some information about driving tendencies.
Of the top 230 golfers in my data set for 2008, 73 of them found the left rough more often than the right rough (table 10.7). David Duval, a right-hander, and Richard Green, a left-hander, lead the way with about two-thirds of their wayward drives to the left. Of Phil Mickelson’s missed fairways, 53.8% of them were to the left. Only 32 of the 230 golfers had over 55% of their missed drives to the left. Three of these golfers (O’Hern, Stankowski, and Green) were in the top ten in 2007, when Bob Tway led the tour with 66.5% of his missed fairways to the left.
Fully 151 of the 230 golfers missed more often to the right than to the left. I do not know the percentage of golfers who play fades regularly, but either the right-handed golfers’ fades are slipping out of the fairway or their hooks are not hooking enough. This would be the case with Rocco Mediate, who almost always hits a hook: in 2008, over 58% of Rocco’s missed fairways were missed to the right. For 77 of the 230 golfers, more than 55% of missed fairways were to the right. As advertised, Tiger Woods missed to the right more than twice as often as he missed to the left. (In 2007, 62.4% of his missed fairways were to the right.)
Table 10.7 Top ten players missing fairways in the left rough, 2008
This might be an interesting statistic to study over time, to see if the skew to the right for the Tour golfers is consistent or just a one-year statistical oddity. If it is consistent, the cause could be swing patterns (as with Ben Hogan), course design (fairways sloping right, the hole being a dogleg left, or trouble to the left), or some other cause that I have not thought of. Regardless, many more players miss to the right than to the left.
One lesson you might learn from this is that, if you want to get close to a particular golfer at a tournament, join the gallery to the right of the fairway. Your favorite golfer might be headed your way.
