You create your own luck by the way you play. There is no such thing as bad luck.
—Greg Norman
Greg Norman was in control of the 1986 PGA Championship. He and Bob Tway were tied for the lead on the 72nd hole, in what amounted to a sudden-death playoff. Tway had played his second shot on the par 4, gouging a 9-iron out of heavy rough and short-siding himself in a bunker. Norman’s approach shot spun back into the fringe 25 feet below the hole, in position for an easy par that could win the tournament. And then ... Tway’s bunker shot came out cleanly but appeared to be headed well past the hole until it rattled the pin and dropped in the hole for a miraculous birdie that Norman could not match. Unbelievably, this scenario repeated itself in the very next major, when Larry Mize holed a 140-foot pitch in a playoff with Norman to win the 1987 Masters.1 While Norman never did win the Masters, “the Shark” did win the British Open twice; he also held the 54-hole lead and finished third at age 53 in the 2008 British Open.
In this chapter, we will look at PGA statistics for various shots from inside 50 yards. We will see whether, 20 years later, Bob Tway is unusually skilled from bunkers and Larry Mize is a wizard with chips.
For official PGA statistics a putt is a stroke taken from somewhere on the green. Often, strokes are taken with a putter from just off the green, but these are not officially classified as putts. Unfortunately, the ShotLink data does not record the club used on a shot, so we do not know whether or not a putter was used from just off the green. Figure 8.1a shows the average number of strokes needed to hole out when starting in the fringe from 12 to 20 feet, along with the curve y = 0.88 + 0.337 ln (x − 2). Recall that this curve closely matches the average number of putts from different distances. The performance from the fringe is noticeably worse than from the green, where the data closely match the curve.
Figure 8.1b shows the average number of strokes from the fringe from 80 to 100 feet, along with the curve for putting averages. From the longer distances, the fringe performance seems to be very close to the averages achieved from the green. A possible conclusion is that hitting from the fringe decreases the pros’ chance of holing a short shot but does not strongly affect their ability to lag close to the hole from longer distances.
Figure 8.1 Average number of strokes needed when starting in the fringe from distances ranging from (a) 3–20 feet and (b) 80–100 feet. The points represent averages from the fringe, the curve fits averages for putts.
The PGA Tour keeps a scrambling statistic to measure players’ abilities to save par. The general success rate for scrambling of any sort from 2004 to 2008 was 56%. For example, out of the 120,882 times that a green was missed in 2007, the golfers made par or better 67,560 times, or about 56% of the time. Even the pros sometimes have to get up-and-down (one shot “up” on the green, one shot “down” into the hole) for bogey, so this percentage underestimates the actual success rate for getting up-and-down.
While scrambling rates may be interesting, the focus here is on individual skills. In particular, I want to isolate chipping ability from putting ability. Unfortunately, the data set does not classify which of the many possible types of short shots (chip shot, pitch, flop shot, putt) the player attempted. We can analyze only the result of the shot. For shots taken from off the green, the goal is to get close to the hole; like many painfully obvious statements, this one has some ramifications. In particular, golfers often talk about getting up-and-down. However, an up-and-down consists of two distinct shots, the chip (or pitch or flop) and the putt. The purpose of the chip is to get close, and the purpose of the putt is to go in. So, it makes sense to measure putting by percentage made or average number of putts, while an appropriate measure for chip shots is remaining distance to the hole after the shot.
In 2008, 225 shots were hit from the fairway from distances between 49 and 50 yards from the hole. These shots stopped a total distance of 40,134 inches from the hole. This means that, from 50 yards out, the pros hit the ball an average of 
inches, or about 14.9 feet, from the hole. Figure 8.2 shows the results of analogous calculations for distances ranging from 4 yards to 50 yards. The points form a fairly smooth curve, looking like part of a downward parabola. The curve y = −0.0033(x − 3)2 + .43(x − 3) + 1.57 is superimposed on the data points.
Figure 8.2 Average distance to the hole after shots from the fairway, 4–50 yards
The curve illustrates the obvious rule that the closer you are to the hole, the closer you tend to hit your approach. The downward trend indicates that the decrease in accuracy lessens as you approach 50 yards. For example, the loss in accuracy changing from 10 to 20 yards out is 3.79 feet, which is almost twice the loss in accuracy backing up from 40 to 50 yards (1.98 feet). (See table 8.1.) A likely explanation is that the ability to spin the ball better from the longer distances partially compensates for the increased distance of the shot.
Figure 8.2 summarizes all shots from the given distances from the fairway in 2008. Obviously, the approaches will not be as accurate from other lies, such as the rough. To show how much missing the fairway costs pro golfers, table 8.1 lists averages from several distances for shots from the fairway, intermediate rough, and primary rough. The intermediate rough values tend to be much closer to the fairway values than to the primary rough values. Far more balls in the data set are marked as being in the primary rough than in the intermediate rough, so the intermediate rough data are spotty and, for this reason, mostly ignored in what follows.
Table 8.1 Differences in accuracy of approach shots from various distances and lies
The data do not account for trees at all. The very large values from the primary rough could be due partly to having to chip sideways to avoid trees. The evidence seems to show that being in the intermediate rough does not cost the pros much but that being in the primary rough is indeed a disadvantage. Figure 8.3 shows the averages from the rough for distances ranging from 4 yards to 50 yards, with the curve y = −0.0037(x − 3)2 + .59(x − 3) + 2.84 superimposed.
At first glance, figures 8.3 and 8.2 look very similar. They show the same characteristics of rising from left to right but gently curving along a downward parabola. The most important difference in the figures is the vertical scale. Shots from the rough in figure 8.3 end up farther from the hole than the equivalent shots from the fairway in figure 8.2. To make this comparison more obvious, the two sets of data are combined in figure 8.4.
Figure 8.3 Average distance to the hole after shots from the rough, 4–50 yards
Figure 8.4 Average distance to the hole after shots from the fairway and the rough, 4–50 yards
For casual golfers, sand bunkers are terrifying hazards to be avoided at all costs. The pros are so good from the sand that they often think of bunkers as safe havens from gnarly rough. How good are they? Out of 30,288 trips to greenside bunkers in 2007, the pros recorded 14,568 saves, a 48% recovery rate. Of course, a sand save can be as much a function of great putting as great bunker work. To isolate ability to hit out of the sand, we look at how close to the hole pros are getting their sand shots. For bunker shots that start between 20 feet and 150 feet from the pin, the average shot finished 12.35 feet from the hole. Figure 8.5 shows average distances from 20 to 150 feet, with the curve y = 0.0008(x − 20)2 + 0.035(x − 20) + 9.13 superimposed. Broken down by 30-foot distance ranges, table 8.2 shows the average distances to the hole after bunker shots of different lengths over the five-year period from 2004 to 2008.
Even from 100 feet, the pros are giving themselves a decent putt at par.
Do the pros do better out of the sand than out of the rough? The averages computed here cannot completely answer the question, but they do provide a starting point for discussion. For shots from both the rough and sand, smooth curves can be fit to the data. The two curves, shown in figure 8.6, demonstrate different characteristics. For sand shots, the parabola curves up, indicating that longer bunker shots are much harder than shorter shots. By contrast, the curve for shots from the rough is a downward parabola, indicating that longer shots from the rough are not that much harder. This should match your experience on the course. From 20 yards to 35 yards, the average distances are about the same, with the pros doing slightly better from the sand than from the rough.
Figure 8.5 Average distance to the hole in feet after shots from the bunker, 20–150 feet
Table 8.2 Average distances to the hole after bunker shots of various lengths, 2004–2008
Figure 8.6 Fitted curves for average distance to the hole from the rough and sand, 7–50 yards
Having seen Tour averages for various short-game statistics, we now want to see which individuals perform the best. Imagine a pro golfer needing to get up-and-down at a crucial moment in a tournament. Who would be most likely to get the ball close? See if your choice shows up in the tables that follow.
We start with the 2009 top tens for average approach distance for shots made from 4–50 yards, from both the fairway and rough. You may notice that nobody made both top tens. There is a problem with the statistic shown in table 8.3: averaging all of the shots from 4 to 50 yards could be unfair. If one player was always less than 10 yards away and another player was always more than 40 yards away, a comparison of the averages is misleading. Because the distance range of 4–50 yards is broad, we need a way to adjust for the actual distances faced by each player. The solution depends on a technique that I use repeatedly when determining player ratings. We have average approach distances for 4 yards, 5 yards, and so on. (Figure 8.2 shows these averages.) For each shot from the range 4–50 yards, compare the distance of the player’s approach to the Tour average from that distance. For example, an approach played to 6 feet from 10 yards away is 1.79 feet worse than the Tour average of 4.21. However, an approach played to 7 feet from 50 yards away is 7.55 feet better than the Tour average of 14.55. The second approach is farther from the hole but is a more impressive shot compared to the Tour average. For the two shots, the player would be −1.79 + 7.55 = 5.76 feet better than average. On average, the player is 
feet per shot better than average.
Table 8.3 Top ten average approach distances for shots made from 4–50 yards, 2009
In table 8.4, we see the top tens from the fairway and rough for 2009, compared to tour averages. Most of the names in the top tens are the same as in table 8.3, but considerable shuffling has occurred. Tiger Woods now heads the list from the fairway; the rating of 2.47 means that Tiger’s average shot from the fairway (4–50 yards away) finished 2.47 feet closer to the hole than the average Tour shot from the same distance. Because Woods moved up the list and Omar Uresti dropped down three slots, we can infer that Tiger played from longer distances throughout the year than did Omar. The top tens for 2006–2008 are given in Appendix A, tables A8.1a–A8.1c.
Table 8.4 Top ten average approach distances for shots from 4–50 yards, compared to Tour averages from the same distances, in 2009
One conclusion that can be drawn from the data in these tables is that there is little consistency from one year to the next. It is also interesting that the players who are strong from the rough are not necessarily the ones who make the top ten from the fairway. The need to accurately judge the effect of the lie in the rough makes it a different type of shot.
The same analysis can be applied to sand shots. Table 8.5 shows the top tens in bunker play for 2009, both in terms of raw distance and distance better than Tour average. There is not a large difference in the two rankings. Table 8.6 shows five-year totals, broken down by distance. About 55% of the bunker shots in 2004–2008 were taken from the 30–60 foot range, and 27% were taken from the 60–90 foot range. More lists are given in Appendix A, table A8.2. Based on the data in these tables, we conclude that Mark Wilson, Mike Donald, Mike Weir, Nick Price, Jeff Sluman, Omar Uresti, and Phil Mickelson were among the top bunker players in the 2000s.
Table 8.5 Top ten average distances from bunkers, raw and compared to Tour averages, 2009
Table 8.6 Top ten average approach distances for shots from bunkers, 2004–2008
In the introduction to the chapter, I referred to Greg Norman’s difficulties in the majors in 1986 and 1987. ShotLink data are not available for those years, so there are not valid conclusions to be drawn, but I thought it would be interesting to see where Larry Mize and Bob Tway ranked in the short-game stats. In 2006, from 4–50 yards in the fairway, Larry Mize ranked 68th out of 230 golfers, well above average. In 2007, Bob Tway ranked 190th out of 230 golfers in bunker play, averaging nearly 3 feet per shot farther from the hole than the average PGA golfer. Nevertheless, in 1986 he made the one that mattered the most.
The ShotLink data sets are stunningly detailed but not complete. For example, in the first round of the 2008 Barclays tournament, on the 5th hole, Steve Stricker faced a second shot of 2,312 inches (about 64 yards). The ball traveled a distance of 2,293 inches, coming to rest 59 inches (about 5 feet) from the hole. (In case you’re wondering, Stricker made the putt for birdie. The time of day was 4:56 p.m.)
Stricker’s shot finished 5 feet away, but was it long or short? Left or right? The data set does not record the answers, but the first question is easy enough to resolve. The ball started 2,312 inches from the hole and flew only 2,293 inches, so the fact that 2,293 < 2,312 tells us that the shot came up short. We can also determine how far off-line the shot was, but whether the shot missed left or right will remain unknown. Start with the fact that the ball stopped 59 inches from the hole. The ball must be on a circle of radius 59 inches centered at the hole. Similarly, the ball traveled 2,293 inches from its starting point, so the ball also lies on a circle of radius 2,293 inches centered at Steve Stricker. As seen in figure 8.7 (representing an overhead view of the hole with Stricker standing somewhere below the bottom of the page), the two circles intersect in two locations.
The ball must be located at one of the two intersection points in figure 8.7. The possible locations of the ball can be found using some algebra. Locate the center of the hole at (0, 0), and let the distance to the hole after the shot be H. An equation of the circle of radius H centered at the origin is x2 + y2 = H2. Place the golfer d inches away at (0, −d), and let B be the distance the ball travels in inches. An equation of the circle of radius B centered at (0, −d) is x2 + (y + d)2 = B2. To find the intersections of the circles, expand (y + d)2 and subtract one equation from the other. You should find that
Figure 8.7 Intersections of two circles, showing possible ball locations
and then 
. The y-value tells whether the shot is long (y > 0) or short (y < 0). We can’t distinguish whether x is positive or negative, so we can’t determine whether the ball is to the left or right. In what follows, we use both possibilities.
Figure 8.8 shows the computed stopping points for approximately 2,000 shots from the fairway from 10–20 yards out. The stopping points range from about 20 feet short to 20 feet long, with a fairly symmetric distribution. (The left-right symmetry is artificial, as I have plotted each point twice—once to the right and once to the left.) The shots were up to 8 feet off-line. Two features stand out to me in figure 8.8. First, many of the shots seem to have been exactly on-line. Far more worrisome are the bands of points branching out like parabolas. If you’ve read too many Knights Templar conspiracy novels lately, you might be trying to decipher the secret message embedded in the PGA Tour data sets. As we will see, the real message is to beware of data that are rounded off.
Figure 8.8 Computed locations of shots from the fairway, 10–20 yards away
In our example, Steve Stricker’s approach finished 59 inches from the hole. It is unlikely that the ball was exactly 59 inches away, but rounding off 58.7 inches to 59 inches seems harmless enough. In particular, rounding off to the inch couldn’t produce the gaps in figure 8.8, could it? After all, at y = 10, there are no points plotted for x-values between 0 inches and nearly 1 foot.
Surprisingly, the gaps in figure 8.8 are caused by rounding the data to the inch.2 To illustrate this, I took the data used to produce figure 8.8 and “unrounded” it by adding random decimals. The result in figure 8.9 shows that the gaps have been mostly filled in. The stopping points for shots from the fairway from 10–20 yards out should look like those in figure 8.9. Rounding off the location of each shot to the inch has the surprising effect of producing the bands seen in figure 8.8. Simple mathematical processes like rounding can have unexpected consequences.3
Figure 8.9 Locations of shots from the fairway, 10–20 yards away, with unrounded data
This technique yields some interesting conclusions, which are presented in the next chapter, where we look at longer approach shots.
