He’s a good player. Of course, when I was his age,
I won the U.S. Open and PGA Championship.
—Gene Sarazen, talking about Tiger Woods at age 20
Who was the best golfer in the world in the decade of the 2000s? In most sports, “Who is the best?” is a provocative question that can carry a bar conversation or an hour of talk radio. In golf, however, there was no doubt who was the best from 2004 to 2009. The rating system developed in this chapter will clarify how dominant Tiger was in the 2000s, while providing a way to answer future questions of who the best golfer is.*
Much of mathematics is devoted to determining a “best” answer, whether it is the best design of a golf club, the best investment strategy for a company, or the best golfer. What can distinguish mathematics from other methods of answering these questions is that mathematicians start with clearly defined objectives and logically progress to the answer. There can be disagreement as to whether the objectives are well chosen, but the objectives are the only sources of hidden agendas or biases that influence the answer. The task in this chapter, then, is to precisely define what we might mean by “best” golfer and to follow through to see how these objectives identify Tiger as the best in the 2000s. This, of course, is not exactly the way the scientific method is supposed to work. It is poor form to decide in advance what the answer is. Instead, the evidence is supposed to speak for itself, with the rational mathematician merely recording the answer.
Many of the issues that arise when thinking about a player rating system also apply to determining the best putter. This narrower question has the advantage of not having an obvious answer. A raft of putting statistics was dumped on you in chapter 7, and there are many more rafts of similar statistics afloat. What is a reasonable way to combine all of these numbers into a single rating?
One way would be to take the most important of the statistics and base rankings on that. Putts per green in regulation predicts scoring the best, so we could use only that. The problem is that putts per green in regulation does depend on ball-striking ability. If you always aim at the pin and either miss the green or stick it tight, your putts per green in regulation score might be lower than a better putter who is constantly having to two-putt from 40 feet. An important objective in defining a putter rating is to try to avoid dependence on other skills, such as iron play and chipping.
The percentages of putts made from different distances are important statistics, and a player who rates highly from every distance must be a good putter. But how can this information be combined into a single number? We could add rankings like the PGA does for driving distance and accuracy. However, this assumes that all of the statistics that we add together have equal importance. Plus, in many cases, the difference between ranking 40th and ranking 80th is only one or two made putts. More important, along with listing the top ten, it would be nice to be able to say something about how much better the number-one player is than the number-ten player. The best way to guarantee that the final rating is meaningful is to insist that each statistic that enters into the calculation relates to the same, meaningful quantity. So, what is the most meaningful quantity in golf?
When he was contending to be the best player in the world, Nick Faldo trundled down to Fort Worth to visit the great Ben Hogan. Over lunch, Faldo asked Hogan what he needed to do to win the U.S. Open.
“Shoot the lowest score, Nick,” said Hogan.1
As with many of Hogan’s rare utterances, this has a pure logical base that turns out to have profound consequences. The question that Bill James and others constantly pondered about baseball statistics was how they relate to winning. Obviously, a home run is better than a single, but how much better? The answer, they insisted, had to come in some fashion from an analysis of how games are won. Similarly, to do anything with the avalanche of golf statistics that is now available, we must have a good criterion for evaluating the statistics. Any reasonable statistic (cue the Hogan tape) must relate directly to the golfer’s score.
For example, in 2007 Tiger Woods made 12 out of 27 (44%) of his putts between 7 and 8 feet. We saw in chapter 7 that this is below the Tour average. In fact, that percentage placed Tiger at 181 out of 230 golfers for putts between 7 and 8 feet. But what does this really mean? You could talk about Tiger not being that great a putter or about 27 putts not being enough from which to draw a conclusion2 or any number of provocative discussion points. The bottom line, however, is simply that Tiger took more strokes than he should have when putting from 8 feet. The exact number of strokes can be measured. The Tour average from 8 feet was 53.1%. If Tiger had made that average percentage, he would have made 0.531 × 27 ≈ 14.3 of his putts. Since Tiger made only 12 putts, he lost 2.3 strokes when putting from 8 feet.
We can do this from every distance. However, there is a crucial element missing from this line of logic. From longer distances, the important statistic is not so much percentage of putts made but total putts needed to hole out. A player who makes 20% of his 20-foot putts is not really ahead of the game if he three-putts 25% of the time from 20 feet.
Here is my system. For each distance up to 100 feet, compile the Tour’s average number of putts taken when the first putt is of that distance. Then, for each player, take every green for the entire season and compare the number of putts on that hole to the Tour average. From 60 feet, the Tour average is 2.248 putts. If a player starts 60 feet away and two-putts, he gets credit for being 0.248 strokes better than the Tour average on that hole.3 Add them all up, and you have the total number of strokes saved (or lost) by that player when putting. If you divide by the number of holes played and multiply by 18, you get the number of strokes saved or lost per round.
Table 11.1 Sample calculation of putts better/worse than average
A sample calculation is given in table 11.1. This golfer starts by making a 13-foot putt on the 1st hole. I don’t know or care whether it was for birdie, par, or bogey. A one-putt from 13 feet is 0.725 strokes better than the Tour average of 1.725 putts from that distance. On the next four holes, the golfer two-putts from a variety of distances. From 35 and 41 feet, a two-putt is better than the Tour average, and the difference is added to the golfer’s running total. From 7 and 22 feet, a two-putt is worse than the Tour average, so the difference is subtracted from the golfer’s total. Through five holes, the golfer is 0.133 putts better than average, for an average of 0.0266 putts better than average per hole. Multiplied by 18, the golfer’s average is 0.0266 * 18 = 0.479 strokes per round better than average. The number that would appear in table 11.2 for this golfer is 0.48.
Before looking at the ratings, think for a moment about what is involved and not involved in each player’s putting rating. Most important, the rating does not depend directly on how the player gets to the green. Regardless of whether it was a green in regulation or a great shot or a conservative shot, the rating shows how the player putted on each hole compared to the Tour average. Therefore, the rating has isolated the putting performance of each player for the year. The main element that the rating does not control for is difficulty of greens. A player who plays only the most difficult courses (Tiger, perhaps) might be underrated here because short putts could be more difficult to make, and two-putts from long distances could be much more difficult.4
Table 11.2 Putting efficiency ratings: top ten putts per round better than Tour average from the same distances, 2007–2009
Table 11.2 shows the top ten in putting efficiency from 2007 to 2009. (Ratings for 2004 to 2006 are in Appendix A, table A11.1.) The ratings represent strokes per round. This means that in 2009 Luke Donald took one putt per round less than an average Tour putter would have taken from the distances that Donald faced. Although he never rated as the best putter for the year, Tiger Woods made the top ten in five out of six years. (He was ranked 17 in 2006.) Tiger is the only golfer in the top twenty in all six years. Brad Faxon, Ben Crane, and Stewart Cink made the top ten three out of six years. Table 11.3 shows putting efficiency ratings for the six-year period from 2004 through 2009. Brad Faxon, Aaron Baddeley, and Steve Stricker are often cited as the best putters on the Tour. In 2005, Stewart Cink (no. 8) fell just short of breaking Faxon’s record of 362 straight holes without a three-putt when one of the slick greens at Pinehurst (North Carolina) ended his streak at 351 consecutive holes.
The logic behind the putter ratings can be followed back onto the fairway and then all the way back to the tee. For example, how could we isolate chipping ability? The ShotLink data might tell us that players X and Y were both in the fairway 90 feet from the pin on a given hole. Player X chipped up 11 feet from the hole, while player Y got a foot closer. One way to rate these shots is to say that player Y did 1 foot better, but our goal is to relate everything to scoring. Do you really have an advantage putting from 10 feet over putting from 11 feet? There is a very slight advantage, and comparing the Tour putting averages from these two distances shows us exactly how much of one.
There is a larger problem with using proximity of shot as the criterion. Suppose player X hits two chips to within 12 feet. Tour average from 12 feet is 32%, so there is no guarantee that he makes either putt. Player Y averages 12 feet on two chips, but one of them is 4 feet away, and the other is 20 feet away. The Tour average from 4 feet is over 91%, so he almost surely has one up-and-down. Since the Tour average from 20 feet is about 16%, he has a much better shot at two saves (about 14%) than does player X (about 10%).
Table 11.3 Top ten putts per round better than average, 2004–2009
The solution is to convert each shot to a score. For example, a chip to within 12 feet is assigned a score of 1.69, equal to the average number of putts from that distance. A chip to within 4 feet receives a score of 1.11, the average number of putts from 4 feet. So, you can see that the importance of chipping closer is not having a better view of the hole; it is having a better chance of making the putt. In this case, 4 feet is not just 8 feet better than 12 feet; it is 0.58 strokes better.
The ratings in table 11.4 are not necessarily for chipping. Shot-Link does not record the type of shot played (chip, pitch, flop, putt). The ratings average the results of playing shots from between 4 and 50 yards from the hole, from the fairway. For each player, each shot from this range is compared to the Tour average from that distance and lie. As discussed above, the comparison is not about feet from the hole; it is about how many putts would be needed from that distance. A positive value means better than average (fewer strokes needed), and a negative value means worse than average. All of a player’s shot values are added together; this number is then divided by the total number of shots, producing the rating for that player.
Phil Mickelson is known for his imagination and spectacular recoveries. “Lefty” ranks second for the six year span of 2004 through 2009. In 2007, his average shot from the fairway from 4–50 yards out finished in a position that was 0.136 strokes better than the position for the average Tour player hitting from the same distance. By making eight such shots, Phil would save a full stroke compared to the average Tour player, assuming that they both are average putters. The six-year top ten would make a very strong leader board.
Table 11.5 shows the corresponding ratings for shots from the rough from 4–50 yards. These are the spectacular flop shots and delicate pitches from gnarly rough that the pros execute so well. Chris Riley has the best six-year average. His average shot from the rough from 4–50 yards out finished in a position that was 0.071 strokes better than the position for the average Tour shot from the same distance. Given 14 such shots, Riley would save a full stroke compared to the average Tour player.
Table 11.4 Strokes per shot better than Tour average, 4–50 yards, from fairway
Table 11.5 Strokes per shot better than Tour average, 4–50 yards, from rough
The same procedure can be implemented for any range of distances. The results are not dramatically different from the tables of average distances recorded in chapter 10. The advantage of the new calculations is that they all use strokes as the unit of measurement. This allows me to use larger distance ranges. For example, suppose that Phil averaged 2 feet closer to the hole than Tiger from one distance, and Tiger averaged 2 feet closer to the hole than Phil from a different distance. Are they even? Not necessarily. An improvement from 6 feet away to 4 feet away gives a large putting advantage, while an improvement from 46 feet away to 44 feet away hardly makes a difference. By contrast, an improvement of 0.1 strokes is the same from any distance. For the rankings in table 11.6, I grouped all of the distances that might correspond to full swings with irons into a 50- to 200-yard category. The results from the fairway are given first, then the results from the rough.
At the risk of being redundant, I will give another example of what the above statistics measure. Let’s say that Tiger is in the fairway 146 yards from the pin. ShotLink recorded 1,098 shots from this distance in 2007. On average, these 1,098 approach shots finished 25.3 feet from the hole. In turn, from 25–26 feet the Tour averaged 1.93 putts. Therefore, an average player would take 2.93 strokes from Tiger’s position. If Tiger were to hit his shot 6 feet from the hole, we would use the Tour putting average from 6 feet of 1.32 putts and conclude that Tiger’s iron was worth 2.32 strokes. Tiger’s shot saved him 2.93 − 2.32 = 0.61 strokes from the Tour average. By comparison, a conservative play to 35 feet away would be worse than average. Since the Tour averaged 2.02 putts from 35 feet, Tiger’s iron was worth 3.02 strokes, and this shot cost him 2.92 − 3.02 = −0.10 strokes. For the two shots, Tiger is ahead by 0.61 + (− 0.10) = 0.51 strokes, an average of 
strokes per shot. The rating shows that for all of Tiger’s fairway shots (2004 through 2009) from between 50 and 200 yards, his average was 0.0575 strokes better per shot than the Tour average. Given 17 of these shots, Tiger would save a full stroke over the average PGA golfer. For all of Tiger’s shots from the rough (2004 through 2009) from 50 to 200 yards, his average was 0.0431 strokes better per shot than the Tour average.
Table 11.6 Strokes per shot better than Tour average, 50–200 yards, from fairway and rough
Choosing the distance range 200–250 yards from the fairway, we get the results in table 11.7. Many of these shots would be second shots to par 5 greens.
One interesting consequence of tying all ratings to strokes is that it becomes meaningful to compare different categories. For example, is it more important to be strong inside of 50 yards or outside of 200 yards? While the ratings do not fully answer the question, some useful calculations can be made. For instance, in 2009 Bob Estes saved 0.139 strokes per shot inside of 50 yards. If he had had six such shots in a round, he would have saved 0.834 strokes for the round. If Robert Garrigus had had six shots in the 200–250 yard range from the fairway in a 2009 round, he would have saved 6 × 0.093 ≈ 0.56 strokes for the round. Estes would have saved more with his short game than Garrigus with his long game. This analysis, of course, is valid only if both players were hitting six of these shots in a round. However, it does give you a basis on which to make comparisons.
For the most part, the top ten in each category of iron shot have similar strokes-saved ratings. There is not an obvious bias toward either the short or the long game. As well, the numbers quoted above for a round are slightly less than but similar to the strokes saved in a round by the best putters. If Tim Clark had had 12 shots from the fairway between 50 and 200 yards in a round in 2009, he would have saved 12 × 0.067 = 0.804 strokes for the round, more than all but the top four putters in the putting efficiency ratings in 2009.
Table 11.7 Strokes per shot better than Tour average, 200–250 yards, from fairway
Table 11.8 Strokes per tee shot better than Tour average, par 3s only
The same system can be applied to ratings for bunker shots. These are given in Appendix A, table A11.2. Table 11.8 shows the ratings for par 3 tee shots.
So far, we have analyzed putts and almost every shot with which the golfer would expect to hit the green. This leaves tee shots on par 4s and par 5s as the only common shots for which we have not accounted. These tee shots present a new challenge for relating the quality of the shot to strokes saved or lost.
The concept for rating tee shots on par 4s and par 5s is the same as for the various shots discussed above. Take the position the golfer is in: in this case, it’s the tee box for a hole of a certain length. From the ShotLink data for the Tour, compute the average score that the average pro would make from that position. We did this computation for par 4s in chapter 10 for holes of different lengths. Then look at the drive that the golfer hits and compute the average score from where the ball finishes. Comparing the two averages, we get the number of strokes better than or worse than average for that drive. Add up all of the net strokes for all of the golfer’s drives and divide by the number of drives.
To give one example, suppose that a golfer is on the tee for a par 4 hole of length 420 yards. The average score for that position is based on an average drive length of 274 yards and an average of 66% for drives landing in the fairway. As computed in chapter 10, this leads to an average score of 4.02. Suppose the golfer smokes a 300-yard drive in the fairway. From 120 yards out in the fairway, the average Tour approach shot lands 20.6 feet from the hole; from there, the pros average 1.87 putts. Following this drive, then, average shots will produce an average score of 3.87. The drive is therefore 4.02 − 3.87 = 0.15 strokes better than average.
Table 11.9 shows top tens for driving on par 4s. If you glance down the lists, you may be struck by how they are dominated by the long hitters. However, the system is not intentionally biased. The ratings show us that, from a typical Dustin Johnson drive on a par 4 in 2009, the average player would score 0.058 strokes better than he would from his own average drive. If the long hitters dominate the list, that is good evidence that distance is more important than accuracy. This is well known to the top golfers, and it explains why the modern game is all about bombing away from the tee. It’s fun, and it also pays off with better scores.
The other important fact to note about table 11.9 is that the stroke ratings are not very high. That is, nobody other than Dustin Johnson was gaining even a half stroke per round on the average golfer based on driving ability. This supports the findings in chapter 6 that driving stats do not correlate highly with scoring.
Table 11.9 Strokes per tee shot better than Tour average, par 4s only
The ratings for par 5 drives are computed in a similar way, except that they are based on a hole length of 520 yards. All of a player’s drives on par 5s are applied against this standardized hole length and compared to the average Tour drive on par 5s. The results are given in Appendix A, table A11.3. and are similar to the driving stats for par 4s.
The original goal in this chapter was to identify the best player on the PGA Tour. What we have accomplished so far is to evaluate individual skills in terms of strokes better or worse than average. Using the common measurement unit of strokes, all we need to do to get an overall rating is some simple arithmetic. As an example, let’s take Tiger Woods, who (surprise!) ranked number one overall for the 2009 season.
Tiger tees off on all 18 holes. On average, eleven of these are par 4s, four are par 3s, and three are par 5s. For 2009, his ranking of 0.0145 strokes better than average for par 4 tee shots is multiplied by 11, his rating of 0.0531 strokes better than average for par 3 tee shots is multiplied by 4, and his rating of 0.0932 strokes better than average for par 5 tee shots is multiplied by 3. Adding these values, Tiger is
11(0.0145) + 4(0.0531) + 3(0.0932) = 0.65
strokes better than average on tee shots. Fourteen of the holes are not par 3s; let’s assume that on ten of these holes an approach shot between 50 and 200 yards from the fairway is required, and that the other four require a shot from the rough from 50–200 yards.5 Let’s also assume that six shots between 4 and 50 yards are required, with three from the fairway and three from the rough. These might be third shots on par 5s or recovery shots on par 4s. Tiger’s 2009 ratings from 50–200 yards are 0.0419 strokes better than average from the fairway and 0.0109 strokes worse than average from the rough, while from 4–50 yards he rates 0.1271 strokes better than average from the fairway and 0.0401 strokes better than average from the rough. Add in one sand shot (rating 0.0597) and two shots on par 5s between 200 and 250 yards (rating 0.0527), and you have an overall approach shot rating of 10(0.0419) + 4(− 0.0109) + 3(0.1271) + 3(0.0401) + 0.0597 + 2(0.0527) = 1.08 strokes better than average. His putting rating for a full round is 0.99 strokes better than average. Adding this all together, for a round consisting of the shots described above, Tiger’s expected score is 0.65 + 1.08 + 0.99 = 2.72 strokes better than average (rounded to two decimal places). Notice that in the one category in which Tiger rated worse than average, the rating is negative and points are deducted from his score.
Table 11.10 Strokes per round better than Tour average, all shots considered
This calculation is done for each of the players. The top ten for the 2004 through 2009 seasons are shown in table 11.10. Second tens for 2007 through 2009 are given in Appendix A, table A11.4.
Recall again that these ratings are based on ShotLink statistics, which only cover PGA Tour tournaments and exclude the four major tournaments and other tours. The rating is for one round of golf. So, the ratings say that in 2009 Tiger was 1.2 strokes per round better than anybody else on tour. In 2008, he was 1.29 strokes per round (more than 5 strokes for a tournament) better than anyone else. In each of the years from 2004 through 2009, Tiger rated more than 2 strokes better than average. He is the only golfer to reach the 2-stroke rating level. His dominance is spectacular. A brief analysis of the ratings for a small number of golfers is given in Appendix B. You will notice that Tiger is unusual in that in some years he rates as better than average in every category considered. Most golfers have definite strengths and weaknesses, at least according to the numbers.
We humans have a love-hate relationship with numbers. While many of us can spend hours absorbed in the types of statistics detailed in this chapter (and may find ourselves in full command of Tiger’s career statistics while stumbling when asked how many years we have been married), we also have a distrust of the avalanche of numbers that threaten to take away the mystique of the game. When a crucial putt slides in the side door, we tend to talk more about the golfer’s strong will and determination and less about luck. Mathematicians as a group fall squarely on both sides of this issue. The profession is devoted to looking for patterns, many of which are numerical in nature. As descendants of Pythagoras and Galileo, we think that the world can be quantified. At the same time, mathematicians have more experience than most with various forms of “numerology” in which numerical coincidences are falsely given great significance. An essential part of a good mathematical analysis is an evaluation of the likelihood of errors in the analysis.
So, I am not going to point to Vijay Singh’s 2008 putting rating of −0.41 and say that he is a poor putter. For one thing, I have a healthy respect for two of Vijay’s other statistics: at 6 foot 3 and 208 pounds, he is a lot bigger than I am. More importantly, the putter rating is only for 2008 and is relative to the other players on the PGA Tour. This is a high standard. For his part, Singh has the right attitude. After winning the 2008 FedEx Cup, he said in an interview that, “I believed in myself that I’m the best putter.”6
There are two main criticisms I have of the numbers presented here. One is the lack of consideration of course difficulty. Putting statistics and all of the other stats we’ve looked at can be affected by course setups, weather conditions, and other factors. Presumably, this means that golfers who tend to play in the toughest events (Tiger, Phil, most of the non-PGA–Tour players) are underrated in this rating system. The second criticism involves location. The system I have used so far is fairly coarse in that I have not taken into account finely detailed aspects of position. For example, when a shot finishes 30 feet from the hole, I use that distance in the calculation but don’t factor in whether the ball is actually on the green. A better system would account for whether the ball is on the green or in a bunker. This type of fine-tuning is high on my to-do list.
The end-of-chapter discussion that follows is a first look at possible corrections for course difficulty. It is essentially a case study to assess how important such corrections might be.
The basic question is whether there is evidence that some courses play harder than other courses. This is not as silly a question as you might think. Scores are much higher in some tournaments than in others. Isn’t that evidence enough? Actually, no. My real question is whether the course difficulty would affect the statistics that I’ve used in my rating system. If a course is just brutally long, then the pros will be hitting approach shots from outrageous distances and will score poorly. However, my ratings are based on comparing shots of the same distance. This will not hurt anybody’s rating. Still, it would be naive to expect that all courses putt the same. Some greens are trickier to read, some greens are contoured more, and some greens are in better shape than others. Is there a way to measure the effects of difficult greens and adjust for them?
As a first attempt at assessing how strong the effects are, I computed a putting rating for each tournament in 2007 and 2008. That is, I treated each tournament as a golfer and ran all of the putting outcomes through the putting efficiency formula. The results were more dramatic than I was expecting. In 2008, all four tournaments in California rated as harder to putt than average. Six of the seven Florida tournaments rated as easier to putt than average. Three of the four Texas tournaments rated as easier to putt than average. Two-thirds of the tournaments showed the same effect (harder or easier) in 2007 and 2008. My conclusion is that there are some stable, measurable putting effects that should be incorporated into a good rating system for putters. In fact, a group of MIT professors has done just that. Douglas Fearing, Jason Acimovic, and Stephen Graves created a statistic called “putts gained per round” using ShotLink data.7 Their approach is similar to what I have done, but they corrected for course difficulty.
My computations indicate that Pebble Beach is the most difficult course for putting, rating 1.0 and 0.7 strokes per round harder than average in 2007 and 2008, respectively. In both years, Riviera rated 0.4 strokes per round harder to putt than average. Torrey Pines rated 0.3 and 0.5 strokes per round harder than average in 2007 and 2008, respectively. Note that all three of these courses are in California. I was curious about what was common to the courses that rated as difficult. Any course can be made impossible to putt; we have all suffered through “greenskeeper’s revenge” days where the pins are all placed in the worst locations imaginable. But the fact that the difficulty ratings seemed to depend on geography made me suspicious that the type of grass might have something to do with it. Although I am not at all knowledgeable about how different strains of grass might affect putting, it is interesting that Pebble Beach, Riviera, and Torrey Pines list poa annua as a primary grass used on the greens. In fact, five of the six courses that list poa annua for greens rate harder than average in both 2007 and 2008.
One hard course that does not feature poa annua greens is the Plantation Course at Kapalua, Hawaii. The rating was 1.2 strokes harder than average in 2007 and 0.3 strokes harder than average in 2008. The Plantation Course is often buffeted by high winds, which can make putting a nightmare. Without having weather reports, I would suspect that wind played a role in the very large difference in the difficulty ratings for the two years. Also, the season-opening tournament played at Kapalua has a small field, which could lead to large deviations in statistics.
The easiest course in the ratings was the TPC Deere Run, rating 0.4 and 0.5 strokes per round easier than average in 2008 and 2007, respectively. The largest single-tournament effect on the easy side was the 2007 Colonial, at 0.7 strokes per round easier than average. The 2008 rating dropped to 0.2 strokes easier than average. Both Deere Run and Colonial have bent grass greens. Of the 23 courses listed as having bent greens in 2008, 15 of them rated as easier than average to putt.
Further studies into the effects of physical course characteristics on putting and all other shots are needed. The rating system presented here is a skeleton that will be fleshed out as the analysis of golf statistics progresses.
