Unbelievable. I knew he’d make it.
—Rocco Mediate, after Tiger Woods made a 12-foot putt to tie him in the 2008 U.S. Open
The 2008 U.S. Open at Torrey Pines was one of the most dramatic tournaments ever. Tiger Woods, hampered by a torn anterior cruciate ligament and stress fractures of his tibia, made a 12-foot birdie putt on the 72nd hole to tie Rocco Mediate and force a playoff. In the playoff the next day, Tiger again birdied the 18th hole to tie Mediate and finally won in sudden death on the 91st hole. The video of Tiger’s tying putt on the 72nd hole is one of the most-played golf videos online, featuring replays from several angles and a wild celebration by Tiger and caddie Steve Williams.
One of the replays shows a close-up of the ball rolling and bouncing on its way to the hole, then catching the right edge and (barely) curling into the hole. This replay is a great illustration of the randomness involved in putting, which we first discussed in chapter 3. My favorite quotation about this putt is Rocco’s comment in the epigraph. “Unbelievable” and “I knew he’d make it” do not logically go together, but somehow they capture many aspects of the scene. A normal golfer would not make that putt, yet everybody expected Tiger to make it.
There were conflicting aspects about that putt for Rocco. If Tiger had missed, Rocco Mediate would have become a U.S. Open champion, yet the playoff gave him another day in the national spotlight, going one-on-one on national television with the best player of our time and matching him stroke for stroke. To his great credit, Rocco has consistently emphasized the thrill of the playoff rather than regret over his near miss.1
Thinking of all the dramatic putts that Tiger Woods has made over the years, the obvious question is whether he is the best putter on the Tour. We’ll get part of the answer in this chapter and a more complete one in chapter 11. Compiling ShotLink data can show us that, in 2007, Tiger made 12 out of 27 (44%) of his 7- to 8-foot putts. But here is the big question. How good is 44% from 8 feet? Is it better than most PGA players typically do, right at the Tour average, or well below average?
These and related questions are the focus of this chapter.
Looking at all putts on the 2007 PGA Tour (not including the four majors), it turns out that 165,026 putts were attempted from distances of 3 feet or less. A full 163,676 of these putts were made. This makes the Tour average of putts made from 3 feet or less 
or about 99%. From distances greater than 3 feet but less than or equal to 4 feet, the Tour made 30, 653 out of 33, 532 putts, or about 91%. Other percentages are given in table 7.1. Note that Tiger’s 44% from 8 feet is well below the tour average.
The Tour percentages in other years are similar. One way to summarize the data in table 7.1 is to break it into quartiles of percentages. We see that:
• At all distances beyond 5 feet, the percentage of putts made is less than 75%.
• At all distances beyond 8 feet, the percentage of putts made is less than 50%.
• At all distances beyond 14 feet, the percentage of putts made is less than 25%.
A scatter plot showing distance in feet on the horizontal axis and percentage of putts made on the vertical axis is given in figure 7.1. Up to about 20 feet, the data points line up nicely on a curve. The curve y = 148e−.122x matches the data fairly closely and is superimposed on the data points.2
Table 7.1 Percentage of putts made from different distances, 2007
Figure 7.1 Percentages of putts made from 3–20 feet in ShotLink tournaments in 2007
A different way of looking at the putting statistics is to record the distance of the golfer’s first putt and then compute the number of putts needed on the hole. For example, there were 12,833 times in 2007 that a PGA Tour player’s first putt was between 4 feet and 5 feet long. In these 12,833 instances players took a total of 15,614 putts. The average number of putts starting between 4 and 5 feet away is then 
putts. Figure 7.2 shows the scatter plot of data points for average number of putts from different distances, along with the curve y = 0.88 + 0.337 ln (x − 2). There is a milestone reached at 32 feet. Specifically, at all distances beyond 32 feet, the average is more than 2 putts.
Figure 7.3 shows the average number of putts for distances out to 100 feet, along with the curve y = 0.88 + 0.337 ln (x − 2). The figure illustrates a principle that we need to keep in mind in any discussion of statistical information. Starting at about 60 feet, the data points no longer follow a nice curve. The points seem to bounce above and below a curve running through the center. The reason for the random ups and downs of the data points is, in fact, randomness. Whereas there were 5,453 putts taken from 20 feet, there were only 8 taken from 100 feet. A lucky 1-putt or an unlucky 3-putt from 20 feet will not affect the average noticeably, but if one of the 8 putts from 100 feet goes in, the average can change dramatically. The sample size is critical.
Figure 7.2 Average number of putts needed from 3–20 feet in ShotLink tournaments in 2007
Figure 7.3 Average number of putts needed from 3–100 feet in ShotLink tournaments in 2007
Having looked at overall Tour averages, we now want to see which golfers make the most putts. The ShotLink data allow us to break the putting distances down to the inch. We will start with a broader range, looking at all putts of less than 10 feet. This, you may recall from chapter 6, was the distance range with the highest correlation to scoring. Table 7.2 shows the top 10 putters on the Tour from less than 10 feet, for each of the years from 2007 through 2009.
What can we learn from these lists? Amazingly, only one player, Stewart Cink, made the top ten in more than one year, finishing 8th in 2007 and 9th in 2009. This is an indication that the percentage of putts made from less than 10 feet is not a stable statistic. One of the studies that I would like to do (or see done by others) is to analyze more thoroughly how putting statistics vary over time. During a season, are some putters streaky, with four or five good tournaments followed by several bad ones? Are the ups and downs in putting efficiency predictable or apparently random?
Table 7.2 Top ten percentages of putts made from 0–10 feet, 2007–2009
Table 7.3 Top ten percentages of putts made from short distances, 2004–2009
Another way to parse the short putt data is to compile averages of made putts from smaller, 1-foot distance ranges. As you might expect, the results are even more variable than for the range of 0–10 feet shown in table 7.2. To try to get some meaningful information, I have averaged over multiple years to smooth out some of the year-to-year variability. Results are shown in table 7.3.
The fact that Tiger Woods and John Cook made all three lists means that they are among the best short putters in the game. Interestingly, Tiger does not necessarily jump out as a great putter if you look at year-by-year statistics. For example, from 2004 to 2008 his yearly rankings from 0–3 feet were 62, 1, 12, 80, and 153, respectively. From 3–4 feet, the rankings were 17, 57, 2, 56, and 160, respectively. As far as rankings go, then, an individual player’s performance can vary substantially from one year to the next.
We next look at six-year averages from longer distances. One important issue here is the extent to which putting is a single skill. That is, do players who putt well from short distances automatically putt well from longer distances? Some golfers have been quite successful at short range with putting strokes that are essentially jabs, but this short, choppy stroke might not work well from longer distances. In addition, the ability to read breaks is more crucial from longer distances. From 3 feet away, you can often hit a putt hard enough that it does not break significantly. That is not typically an option from 15 feet away.
Table 7.4 Top ten percentages of putts made from medium distances, 2004–2009
The top ten putters for distances longer than 10 feet over the combined years of 2004 through 2009 are shown in table 7.4. The percentages are amazing. Over a six-year span, Len Mattiace made nearly 40% of his putts from 10 to 15 feet. Bob Heintz made one-fourth of his putts from 15 to 20 feet. Tiger Woods, Brad Faxon, Brent Geiberger, and Ben Crane make the top ten on two of these lists.
Over longer distances, the number of putts attempted by any given player decreases to the point where yearly percentages mean little. Still, table 7.5 may give you some idea of who is dropping bombs on the greens. The small sample sizes make these rankings suspect. For example, the top three putters in the percentages of putts made from over 50 feet all made exactly 6 putts from long distance. The differences in percentages result from Wes Short attempting 100 putts from over 50 feet compared to Bernhard Langer’s 154 attempts. John Senden also made 6 putts of over 50 feet but did not make the top 10 because he attempted 377 at these distances (the most in the data set). Several of the 205 pros in my five-year lists did not make a putt over 50 feet. Geoff Ogilvy (who ranked all the way up at no. 8 from 25–30 feet) went 0 for 191, the most attempts without a make.
Table 7.5 Top ten percentages of putts made from long distances, 2004–2008
From long distance, the goal even for pros is to limit the damage to two putts. The final two putting lists are for percentages of 3-putts (a small percentage is desired) and 1-putts (where a high percentage is good) in 2008. Surprisingly, four golfers made both lists. Apparently, having your last name rhyme with “okay” makes for success on the greens.
Table 7.6 Top ten percentages of 1-putts and 3-putts, 2008
Jack Nicklaus has said that he never missed a putt from inside 5 feet on the last hole of a tournament.3 He did, of course, miss important putts, but in his mind he was never the one who was at fault. It certainly appeared to us fans that Nicklaus never missed a crucial putt. There is evidence that touring professionals as a group have the same mindset.
Figure 7.4 shows percentages of putts made from different distances in 2007. In (a), shots are broken down by par and birdie, in (b), by par and bogey. The actual percentages for distances up to 10 feet are given in table 7.2. The corresponding values for distances up to 15 feet are given in appendix A, table A7.1.
The results are remarkable. For every single distance, the percentage of putts made is higher for par than for birdie. Significantly higher.4 For every distance up to 9 feet, the percentage of bogey putts made is even higher. Among conceivable explanations, the differences are small enough that they could be due to extra knowledge gained from watching birdie putts slide by the hole. Phil Mickelson is especially blatant about watching the break as the ball goes past. This explanation seems to be borne out in figure 7.5, which shows percentages of putts made from different distances, broken down by whether the putt was a first or second putt. The corresponding values for distances up to 15 feet are given in appendix A, table A7.2.
Figure 7.4 Percentages of putts made from different distances when putting for (a) birdie and par and (b) bogey and par
Again, the comparison is clear. At every distance, more second putts are made than first putts. The differences are larger this time. This result is at least partly due to small sample sizes. There were only 112 15-foot second putts attempted in 2007, with 52 of them made. (Personally, I feel better about my own putting knowing that there were this many bad first putts.) The improved “read” from watching the first putt likely increases the second-putt percentages.
There is more. The graph in figure 7.6 shows percentages of first putts made from different distances, broken down by whether the putt was for birdie or par. The corresponding values for distances up to 15 feet are given in appendix A, table A7.3. The differences persist. Even for the first putts taken by players, the percentage of makes is higher when putting for par. So it is not just personal experience on the green that explains the difference. It could be that watching a chip roll on the green is a benefit. However, the best explanation I can offer is that par putts and second putts are more important, in psychological if not numerical value. Nobody wants to 3-putt or make a bogey or double-bogey. The pros may grind a little harder on bogey putts, par putts, and second putts, with the result that they make slightly higher percentages.
Figure 7.5 Percentages of putts made from different distances, first putt versus second putt
Figure 7.6 Percentages of first putts made from different distances when putting for birdie and par
Chip Sullivan, a top club professional with PGA Tour experience, is not surprised by the difference in percentages. According to Chip, it is less about grinding or focus and more about a total mindset: “It’s great to make a birdie putt, but you absolutely have to make the par putt.”5 He says that he sometimes tries to convince himself that a putt for birdie is, instead, for par.
Apparently, even for pros, not all shots are created equal. Perhaps what makes them pros, as distinguished from the army of amateur wannabes, is that they convert the extra pressure into an increased percentage of putts made.
From 0-3 feet, the pros made 97.9% of their birdie putts, 99.2% of their par putts, and 99.3% of their bogey putts. Are these differences significant? An increase from 98% to 99% seems dramatic, but is the change from 99.2% to 99.3% worth mentioning? Statistics gives us a way of objectively measuring the likelihood that such differences are purely random.
The conclusions that are drawn from any statistical analysis depend on the assumptions made. Here, I assume that the putts are Bernoulli trials. This means that every par putt from 0–3 feet by every player is an independent event with the same probability p of being made. Every birdie putt from 0–3 feet is also an independent event, with probability b of being made. The question is whether the evidence points to p and b being different.
If you were under the misconception that I am a professional golfer, you could watch me play a few holes and decide otherwise. After each smothered iron and chunked chip, you might think, “It’s not likely that a pro would do that.” Eventually, enough of these unlikely events would accumulate that you would conclude that this isn’t a pro having a bad day; this guy is just not a pro. Similarly, we assume (the null hypothesis) that the probabilities p and b are equal and then compute how likely or unlikely the observed data are. If the actual results are unlikely enough, we conclude that the probabilities are not equal. The assumptions made allow us to compute probabilities to precisely quantify “likely” and “unlikely.”
To compare par and birdie putts from 0–3 feet, we use the following data: 114,476 out of n1 = 115, 347 par putts were made (a success rate of p1 = 0.992449) and 8,962 out of n 2 = 9,150 birdie putts were made (a success rate of p2 = 0.979454). The common probability of making a putt is estimated as 
, with a standard error of 
. The z-score for the data is 
.
The z-score is associated with a probability. In this case, the probability of being at least 13 standard deviations above the mean is essentially zero and much less than 0.01. If the probability is less than 0.01, we say that the result is significant at the α = .01 level and reject the null hypothesis. In other words, the difference in performance for par versus birdie is too unlikely to be attributable to chance. There is enough evidence to conclude (with almost no doubt) that the pros make significantly more putts from 0–3 feet for par than for birdie.
What about the difference between par and bogey putts from 0–3 feet? The difference between 99.2% and 99.3% does not seem impressive, but is it significant? Repeating the above calculations with n2 = 34,058 and p 2 = 0.993452 (the pros made 33,835 out of 34,058 bogey putts from 0–3 feet), we get a z-value of about 1.9, with an associated probability of 0.028. The probability is larger than 0.01, so we do not find the result significant at the α = .01 level. However, the probability is less than 0.05, so the result is significant at the α = .05 level. If we are willing to accept a 5% chance of being wrong, we can reject the null hypothesis and conclude that the pros make significantly more 0- to 3-foot putts for bogey than for par. It turns out that 99.3% can be significantly larger than 99.2%.
Contrast this with the following example that 44% is not significantly higher than 42%. For putts from 9–10 feet, the pros made 1,721 out of 3,869 (44.48%) putts for par and 166 out of 389 (42.67%) putts for bogey. The z-value here is 0.684 with an associated probability of 0.248, well above the usual significance levels of 0.05 or 0.01. The difference between this example and the previous example is the much smaller number of putts involved from 9–10 feet. A handful of lucky putts can make the difference between 42% and 44%. This is one of the few cases in which a higher percentage of par putts than bogey putts is made but the number of putts is not large enough to rule out randomness as the cause.
A nice way of visualizing significance of differences is to use error bars. Here, we compute a standard error that quantifies the accuracy of a measurement. Instead of simply plotting the corresponding data point, we frame the point with error bars that vertically add and subtract the standard error. If error bars from two measurements do not overlap, we have visual evidence that the differences in values are significant and are not likely to be due to random fluctuations.
Figure 7.7 Percentages with error bars of putts made from (a) 0 to 3 feet and (b) 3, 4, and 5 feet
Figure 7.7a shows the percentages for made putts from 0–3 feet with error bars. The birdie bar is well below the other two bars, and we saw that there is no question that the birdie percentage is significantly lower than the par percentage. The par and bogey bars are close together but do not overlap. The calculations showed that these percentages are significantly different at the α = .05 level but not at the α = .01 level. Also, notice that the error bars are much larger for birdie putts (small sample size) than for par putts (large sample size). Thus, error bars contain good information.
In Figure 7.7b, percentages of made putts for 3, 4, and 5 feet are shown on the same graph. At this scale, it is very hard to discern that the bogey and par bars are separate for 0- to 3-foot putts. However, it is apparent that the percentages at distances of 4 and 5 feet are significantly different, with birdie percentages being less than par percentages, which are less than bogey percentages.
For almost all distances less than 10 feet, the differences in made putt percentages are significant when comparing putts for birdie, putts for par, and putts for bogey. In general, the pros do make a higher percentage of their putts for par than for birdie and a higher percentage of their putts for bogey than for par.
