Shooting your age in golf

There are many memorable moments in a golfer’s life. Breaking 90, 80 or 70 for the first time, striking a hole-in-one or scoring an eagle on a par-5 hole are some examples most golfers never forget. One cherished moment that few attain, it would seem, is shooting an 18-hole score equal to or less than one’s age on a regulation, full-length, 18-hole course. The beauty of this feat is that it comes later in life, and it is to be savored.

George Peper, editor of LINKS Magazine, former editor of Golf Magazine and arguably one of the most inventive writer/thinkers in the game, asked me, “What’s the probability of a golfer shooting his/her age at least once in a lifetime?” The question was too intriguing to ignore. The following is an analysis of the problem.

Background

In 1994, this author presented a paper at the Second World Scientific Congress of Golf at St. Andrews, Scotland, entitled, “The Aging of a Great Golfer: Tom Watson’s Play at the U.S. Open” [1]. Long before the PGA Tour’s Shotlink system, I had tracked every shot Watson took in all his rounds at the U.S. Open for almost two decades using the Golf Analyzer Scorecard, and his play was statistically analyzed with the Golf Analyzer software. It was clear that over the years, Watson improved quickly (mid-1970s), then stayed at a high level (late 70s through the 80s), then his scores slowly deteriorated beyond that period as he aged.

As a result of the interest in that paper, the Trust (organizers of the Congress) offered several sessions on aging and golf at the next Congress in 1998, and I chaired one of those paper sessions, as well as the plenary session on the subject. Two papers of note were presented, one by Berry and Larkey [2] on the play of touring pro golfers, and the other by Lockwood [3], who studied amateur golfers of all skill levels. Both papers indicated that there are three phases to a golfer’s life: a period of rapid improvement while the player learns the game, a long plateau where his/her scores change little, and then a slow decline in scoring. Lockwood actually calculated that for all skill levels; the loss of ability for male golfers translates to a decline of one stroke for every eight years once a player reaches “senior” status, but it accelerates after the player reaches age 75.

Every older golfer knows scoring deteriorates with advanced age. Image © Thinkstock

Every older golfer knows scoring deteriorates with advanced age. Even pro tour players can see their competitive edge dropping on the regular PGA tour as they reach their 40s. As Lee Trevino reached 50 years old, he often said he could no longer compete with the “flat bellies.” Even on the Champions Tour, the “younger” players win most of the events. Gary Wiren, the noted golf teacher, said (at the plenary session on aging and golf at the 1998 Congress mentioned before) that the teaching pros he would see each January when they came to the PGA Annual Meeting in Florida still played a good game but not quite as good. He could recognize a small drop in performance, until an injury, illness or just plain weakness changed their games [4].

Also, at that session, Dr. Archie Young of London’s Royal Free Hospital [5] noted that after the age of 60, men lose about 2 percent of their muscle mass and about 3 percent of their dynamic force each year. Just as aging is inevitable, so are higher average scores. The facts and anecdotes are supportive of Lockwood’s analysis, which makes his work quantitatively instructive for the study at hand.

Of course, this all begs the question: Can a golfer play well enough into his/her senior years such that his/her age “catches up” with his/her scoring ability? This article provides a description of a model to estimate the probability of scoring one’s age given a number of factors.

Who Shoots Their Age

Realistically, it is unlikely that a player younger than 66 will shoot their age unless they are an exceptional super senior. Many players on the PGA Champions Tour have done it, but they are the best senior players in the world. The story goes that Sam Snead did it almost every time he teed it up for a casual round once he reached 75 years old.

According to websites that keep track of this matter, the youngest golfer to have done it was Bob Hamilton. “Hamilton, the 1944 PGA Championship winner, who shot his age of 59 at Hamilton Golf Club in Evansville, Ind., in 1975, … and the oldest was 103-year-old Arthur Thompson of Victoria, British Columbia. Thompson was playing the Uplands Golf Club in Victoria when he accomplished the feat in 1972” [6].

Sam Snead, at age 67, shot a 67 one day then a 66 the next at the 1979 Quad Cities Open for the record on the PGA Tour. Walter Morgan, at age 61, shot a 60 on the Champions Tour in the AT&T Canada Senior Open Championship [7].

Once done, apparently it is not hard to do it again: “The record for most times shooting your age belongs to T. Edison Smith of Moorhead, Minn. Frank Bailey of Abilene, Texas, had long held this record, matching or beating his age 2,623 times, from age 71 until age 98. But in 2006, Smith passed Bailey and continues on with the record” [8].

As for the general question of who shoots their age, these anecdotes are not particularly helpful. Unfortunately, there is very little data available for regular golfers to indicate which type of golfer actually shoots his/her age and how often they do. The USGA has 18-hole scores for millions of golfers and hundreds of millions of rounds. Alas, what they do not have is the age of the golfers.

Some clubs keep track and honor players who accomplish the feat. For example, Lake Nona Country Club (Florida) has a plaque in the clubhouse with player names when they have done it. The club is 31 years old and has the typical number of members (about 300), approximately half of whom are over 65. The plaque has 30 entries, which means 30 times someone in the club has shot their age. Interestingly, 29 of those entries are a single player. As such only two players have done it, one once and one 29 times. That supports some of the speculation mentioned above: Shooting one’s age is hard to do for most senior golfers, but for those who can do it, they do it many times. But without the ages of the members and the number of rounds they play, it is nearly impossible to make an estimate of the sought-after probabilities [9].

Without data for more actual amateur experiences, the probabilities cannot be directly estimated. As such, the following is an effort to build a hypothetical model to estimate as accurately as possible the probability of shootings one’s age given all the relevant factors.

The Model

Several studies have suggested that if all the scores a golfer shoots over a multi-year period are plotted, they take on a mound shaped form. That distribution likely can be fit to a normal distribution with a slightly longer tail on the high side. (It’s easier to shoot a higher than average score than a lower one.) Typically, as the average score goes up, the spread of scores goes up as well. That is, the higher a player’s average score, in general, the higher the standard deviation (the higher the spread of scores from lowest to highest, in less technical language).

This study considers regular players age 66 and up. As Lockwood pointed out, as a player enters his (he only studied male golfers) senior years, his average score slowly increases and then accelerates at the age of 75. Interestingly, Lockwood’s calculations would indicate that as a player gets one year older, his average score increases only an eighth of a stroke. If so, with age going up one year at a time and average score going up as little as an eighth of a stroke a year, there is some reason to believe that as one gets older, his/her chances may improve. At least Lockwood gave us a baseline of deterioration to consider in a model.

Since this article is about shooting a low score, the normal distribution assumption is likely a valid one to use for making probability estimates of good scores. As such we can estimate the probability of any player’s chances of shooting a low score if we have historical data, which provide an estimate of that player’s average score and standard deviation when playing on a particular course. Knowing a player’s age, we can calculate the probability of shooting a score equal to or less than the player’s age using the normal distribution based on the player’s average and spread of scores.

A player’s historical record of scores can be used to predict the scores they will shoot. Using Lockwood’s baseline deterioration rate, predictions of average score as a player ages can be made given an initial average. Of course, predicting a player’s average score is easier than predicting the exceptional score. To predict the exceptional score of a specific player, we would also need a good prediction for a spread of scores, or standard deviation of scores. Data for golfers of all skill levels show quite a range of standard deviations. Generally, but not always, the higher the handicap, the higher a player’s range of scores will be. According to a study of male golfers by Simmons [10] for the USGA, a standard deviation of three strokes is quite common for low, single-digit handicappers, four or five strokes for bogey golfers, and as much as six or more strokes for double bogey golfers. (To estimate your own standard deviation, you can divide the difference of your highest score to your lowest by 6.)

Probability of Shooting One’s Age in a Round

Obviously, a 75-year-old player who averages 75 strokes a round shoots his/her age about half the time. More interestingly, using the normal model, that probability can be calculated for any age and any average score/standard deviation combination. Now let’s assume the standard deviation is about three strokes. That means that the golfer’s maximum potential is about three standard deviations below his/her average. So, a 71 is likely the best score for someone who shoots 80 on average. Not that they couldn’t have a miracle round better than that, but it is highly unlikely.

Using the information about standard deviations, to have any chance of shooting their age any time they play, a golfer should play a course for which their average score is no more than nine strokes higher than his/her age, although that probability is (depending on the spread of scores) well below 1 percent. Beyond nine strokes, unless the player’s spread is much higher than three strokes, the probability is essentially zero. On the other hand, the closer the average score is to age, the probability goes up quickly to where, as noted, when age equals average score, it is approximately 50 percent.

Table 1 shows that for a 70-year-old, if their average score is 80, the probability is essentially zero. The nine-stroke rule applies.

Table 1: Standard deviation of scores.

Table 2: Probability of shooting age by age with average score of 80 at age 66 with three deterioration rates.

It could also be said that if a player maintains their average score as they get older, if they live long enough, the probability becomes non-zero. Of course, as previously noted, maintaining an average score while aging is highly unrealistic. It can be assumed that the average score increases at a minimum, according to Lockwood, one-eighth stroke per year. For many golfers the deterioration rate is probably higher. Table 2 shows how the probability of shooting one’s age for any one round changes as the player gets older for a golfer who averages 80 at age 66 and whose average score increases either one-eighth stroke per year, one quarter or one-half.

Understandably, these probabilities are somewhat smaller than those for the same golfer in Table 1 because in Table 1, the average score did not increase. Also, obviously the probability is quite sensitive to the deterioration rate. Of course, even this is somewhat unrealistic in that it assumes a constant increase in scores. Wiren, even Lockwood, recognized that the increase probability increases with age. One would have to make assumptions of that rate before making calculations. Clearly the probabilities would significantly drop if the rate increased even just a bit. Table 2 can be thought of as upper bounds on the probabilities.

Probability of Shooting One’s Age in a Year

The more rounds a golfer plays per year, the better the chance at least one exceptional score is recorded. Assuming a golfer plays many rounds a year, the probability of the player shooting a score less than or equal to his/her age at least once in that year can be calculated using the binomial distribution. The simple way to calculate that probability is “one minus the probability the player does not shoot his/her age in all the rounds played that year.” If the golfer plays N rounds in a year all with the same scoring probability distribution, the probability is one minus the binomial of zero successes in N trials each with a probability of success equal to the single-round success probability calculated in the previous section.

Table 3 shows the probability of doing it at least once in a year given the average score and the age. For a golfer who averages nine strokes above his/her age who plays 40 rounds a year, the chance of doing it at least once in a year jumps to about 5 percent, but it is zero for more than nine strokes. Interestingly, at an average score of about six strokes above the player’s age, the probability of shooting it at least once in a year jumps up to about 50 percent. Said more concisely, if you average three or less strokes above your age, there is a near certainty you’ll shoot your age once in a 40-round year. Six strokes above your age, it’s about a 50 percent chance, and at more than nine strokes, it is essentially zero: 3 strokes – 100 percent, 6 – 50 percent, 9+ – 0 percent.

Table 3: Probability of shooting age at least once in a year of age (40 rounds).

At this point, it is obvious that there are many factors working together that must be assumed before an estimate can be made. To make these calculations, one has to assume the average score, standard deviation and number of rounds. Any variation from the numbers presented for those factors significantly changes the probabilities. As such, the general rule of nine strokes is dependent on not only the standard deviation, but also the number of rounds played by a particular golfer in a year.

Shooting It at Least Once in a Lifetime

To estimate the probability of shooting your age it at least once in a lifetime requires even more assumptions. First, it requires an estimate of the deterioration rate of average scores as a player ages. Second, it depends, for computation purposes, significantly on when the golfer essentially stops playing. At some point, the golfer no longer plays the game.

Once those assumptions are assigned, the calculation for “at least once in a lifetime” is quite similar to the “at least once in a year.” It is found by using the binomial distribution a second time. It is “one minus the probability of not shooting it at least once in any of the remaining golfing years left in a golfer’s lifetime.” In this model, the probability of shooting it at least once in any year changes as the golfer’s average score deteriorates as he/she gets older. The Lockwood data is instructive in this matter.

For a golfer playing 40 rounds per year, with a normal distribution of scores with a three-stroke standard deviation, and a constant deterioration rate of one-eighth stroke per year, Table 4 shows the probability of shooting one’s age at least once in a lifetime given the player’s average score at age 66 and given the player’s age at the final year of playing.

Table 4: Probabilities of shooting one’s age at least once in a lifetime.

Clearly, the lower the initial scoring average, the better the chances. With these assumptions, a player who averages 80 with a standard deviation of three strokes at age 66 who loses one-eighth of a stroke per year and continues playing at least until 80 is virtually assured of shooting his/her age at least once in their playing lifetime. However, even with these assumptions, a player who averages 90 at age 66 has to keep playing until at least 88 to have a good chance of doing it just once.

Calculations for golfers with higher scoring standard deviations are not displayed, but it should be obvious that the higher the spread of scores, the higher the probability. Said another way, models based on this approach would indicate that golfers with higher standard deviations will have better probabilities.

Faster Rates of Deterioration

It is likely that scores go up faster than these predictions, in which case the chances would be lower. The calculations in Table 5 indicate that the probabilities change dramatically if the deterioration rate increases with age. Table 5 displays the probabilities if the acceleration rate is 50 percent higher (1.5 in the table) than the Lockwood base and increases with age by multiplying the increase by the difference in age from 66. That is, the initial rate is 50 percent higher, but it increases by the difference in years from 66. As such, the rate is not only higher than in Table 4, it increases with age as Lockwood suggested.

Table 5: Probabilities of shooting one’s age at least once in a lifetime if the deterioration rate increases with age.

Even with this increased rate of deterioration, single-digit handicappers have a reasonable chance of doing it if they continue playing well into their 80s. Unfortunately, bogey golfers don’t have much of a chance under this assumption.

Clearly, even at this accelerated deterioration rate, the longer a player stays active, the higher the probability of shooting one’s age. Unfortunately, if the deterioration rate was higher, the probabilities would drop dramatically. Table 6 calculates the probabilities with an even faster rate of deterioration. In Table 6, the rate is 100 percent faster (a factor of 2) for the increase in scores. At that rate, the probability essentially goes to zero. Unfortunately, this may be the most accurate prediction. It would explain why so few golfers actually achieve this special accomplishment.

Table 6: Probabilities of shooting one’s age with an even faster rate of deterioration.

It is possible that if the deterioration rate increases with age differently than these calculations project, there may be an “optimal” age at which point the probability peaks and then goes down. But without a considerable amount of real data or a firm understanding of “super senior” deterioration rates, that can only be offered as conjecture at this point.

Table 7: The “age shooting” experience of one very good golfer, Warren Simmons.

However, Table 7 displays the “age shooting” experience of one very good golfer. Warren Simmons (mentioned earlier in the study of variability of scores), a scratch golfer most of his life who qualified and played in the 1956 U.S. Open as an amateur, has shot his age 164 times. In the table, it can be seen that the number of times he shot his age increased as he got older, peaked at age 79 and then went down, all the while playing a substantial number of rounds [11]. Anecdotally, this provides some evidence to the conjecture.

Comfort Zones

These calculations are all based on the normal distribution model of player scores. Of course, there are different psychological understandings of how golfers react later in a round to great play early in a round. Many amateurs complain that they “wasted” a good front nine with a bad back nine. “I could have broken 80 if I had just kept it up” is a common refrain.

What do we know about golfers who are aware of the moment at hand? Sports psychologists often talk about golfer’s comfort zones. As a player approaches a barrier, they may change behavior. There are golfers who “rise to the occasion,” and others who, you might say, choke. These models assume no change in behavior.

Jim Furyk shot a 59 on the PGA Tour. He was quite impressive in his post-round interview explaining how he maintained his “cool” under the pressure of making history. Remaining cool under the pressure of that round must have helped him when faced with a similar situation a few years later when he shot the Tour’s first-ever 58. Apparently, we could all learn from him about comfort zone behavior.

Increasing the Chances

How can a golfer increase his/her chances? As pointed out earlier, to have any chance of shooting one’s age at least once in a year (approximately 40 rounds), a golfer should play a course for which his/her average score is no more than nine strokes higher than one’s age, although that probability (depending on one’s spread of scores) likely is still quite low. To have a 50 percent chance of doing it at least once in a year, a golfer should regularly play a course on which one’s average score is no more than five or six strokes above one’s age.

To increase the chances, a golfer would have to “reduce the deterioration rate.” One way is to practice the parts of the game in which the player is weakest. A statistical analysis is a good way to start a practice plan. Reading instruction material and getting lessons can help provided there is a follow-up practice strategy.

Unfortunately, there is no guarantee those efforts will result in a reduction in deterioration rate. However, there are a couple of choices a golfer can make to help. One is choosing a course that fits their game. Another is, given a course, choosing the set of tees from which to play.

To compensate for the deterioration in scoring and to increase the chances of shooting one’s age, one “remedy” for older players is to seek out courses that fit their game, particularly the ability to hit greens in regulation (GIRs.) Using Riccio’s First Rule (Score = 95 - 2*GIRs, or in nontechnical terms: three greens break 90, eight greens break 80, 13 greens break 70), you can calculate how many GIRs you need to average nine strokes above your age [12]. If you are 71, you need to average 80 or less, in which case you will need to hit an average of eight GIRs in order to, on average, break 80. So, the target for length of course should be the length that allows you to hit at least eight GIRs. Find a course where that is a realistic possibility. Similarly, an 81-year-old golfer would have to average no more than 90. The target then is a course for which the player can regularly hit at least three greens in regulation.

Another way to look at it is to consider the length of the course. Typically, to gain a stroke on average the player would have to play a course whose yardage is about 200 yards shorter than their current course. So, a 71-year-old golfer, for example, who averages 82 would have to play a course about 400 yards shorter to get their average down to at least 80, for the nine-stroke rule. Taking that to an extreme, if that golfer averaged 90, he/she would have to find a course about 2,000 yards shorter to drop their average score to 80.

Conclusions

The chance of shooting your age in your lifetime depends on how good you are to begin with, at what age you play until, how your average score goes up with age, and whether you play shorter and shorter courses as you age. If you are a good player, play a lot of rounds each year, stay healthy and play into your 90s, the probability is good you will do it at least once. Someone who averages 80 while in their 60s and whose average score increases modestly (one stroke every eight years) as they age, and plays into their early 90s, has a strong chance of doing it at least once. But that probability drops substantially with an assumption of a higher rate of scoring deterioration, one in which scoring deterioration outpaces aging. Unfortunately, the latter may be a more realistic prospective.