PREAMBLE

This research analysis into golf's 'Grand Slam'- the winning of all four major championships in a calendar year - was a throwback to the sort of work I did in my former career. Its main purpose was to show what level of success had to be sustained in the Majors for a top golfer to have a realistic chance of taking-out golf's 'holy grail' at some time in his career.

Various golf magazines and a popular scientific journal were sounded-out to determine the likely interest in my analyis. For quite some time, I remained convinced various editors were singing from the same 'Goldilocks and the Three Bears' songbook; too complex for a golf magazine, far too simplistic for any scientific or mathematical journal to contemplate publishing. Despite these concerns, my work finally found its 'just right' home after some seven and a half years of trying. In January 2000. the statistical appraisal was published in "Golf Research News" - the journal of the World Scientific Golf Congress, whose headquarters are located at the University of St Andrews, Scotland.

In June 2001, I was interviewed on Amanda Smith's ABC Radio National Programme, Sports Factor.


GOLF'S 'GRAND SLAM'
- A STATISTICAL APPRAISAL

Jack Nicklaus once observed that professional golf was unique in that one need win only 20% of the time to be rated as the world's top golfer. Jack's observations were presumably based on his own performances in the four Majors. Having fulfilled his enormous potential as an amateur with his second to Arnold Palmer in the 1960 US Open at Cherry Hills, the period from 1962 (the year he won his first Major, the US Open at Oakmont, and first played all four Majors on a regular basis) to 1982 saw Jack win 17 Majors from 84 starts, a record which most golfing commentators believe will never be bettered, see Tables 1 and 2. From 1983 onwards, Jack's dominance in the Majors effectively ceased, in hindsight, his 1986 US Masters championship win, his 18th, could best be regarded as a 'one-off' - a triumphant 'last hurrah'.

If we take a closer look at Jack's Majors record, the manner in which it was compiled over the peak years suggests his season-by-season results were drawn from a binomial distribution - a distribution based on a probability of 'success' in an individual major of 0.2 and, by definition, a probability of 'failure' (or a win to the opposition), of 0.8. From these two probabilities it is possible to make a reasonable estimate of what his actual chances were of winning the greatest prize of all: the Grand Slam.

For this hypothesis to be credible, certain criteria need to be satisfied. They concern:

1. A consistency of performance: In addition to his first and second place finishes, Jack also managed to finish third on some nine occasions. After the '71 season his record in the previous 40 Majors stood at 9 wins and 9 second-place finishes; after a further 44 Majors his record had advanced to 17 wins and 17 seconds. Only in 1969 did he not at least finish second in any of the Majors over these 21 seasons.

2. The quality and consistency of opposition: Even though the names of his chief rivals changed over the 21 seasons, the likes of Arnold Palmer, Gary Player, Lee Trevino and Tom Watson relegated Jack to second spot on some 10 occasions between 1962 and 1982. Between them they won 30 Majors and all four achieved the feat of winning two Majors in a season (see Table 2).

3. An ability to play all Major course venues: Jack competed and won on all types and conditions of courses, not only in the Majors but world-wide. Jack's 1962-1982 split of 5 wins/4 seconds in the US Masters was matched by 4/3 in the US Open, 3/7 in the Open and 5/3 in the US PGA. Whether the Majors were played at Augusta, British seaside Links courses or the toughest courses the US PGA circuit has to offer, Jack was able to take them all in his stride..

The sheer consistency and duration of Jack's record suggests it would not be stretching a longbow too far to conclude that while he won, overall, '20% of the time', his probability of winning any individual major over that period rarely strayed too far away from 0.2 (or a success rate of 20%).

Given the above criteria are satisfied, it is reasonable to assume Jack's season-by-season results were sampled from a binomial distribution. It follows that the probability of all five win/loss Majors single-season outcomes may be obtained from a term-by-term expansion of the function

(p + q)4 = 1.

Thus, if P(M), (M= 0, 1... 4 ) is the probability of winning M Majors in a single season, then P(4) - the Grand Slam - is p4, while P(3), P(2) , P(1) and P(0) are respectively is 4p3q, 6p2q2, 4pq3 and q4.

The number of times Jack would be expected to achieve these five win/loss outcomes for the four Majors, over the period 1962-82, is simply the number of years (21), times these probabilities.

Between 1962 and 1982 there were nine seasons where Jack failed to win a Major, which compares favourably with the expected 8.60 occasions predicted by theory. On seven occasions Jack won just one Major in a season, his expected value was 8.60. On no fewer than five occasions - and in four of the six ways possible - Jack won two Majors in a season, viz:

* US Masters - US PGA in 1963,75
* US Masters - The Open in 1966
* US Masters - US Open in 1972
* US Open - US PGA in 1980.

At first glance this result might be regarded as being appreciably higher than the 3.2 occasions Jack was expected to achieve two wins in a season.

Though Jack came within a whisker of winning the third leg of the Grand Slam in the 1972 Open at Muirfield, the landmark of three Majors in a year was eventually to elude him. The difficulty of achieving this feat is borne out by the expected number of occasions that Jack would win three Majors: 0.54.

Had Doug Sanders made his putt at St Andrews in 1970 or had Watson not chipped-in at Pebble Beach in 1982, Jack would have been left with an almost perfect 20% success rate over both the first and second phases of his career. In a statistical sense, golf's vagaries make it is scarcely surprising that what Jack gained on the two-Majors swing, he lost on the one-Major or three-Majors roundabouts.

Jack would expect to have achieved the Grand Slam on only close-to 0.03 occasions, which is more accurately equated to a 1-in-30 chance. In hindsight, notwithstanding Jack's exceptional record in the Majors, his expectations of winning a Grand Slam were not strictly warranted.

Another way to highlight the magnitude of the task involved in winning a Grand Slam is to determine the cumulative probability of winning at least one Grand Slam over a number of years, n, where the probability of winning an individual major, p, remains constant.

By definition, Prob.( Fails to win Grand Slam in single season ) = [1 - p4].

Likewise, Prob.( Winning at least one Grand Slam over n seasons) = 1 - Prob.( Fails to win a Grand Slam in any season), or [1 - (1 - p4)n].

Of particular interest, the 'break-even', or 50/50 point for the number of years, N, where the probability of not having won a Grand Slam equals the probability of having won at least one Grand Slam during this period is found from:

[1 -p4]N = 0.5.

For Jack Nicklaus, Tom Watson and Arnold Palmer, we find that the 50/50 point greatly exceeded the span of their golfing careers. Also, since the peak years for Palmer, Watson and Hogan were relatively few in comparison to Jack's, with proportionately less data to work with, any conclusions drawn need to be treated somewhat more cautiously.

For Jack Nicklaus, his probability of winning one Grand Slam, or more, at some time in the 21 years was effectively 21/625, or an approximately 1-in-30 chance. However, 433 years would be required for him to reach the break-even point. (If a historical perspective is required, Jack would need to have embarked on his golfing career around the time of the birth of William Shakespeare and all these years on, he would still rate as an even chance of not having won a Grand Slam!)

Tom Watson won seven of his eight Majors from 28 starts over the years 1977-1983 - a probability of success of 0.25 (25%) - however, four of his Majors were Opens and in that period he failed to win a US PGA championship. Nonetheless, if Tom's Majors results are also assumed to be sampled from a binomial distribution, we see, Table 3, that generally good agreement is obtained between his actual record and the number of Majors he would expect to win in a season. His 25% success rate left him with a 1-in-256 chance of a Grand Slam in any one season - almost 2.5 times better than Jack's, and such a level of performance would have given him close to a 1-in-40 chance of having won a Grand Slam at some time in those seven years. Nonetheless, this cumulative probability is but a small step along the way to his 50/50 chance of still not having won a Grand Slam after 177 years.

Over five seasons, 1960-64, Arnold Palmer won six Majors from 19 starts; a probability of success of close to 0.3 (30%). Like Tom, Arnold's expected number of Majors won per season over this period gives shows good agreement with his actual record, Table 3. However, his 30% success rate would still leave him with only an approximate 1-in-125 chance of achieving a Grand Slam in any one year, which reduces to an approximate 1-in-25 chance of his having won one Grand Slam (or more) over five years. His break-even point would be reached in 85 years.

After his only appearance in the 1953 Open at Carnoustie, Ben Hogan had won the first three Majors of that season and eight out of the last 12 Majors he had competed in, a probability of success of 0.67 (67%), and a record no one has even come close to since that time. After his car accident in 1949, Ben was never a serious contender for the Grand Slam, one Open was enough for him and he would have found the 36-holes in a day matchplay format of the US PGA just too physically demanding. His success rate would result in a slightly less than 1/5 probability of winning a Grand Slam in any one year, a success rate which, if sustained over five years, would have given him a probability of winning one Grand Slam, or more, of 0.67.

To whittle down to the break-even point of not having won a Grand Slam at some time in a Majors career typically spanning 10-16 years, a player would require a success rate hovering somewhere between 45%-50% and that was clearly a level of dominance even Jack, with all his great ability, was never able to achieve.

The initial motivation for conducting this analysis stemmed from a newspaper article in which a number of the world's top players were canvassed for their thoughts on what it would take to win a Grand Slam. What struck me as decidedly unusual at the time was that none of them appeared to have given any great thought to how much their fellow competitors might intrude on their own ambitions in this regard.

Such an omission should not surprise unduly, 'self-belief' after all has to be at the top of any top golfer's psychological make-up. In this respect, Jack's observation on his own success rate vis-a-vis other competitors is perhaps as close as anyone is likely to get to acknowledging how other competitors' performances can, and do, intrude on their own record. Yet while Jack would scarcely have relished those occasions where he went so close but did not win, his great record, nevertheless, suggests he was quick to acknowledge and accept such near-misses for what they were: a competitive and positive result.

Finally, and almost inevitably, we have to ask ourselves how does Tiger Woods rate in all this analysis? The short answer is that it is still far too early in his professional career to draw any meaningful conclusions based on his present results in the Majors. We can rest assured that having won the 1999 US PGA, his second Major - and a possible career watershed - he will start as hot favourite in the 2000 Masters. Were he to win, a not altogether unlikely event considering he won the 1997 US Masters, speculation would be rife that here is someone with the talent, the game and the raw nerve to take out the Grand Slam. But with many other top golfers keen to ensure that any dominance he does achieve is kept within manageable bounds, quite how he goes about improving his present record of two Majors wins in 12 professional starts will be one of the new millennium's more intriguing sports issues.

FIGURE 1





TABLE 1 : JACK NICKLAUSíS DISTRIBUTION
OF MAJOR CHAMPIONSHIP WINS

YEAR

NUMBER OF WINS

YEAR

NUMBER OF WINS

YEAR

NUMBER OF WINS


1960
1961
1962
1963
1964
1965
1966
1967
1968
1969


0
0*
1
2
0
1
2
1
0
0*


1970
1971
1972
1973
1974
1975
1976
1977
1978
1979


1
1
2
1
0
2
0
0
1
0


1980
1981
1982
1983
1984
1985
1986
1987
1988
1989


2
0
0
0*
0*
0*
1
0*
0*
0*

0*: No second place finish in these years

 

TABLE 2: LEADING MAJOR CHAMPIONSHIP WINNERS, 1946-86

 

US MASTERS

US OPEN

THE OPEN

US PGA

HOGAN

51;53

48;50;51;53

53

46;48

PALMER

58;60;62;64

60

61;62

 

PLAYER

61;74;78

65

59;68;74

62;72

NICKLAUS
63;65;66;72;75;
86

62;67;72;80

66;70;78

63;71;73;75;80

TREVINO

 

68;71

71;72

74;84

WATSON

77;81

82

75;77;80;82;83

 

 

 

TABLE 3: COMPARISON OF EXPECTED/ACTUAL PERFORMANCES
IN MAJORS FOR NICKLAUS, PALMER AND WATSON


Prob.=

 

No wins
q4
One win
4pq2
Two wins
6p2q2
Three wins
4p3q
Four wins
p4
Nicklaus
1962-82
Expected
Actual
8.60
9
8.60
7
3.23
5
0.54
0
0.03
0
Palmer
1960-64
Expected
Actual
1.20
1
2.06
2
1.32
2
0.38
0
0.04
0
Watson
1977-83
Expected
Actual
Expected
Actual
2.21
2
2.95
3
1.48
2
0.33
0







POSTSCRIPT

A few months prior to the article being published, Tiger Woods had won the second major of his professional career, the 1999 US PGA championship held at Medinah. He did not won the first Major in 2000: the US Masters, but had then stunned the golfing world by taking out the next four - an incredible feat which, while not qualifying as a Grand Slam, had never been achieved by anyone else before. His winning sequence saw him establish a dominance over his rivals not seen since the halcyon days of Ben Hogan in the 1950s.

By winning the first two Majors in 2002: the US Masters and the US Open, Tiger gave himself his best chance thus far of taking out a Grand Slam. However, it was literally a case of of 'gone with the wind' when gale force winds struck in the third round of The Open at Muirfield, his score ballooning-out to a most un-Tiger like 81.

In the time since the article was published, one observation can be made re- the relative merits of Jack Nicklaus and Tiger Woods. Taking as origin the point where both started playing the four Majors on a full-time basis, we find that out of 28 Majors played, Tiger won eight, Jack, seven. Yet when stacked up against Jack's seven second-place finishes and four third-place finishes over this period, Tiger's record: one second-place and two-third place finishes, pales by comparison.






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