1. James and his ideas are featured in the controversial Moneyball by Michael Lewis. The Numbers Game by Alan Schwarz gives an enjoyable and evenhanded history of baseball’s statistics and the role of Bill James in changing the game.
2. The 2006 Joint Mathematics Meetings in San Antonio. In an MAA-sponsored Session on Mathematics of Sports and Games, William Branson of St. Cloud State University gave a talk titled “Bill James as an Exemplar of Statistical Writing.”
3. Coauthored with Bob Smith of Millersville University of Pennsylvania. As I write, various versions of the fourth edition wait impatiently on my desk.
5. Don Wade (2001) includes, in his Talking on Tour compilation, a story told by Trevino. Tenison gets more airtime in Michael Bohn’s Money Golf and in articles like Golf Digest’s dialogue with Raymond Floyd.
6. See R. H. Coop (1998).
7. Gummer (2009) tells the story of the remarkable Homer Kelley. I do not have the dedication to tackle Kelley’s book on my own, but golfers such as Steve Elkington and Bobby Clampett give testimony that the man knew what he was talking about.
9. Steve Evans, Senior VP of Information Systems at PGA Tour, Stephanie Chvala, and Mike Vitti were especially helpful in working with me on the statistical questions.
10. Thanks to my friend Bob Schultz for this wonderful phrase.
Epigraph: Tiger Woods Learning Center, www.twlc.org/spotlight.html.
1. See http://www.exxonmobil.com/Corporate/community_ed_math_academy.aspx.
2. These include Werner and Greig’s How Golf Clubs Really Work, Wishon’s The Search for the Perfect Golf Club, and Zumerchik’s Newton on the Tee. Other books of interest are listed in the reference section.
3. Details can be found in Tommy’s Honor, Kevin Cook’s book about Old Tom Morris and Young Tom Morris. Michael Bohn’s Money Golf gives an enjoyable summary of golf’s history, with a particular emphasis on betting. A fictionalized account of how golf was played in the 1400s is given in Bob Cupp’s The Edict, which provides a possible explanation of James II’s 1457 ban on golf in Scotland beyond the stated reason that Scots needed more time for archery practice.
The only hook that Newton is known to have suffered from was Robert Hooke. Hooke was Newton’s elder, in a position to become his mentor as a leader of the Royal Society of London. Instead, Hooke developed a habit of loudly proclaiming that Newton’s results were either inspired by Hooke, stolen from Hooke, or incorrect. This contributed greatly to Newton’s aversion to publishing his results, which had a surprising influence on the development of science. See Jason Bardi’s The Calculus Wars for details.
4. The magnitude of the drag force is proportional to the density of the air that the ball is in. The density of air in Denver is about 14% lower than at sea level, accounting for standard changes in temperature and pressure. The La Paz Golf Club in Bolivia is identified by Duncan Lennard’s Extreme Golf as being the world’s highest course at 10,650 feet. This altitude would reduce air drag by about 26%. The world’s lowest golf course is Furnace Creek Resort in Death Valley, California, at 214 feet below sea level. Air density is one reason that golf here is a drag. The hottest course is Alice Springs Golf Club in Northern Territory, Australia. At its normal 125°F, the air density is 15% less than air at 30°F.
5. See note 4 above. Among the factors that affect air density and, therefore, air drag are temperature and humidity. Air drag is lower in high temperatures, leading to those nice long drives in the summer. Some people find the humidity result counterintuitive: an increase in humidity reduces air drag, giving you even more distance on a hot, steamy day.
6. An exception to this rule is when the spin axis and velocity vector are parallel. In this case, there is no Magnus force. The Magnus force is represented mathematically by a cross product (see Smith and Minton, Calculus), and the cross product of parallel vectors is the zero vector. The most common sports situation to which the exception applies is a spiral in football. It is good news, indeed, that spirals do not slice or hook, or the forward pass may have been so uncontrollable that it would never have been fully incorporated into the game.
7. Thanks to Acushnet Company for permission to use these figures, which can be found at the Titleist website. www.titleist.com/technology/details.asp?id=20.
8. The parabolic shape is also dependent on the gravitational force being constant. For the flight of a golf ball, this is a reasonable assumption.
9. The “gravity only” curve in figure 1.4 shows the solution of the differential equations x″(t) = 0 ft/s2, y″ (t) = −32 ft/s2 with initial conditions derived from a launch speed of 234 ft/s at angle 15° and initial position at the origin. That is, x(0) = y(0) = 0 ft, x′(0) = 234 cos (15°) ft/s and y′(0) = 234 sin (15°) ft/s. The “gravity plus drag” curve in figure 1.4 shows a numerical approximation of the following differential equations with the same initial conditions:
These equations are consistent with the model published by Smits and Smith in Science and Golf II and with the drag data published online by Titleist (shown in figure 1.3). In these equations, the acceleration due to drag has the form c |v| v, where v is the velocity vector and c = 0.0012743 ft−1. A better model would use a nonconstant c with a dependence on speed. Mathematica software generates the numerical approximations shown.
10. The “gravity only” curve in figure 1.5 is the same as in figure 1.4. For the “gravity, drag, and Magnus” curve in figure 1.5, the same launch speed, angle, and initial position are used. The initial spin is backspin with a magnitude of 3,000 rpm (more precisely, 100π radians per second). The differential equations are
which were developed by Roanoke College student Geoff Boyer based on the model of Smits and Smith. The drag term is the same as before except for including a correction for spin. The general form is now c1 |v| v + c2 |ω| v, where ω is the spin rate in radians per second and c2 = 0.0000669. The acceleration due to spin has the form c3s × v, where s is the spin vector (with magnitude ω). In this example, s = < ω, 0, 0 >. The scalar c3 has the form 
where c4 = 0.000936 ft−0.6, showing that the acceleration due to the Magnus force depends on the spin rate and speed. The spin rate ω is assumed to decay exponentially with a characteristic time of 20 seconds. The spin decay used here is from Werner and Greig, How Golf Clubs Really Work, p. 121. A slightly different result can be found in Zagarola, Lieberman, and Smits, “An Indoor Testing Range to Measure the Aerodynamic Performance of Golf Balls,” pp. 348–354.
11. The comparison in figure 1.5 is not entirely fair. While the launch angle of 15° is near the optimal launch angle for a ball with a spin of 3,000 rpm, it is not at all close to the optimal launch angle in a vacuum, which equals 45°. At this angle, the ball would travel 1,711 feet, almost 580 yards!
12. The three-dimensional model is
where ω(t) < ω1, ω2, ω3 > is the spin vector, with magnitude ω(t) and direction < ω1, ω2, ω3 >.
The initial conditions are x(0) = y(0) = z(0) = 0, x′(0) = v cos (θ) sin (β), y′(0) = v sin (θ) cos (β), z′(0) = v sin (α), and ω(0) = ω. For a given club loft angle A and swing speed S, the launch angle of the ball is approximated by
above vertical. For a swing with plane angle Ap and clubface angle Af, the launch angle of the ball compared to the target (β = 0) is
β = 0.2Ap + 0.8Af,
while the launch speed of the ball is given by
and the initial spin rate is
ω = [7.143S] sin A.
Defining δ = tan−1 (sin (Af − Ap) / tan A), the initial spin vector has ω1 = cos (δ) cos (β), ω2 = − cos (δ) sin (β), and ω3 = − sin (δ).
13. For most golfers, the closed clubface for a hook also turns the clubface down, creating a lower launch angle. Similarly, the open clubface often is tilted upward, creating a higher launch angle. Very few aspects of the golf shot exist in isolation!
14. Calculations use the model of note 12, with a swing speed of 132 ft/s, a launch angle of 34°, and a coefficient of restitution of 0.65 (replacing the 0.83 used previously).
15. A launch angle of 42° produces a flat-terrain carry of 412.6 feet. A rise of 30 feet decreases the carry to 388.7 feet, and a drop of 30 feet increases the carry to 435.0 feet. A launch angle of 26° degrees produces a flat-terrain carry of 532.4 feet. A rise of 30 feet decreases the carry to 498.5 feet, and a drop of 30 feet increases the carry to 562.3 feet.
16. These calculations model a stronger player with a swing speed of 144 ft/s and a launch angle of 40°. The coefficient of restitution was increased to 0.7 so that the flat-terrain distances would match.
17. The flat-terrain distance is 482.8 feet, or about 160 yards. A rise of 30 feet decreases the carry to 460.5 feet, a change of 7.4 yards, and a drop of 30 feet increases the carry to 503.7 feet, a change of 6.9 yards.
18. Werner and Greig, pp. 59–60.
19. An equilibrium can be stable or unstable. “Stable” means that, if the current value is close enough to the equilibrium, it will get closer and closer until it is basically at the equilibrium. How close is “close enough” depends on the specific problem. A handy visual aid for stability is a pencil. Resting on its side it is at a stable equilibrium. If you drop the pencil from a position almost on its side, it will slam down and come to rest on its side. There is another equilibrium where the pencil is balanced on its point. This is “unstable”—if you miss the balance point by any amount, the pencil will fall and eventually end up at its stable equilibrium, resting on its side.
20. To be specific, the equilibrium was computed for different spin rates by setting the equations for x″(t) and y″(t) equal to 0 and solving for the two unknowns x′(t) and y′(t). For a spin rate of 527.25 radians per second, the equilibrium is approximately x′(t) = 86.3 ft/s and y′(t) = −79.6 ft/s. This converts to a slope of 
21. In algebraic terms, the equilibrium line for this graph is a horizontal asymptote. This is a real example of a horizontal asymptote that has meaning, as well as a case in which the graph crosses the asymptote. (Most graphs in algebra class do not cross asymptotes.)
Epigraph: Downs Macrury, Golfers on Golf, p. 54.
1. The Wikipedia entry “Moe Norman” states, “Sam Snead … once described Norman as the greatest striker of the ball.” Lee Trevino has said, “I haven’t seen them all, but I don’t know anyone [who] could hit the ball better than Moe Norman” (Macrury, p. 20). Also, in his foreword to Tim O’Connor’s biography of Moe, The Feeling of Greatness, Trevino wrote, “The simple fact is that when people talk about the great ball-strikers, Moe Norman’s name always comes up.” Both www.moenormangolf.com and www.moenorman.org have links to videos and more information.
2. “Moe Norman,” Wikipedia, last modified on 25 September 2011, http://en.wikipedia.org/wiki/Moe_Norman.
3. See www.moenormangolfacademy.org for more information. Thanks to Todd Graves for his help and to John Hamarik for his picture of Moe Norman.
4. Great golfers respected Moe. Tim O’Connor tells of a driving range session in 1971 in which Moe was interrupted by Gary Player to discuss swing mechanics, with the two being joined in short order by Jack Nicklaus and Lee Trevino, all swapping ideas about the perfect swing. (The Feeling of Greatness, p. 172). I personally have warm feelings for someone like Moe Norman, who once said, “Golf and math were the only two things that I figured mattered to me in my life” (The Feeling of Greatness, p. 25).
6. See Werner and Greig, How Golf Clubs Really Work and How to Optimize Their Designs, p. 21.
7. The deviation is measured from the golfer’s average value. That is, the 20-handicapper may have an outside-in swing that averages 5° to the left. In this case, the 3.66° standard deviation would be added to and subtracted from 5° left. The conclusion would be that about two-thirds of this golfer’s swings have a swing angle between 1.34° and 8.66° to the left of the target.
8. See Werner and Greig, p. 8. The variables are impact velocity v, angle k of impact (above the horizontal), and the backspin rate ω. The estimate of roll in yards is 0.868v + 0.00173v2 cos (k + 24.4 + 0.0012ω). For the standard deviations of clubface angle and swing speed, see p. 21.
9. The “basic fade” has swing speed 161 ft/s, driver loft of 12°, swing plane 4.6° to the left, and clubface angle 0.9° to the left.
10. If you speak British English, these would be “weight” and “borrow.” A Canadian friend, Richard Grant, once startled me after I hit a 50-foot putt even with the hole but a full 5 feet to the right. Being a kind person, he said, “Nice weight.” After briefly thinking that he was complimenting my physique, I did manage the appropriate response: “Too much borrow.” But my financial situation is a different story.
11. Werner and Greig, pp. 139–41.
12. Ibid.
13. See B. Hoadley, “How to lower your putting score without improving.”
14. The Stimp meter, invented by Edward Stimpson, is a small ramp used to measure the speed of greens. The Stimp number is the number of feet the ball rolls after leaving the ramp. So, high Stimp numbers (11 or higher) indicate fast greens.
15. The actual amount of break is 15 tan (3.5°) ≈ 0.917 feet.
16. See Dave Pelz, “A study of golfers’ abilities to read greens.”
17. The curve shown is the trajectory for a ball that starts at (6, 0) ft with an initial velocity of < −7, −.74> ft/s. The tangent line to the curve at (6, 0) has slope 
, and the y-intercept of the tangent line is 
(6 ft) ≈ 0.634 ft ≈ 7.6 in. This is shown in the figure below.
18. See “A study of golfers’ abilities to read greens” in Science and Golf II.
19. See Dave Pelz, Putt Like the Pros, pp. 49–59.
20. Richard Goeres, final project for a class in sport science at Roanoke College.
21. The official record previous to Tiger had been 113 by Byron Nelson, although an article in Sports Illustrated claims that the true record is 177 by Ben Hogan. Research found that Hogan had finished in the money in 177 straight tournaments. Although in the early days of the PGA many tournaments did not have cuts as they are now implemented, the important test for a golfer was finishing high enough to get a paycheck.
22. The curves shown are y = e−(x−70.8)2/0.98 in figure 2.10a, and y = e−(x−70.8)2/12.5 and y = e−(x−68.5)2/12.5 in figure 2.10b.
23. The curves shown in figure 2.11 are y = e−(x−70.8)2/12.5 and y = e−(x−68.5)2/4.5
Epigraph: Downs Macrury, Golfers on Golf, p. 33.
1. CBS television special, “Fantastic Finishes at the Masters,” 2007. As I write, a clip of this, titled “Jack Nicklaus birdies 17th for lead— 1986 Masters,” is available on YouTube. The photo on p. 44 is one of my favorites. It shows Jack and son/caddie Jackie Nicklaus in a practice round on the 16th tee. In the background is Greg Norman, who seems to be struggling, perhaps anticipating the cruel future ahead of him at Augusta National.
2. In his book A Feel for the Game, Ben Crenshaw included this tournament in a short list of golfing events, “Where fate lent a hand,” p. 147. In The Grand Slam, author Mark Frost indicates that Bob Jones himself probably would have called Nicklaus’s victory fate.
3. Dave Pelz, Putt Like the Pros, p. 38.
4. See Pelz, Putt Like the Pros, pp. 24–28.
5. There is some evidence that putting percentages drop as the round progresses in PGA tournaments. For example, in 2007 from 6 feet away the pros made 73% of their putts in the first hour of play, 71% in the second hour of play, 69% in the third and fourth hours of play, and 66% in the fifth hour of play. From most other distances, the percentages did not drop steadily, and the overall statistics from all distances show only a slight decline as play progresses. Clearly, the pros try to minimize damage to the greens. A quick watering of the greens may also help repair footprints.
6. Werner and Greig, How Golf Clubs Really Work and How to Optimize Their Designs, pp. 131–32.
7. Ibid.
8. Yogi Berra includes “90% of short putts don’t go in” as one of his Yogi-isms in The Yogi Book. In this case, we may want to take seriously the book’s wonderful subtitle of I Really Didn’t Say Everything I Said.
9. See Pelz, pp. 126–30.
10. Werner and Greig, pp. 143–45.
11. The Stimp meter, invented by Edward Stimpson, is a small ramp used to measure the speed of greens. The Stimp number is the number of feet the ball rolls after leaving the ramp. So, high Stimp numbers (11 or higher) indicate fast greens.
12. From Golf Digest’s “Dialogue on Golf” with Raymond Floyd, February 1994.
13. Kevin Cook, Titanic Thompson: The Man Who Bet on Everything, pp. 139–42.
14. Raymond Floyd, Golf Digest, February 1994.
15. A full solution of the problem appears in Gualtieri, et al., “Golfer’s Dilemma.” See also Littlewood and Bollobás, Littlewood’s Miscellany; and Neimark and Fufaev, Dynamics of Nonholonomic Systems, pp. 76–80.
16. Gualtieri, et al., place the origin at the contact point, while Little-wood and Bollobás and Neimark and Fufaev place the origin at the center of the ball. Thanks to Bruce Torrence for help with the Mathematica code for figure 3.5.
17. See Pelz, pp. 31–36. The phrase “lumpy doughnut” is Pelz’s.
18. This quote is often attributed to Mark Twain, but German author Kurt Tucholsky has also received credit (from the Germans, naturally). More recently, John Feinstein used it as the title of his book on the PGA Tour.
19. Einstein’s paper on Brownian motion was published in 1905, his “annus mirabilis” of unprecedented scholarly production. This paper was important in establishing the atomic nature of life.
20. To be precise, the probability p must be inversely proportional to the distance to the hole, with a maximum probability of q. Then choose b such that tan 
. More realistically, b can vary as long as the expected value of tan b equals 
.
21. See Pelz, pp. 31–37.
Epigraph: Downs Macrury, Golfers on Golf, p. 68. Palmer is discussing his final round collapse in the 1966 U.S. Open, losing a 7-shot lead with nine holes to play.
1. Will Blythe’s To Hate Like This Is to Be Happy Forever, about the college basketball rivalry between the North Carolina Tar Heels and the Duke Blue Devils, is an excellent reflection on the meaning of fierce rivalries. See Tom Stanton, Ty and the Babe, for the story of the rivalry of Ty Cobb and Babe Ruth, first in baseball (not only for the title of best in the game, but for control of the style and soul of the game) and then in golf.
2. In Arnie and Jack, Ian O’Connor explores the careers of and interactions between Arnold Palmer and Jack Nicklaus, starting with a long-driving contest between young professional Palmer and college freshman Nicklaus. Nicklaus outdrove Palmer, making Arnie mad enough that, in the exhibition match that followed, Palmer broke the course record.
3. See G. H. Hardy, “A Mathematical Theorem about Golf.” The rules that I use are exactly as Hardy defined them. Fortunately for me, Hardy contented himself with an approximate calculation for the odds of a player with probability p beating a perfectly consistent player with probability p = 0. Hardy, with his characteristic wit, named what I call excellent and poor shots “supershots” and “subshots,” respectively. The analysis given here explores the problem further.
One of the book’s reviewers suggested a nice alternative way to understand the assumptions. A golfer must accumulate 4 quality points to finish a hole. A P shot is worth 0 points, an N shot is worth 1 point, and an E shot is worth 2 points, so the sequence NEE represents 3 shots with a “wasted” point coming with the second E shot.
4. The exact average (mean) for a par 4 is 
. For p = 0.1, this computes to 4.099984 … and for p = 0.2 we get 4.1992 … For par 5s the mean is 
, while for par 3s the mean is 
. Details are in my April 2010 Math Horizons article and companion paper online at www.mathaware.com (and the proceedings publication Mathematics and Sports).
5. For example, on the second hole, while my partner was making a 6, I dumped my second into a greenside bunker. Unfortunately, the green it was beside was the 17th. I hit a full 9-iron onto the correct green and dropped a 30-footer for par. Our opponents missed 10-foot and 8-foot birdie putts, and we halved the hole. I have no idea why they did not strangle us on the spot. Any judge who golfs would have let them off with a warning.
6. Private communication. Chris Conklin did his work as a junior at St. Olaf College, where he was captain of the golf team.
7. See Kevin Cook, Driven, pp. 45–46, which gives several revealing case studies of parents supporting their golfing prodigies both well (Ivan Lendl) and poorly. It also gives details of the Scott Robertson Tournament, a top junior tournament in my city of Roanoke.
Epigraph: Leigh Montville, The Mysterious Montague. This book details the amazing golfing feats of the legendary John Montague.
1. The USGA website (www.usga.org) has a number of interesting pages, including a description of the handicap system.
2. A number of articles have been written about the shortcomings of the handicap system. In The Physics of Golf, Theodore Jorgensen comes to the conclusion that the better player has the advantage (pp. 105–15). Bingham and Swartz, “Equitable Handicapping in Golf,” posit that weaker players are favored in head-to-head matchups.
3. As a college professor, I cringe at using the phrase “A-game” for the 80% mark. About 20% of your scores will be at or better than your handicap, once it is adjusted for slope rating. For a course with a high slope rating, you might never shoot only 6 over the course rating.
4. See Francis Scheid, “A General Principle in Golf,” pp. 298–304.
5. For more details, see Dean Knuth, “A Two Parameter Golf Course Rating System”; and R. C. Stroud and L. J. Riccio, “Mathematical Underpinnings of the Slope Handicap System.”
6. See Stroud and Riccio, “Mathematical Underpinnings,” pp. 135–40.
7. The 92% rule holds exactly if the standard deviation is 10% of the mean. As noted later in this chapter, estimates of the actual standard deviations are considerably lower than this, calling the 92% rule into question. The rule is quoted, but not justified, in Stroud and Riccio, “Mathematical Underpinnings.”
8. More information can be found at www.ausgolf.com.
9. From Dean Knuth, “History of Handicapping,” at www.popeofslope.com.
10. See Richard Stroud, “Proposed Handicap Study.”
11. Bingham and Swartz, “Equitable Handicapping in Golf,” pp. 170–77.
12. The probability density function (pdf) for a normally distributed random variable with mean μ and standard deviation σ is given by 
13. See F. J. Scheid, “On the Normality and Independence of Golf Scores, with Various Applications,” pp. 147–52.
14. Theoretically, all values are possible. To help you visualize the concept of “spread,” I am cheating somewhat by focusing on the values of x that produce points with large enough y-values to be visually distinguishable from the x-axis.
15. This again uses the 92% rule from note 7, above.
16. Bingham and Swartz, 170–77.
Epigraph: From www.baseball-almanac.com/quotes/stats5.shtml. Wiles is Director of Research at the Baseball Hall of Fame.
1. In 1941, Ted Williams batted .406, and Joe DiMaggio got hits in 56 straight games. In 1998, Mark McGwire hit 70 home runs, Sammy Sosa had 66, and Ken Griffey, Jr., had 56. In 1968, Bob Gibson had an ERA of 1.12, Denny McLain won 31 games, Don Drysdale pitched 
consecutive scoreless innings, and Carl Yastrzemski led the American League with a .301 average. In 1961, Roger Maris had 61 home runs, and teammate Mickey Mantle had 54.
2. Michael O’Keefe and Teri Thompson,The Card: Collectors, Con Men and the True Story of History’s Most Desired Baseball Card, gives historical background on the popularity of baseball cards.
3. The term “sabermetrics” derives from SABR, the Society for American Baseball Research. Since the B stands for “baseball,” the corresponding name for the analysis of golf statistics will have to be different. “Tigermetrics” is my best shot at a catchy name.
4. Many thanks to Mike Vitti of PGA Tour, Inc., for his assistance and guidance. Also, thanks to Steve Evans and Kin Lo for their help.
5. The men’s major golf championships, often referred to simply as “the majors,” are the Masters Tournament, the U.S. Open Championship, the Open Championship (commonly known as the British Open), and the PGA Championship.
6. While this quotation appears on many Internet quote sites, such as brainyquote.com and thinkexist.com, I do not know the original source.
7. The classic example is a positive correlation between ice cream sales and murders, “proving” that ice cream causes murder. Although a brain freeze can be uncomfortable, the real reason for the correlation is that both ice cream sales and murders increase when the temperature rises.
8. I used the available online statistics to compute correlations to score for several 2008 PGA tournaments, including majors. In every tournament I looked at, greens in regulation and putts per green in regulation were far and away the best predictors of score.
9. We will see later that missing a fairway costs a player, on the average, about a quarter of a stroke. Thus, it is important to hit fairways. The lesson to learn from the correlation study is that the number of fairways hit is not the best predictor of score for the pros. The top ten in a given tournament may comprise a mix of players who hit many fairways and players who hit few fairways. By contrast, the top ten rarely includes a player who misses a relatively large number of greens or takes a large number of putts.
10. You might expect that driving distance would at least correlate with proximity to the hole, which in turn correlates highly with scoring. However, average driving distance for a round has a 0.02 correlation (small and positive!) to average proximity to the hole for the round. For a given (par 4 or par 5) hole, the correlation is −0.09, still quite small. The general idea that longer hitters end up closer to the hole on average is not supported by the data. This statement also needs some context. As we will see, for shots from the fairway, the closer the pros are to the hole, the closer they hit their approach shots. Therefore, extra distance translates to better results, as long as you’re keeping the ball in the fairway. Long hitters miss the fairway often enough that distance off the tee does not correlate highly with the quality of the approach shot.
11. The equations can be solved in general using techniques from calculus or linear algebra. The solution can be written simply using vectors and matrices, which are easily implemented on a computer.
Epigraph: From the NBC telecast, June 15, 2008.
1. See Rocco Mediate, Are You Kidding Me?
2. Outside of 20 feet, the match between the function and the data is quite poor. The function approaches 0 far faster than does the percentage of putts made.
3. See Bob Rotella’s “How to Drain Them Like Jack.”
4. As a sample calculation of significance, if there was no difference (other than randomness) between made par and made birdie percentages, then we would expect that 99.2% of the 9,150 birdie putts from inside 3 feet—9,077 putts—would be made. Actually, only 8,958 of these birdie putts were made. The standard deviation of a binomial distribution with n = 9150 and p = 0.992 is approximately 8.5, so the observed value of 8,958 is a full 14 standard deviations away from the expected value if the averages were the same.
5. Thanks to Chip Sullivan for his time and help at Hanging Rock Golf Club, as he prepared for the 2010 PGA Championship at Whistling Straits.
Epigraph: From “10K Truth Quotes on Golf,” www.10ktruth.com/the_quotes/golf.htm. The rest of the quote is: “Fate has nothing to do with success or failure, because that is a negative philosophy that indicts one’s confidence, and I’ll have no part of it.”
1. The 1987 Masters ended in a three-way tie between Greg Norman, Larry Mize, and Seve Ballesteros. Seve dropped out with a bogey on the first playoff hole, number 10 at Augusta. On the par 4 eleventh, Mize left his second shot well to the right of the green. Norman hit a conservative shot to the front of the green, some 30 feet from the hole. Mize played a bump-and-run that took two hops in the fringe before bouncing onto the green. The chip had a fair amount of speed (Mize estimates it would have gone 4 to 8 feet past the hole) when it hit the pin and dropped.
2. We can derive an equation of the most visible of the parabolas in figure 8.8. Start with a point for which x = 0 and y = H > 0. Suppose the real value of B changes very slightly so that it rounds down instead of up. In effect, we have a sudden change from a reported value of B to a reported value of B − 1. By our formula for y, the new y-value is 
Assuming that B and d are both much larger than 1, then the new y-value is approximately 
=y – 1. The new x-value will be obtained from the equation x2 = H2 – 2=y2 – 2. Substitute the value for found above, square it, and simplify to get 
. Solving for y gives 
. Assuming that B and d are much larger than 1 and that they are approximately the same (for example, a shot from 300 feet out will travel approximately 300 feet), then y 
, which (when converted to feet, giving 
) matches the innermost parabola nicely. A change from a value of d to d + 1 produces the same approximate equation.
3. Rounding off is at the center of a long-standing controversy over how to apportion the United States House of Representatives. The first presidential veto in U.S. history was George Washington’s veto of Thomas Jefferson’s method for apportioning representation in the House in favor of a different method of rounding. Mathematically, all methods of rounding produce undesirable paradoxes in the apportionment of the House. For example, increasing the size of the House can result in a state losing a representative (known as the Oklahoma paradox). The general effect seen in figure 8.8 is related to the butterfly effect. This refers to the sensitivity and unpredictability of the weather, in that a small change (the air disturbed by a butterfly flapping its wings in Brazil) can have a large effect (the formation of a tornado in Texas).
Epigraph: From Thinkexist.com. Hogan also chided an overly exuberant Lanny Wadkins with the warning, “I don’t play jolly golf.”
1. Screenplay by Mark Frost, based on his book of the same title.
2. See Mark Frost, The Match, pp. 2–4.
3. See Tom Wishon, The Search for the Perfect Golf Club, pp. 7–9. Wishon calls this the “dreaded vanishing loft disease.”
4. The equations are as in note 10 of chapter 1, with x(0) = y (0) = 0, x′(0) = v cos (θ), y′ (0) = v sin (θ), ω (0) = ω. For a given club loft angle A and swing speed S, the launch angle of the ball is approximated by
and the launch speed of the ball is given by
with an initial spin rate of
ω = [7.143S] sin A
The adjustments for club length were based on a half-inch difference in lengths of clubs. Clubhead speeds were S = 132 ft/s for the 5-iron, S = 130.3 ft/s for the 6-iron, S = 128.6 ft/s for the 7-iron, S = 126.9 ft/s for the 8-iron, and S = 125.1 ft/s for the 9-iron. See Theodore Jorgensen, The Physics of Golf, pp. 123–31.
5. The top tens for 2008 for average approach distances from the rough (in ft) are given Appendix A, table A9.1.
6. The best fit quadratic function is 0.049x2 + 0.171x + 2.56. The fit is not very good for the first few data points, but beyond 50 yards, the fit is excellent. The best fit quadratic for the standard deviations of the amount offline is 0.035x2 + 0.178x + 3.261.
7. The standard deviations of the amounts short and long increase as the distance increases, but in more of a linear than quadratic fashion.
Epigraph: From Downs Macrury, Golfers on Golf, p. 59. My favorite golf moment is a mid- or long-iron hanging in the air, with time seeming to stop while my eyes go down to the pin and up to the ball to verify that the ball is heading right at the pin.
1. However, you do get a free drop if your ball lands in the sheep droppings that litter the fairways.
2. From Paul DiPerna and Vikki Keller, Oakhurst: The Birth and Rebirth of America’s First Golf Course, p. 156. Many thanks to Nancy Midkiff for her tour of the Oakhurst museum. She cleared the way for me to play the course when it was officially closed to the public, took excellent care of me and my friend John Selby, and mailed me an autographed copy of the book. She and Mr. Keller made us feel very welcome.
3. Unfortunately, the only rhythm I developed was a waltz tempo of backswing–downswing–“Fore, right!” Mr. Keller and Ms. Midkiff had warned me that my fast swing was not likely to work well with the head-heavy hickory driver.
4. See John Andrisani, The Bobby Jones Way, p. 32.
5. On a recent visit to Golf Mart (many thanks to Randy Agee), I hit several shots with my 10-year-old driver and a variety of new drivers. Each of the modern drivers had a clubhead about twice the size of my “midsize” driver. Clubhead speed (CS), ball speed, launch angle, spin rate, and estimated carry distance for representative swings of five different clubs are shown below.
Notice that while the clubhead speeds stayed fairly constant, all of the other variables changed dramatically. Unfortunately for my bank account, my driver (club A) did not fare well. Driver B rated the highest. Although the carry distance for driver D was larger, the lower spin rate of driver B would likely produce more roll and, therefore, the largest overall distance (in spite of driver B having the lowest clubhead and ball speeds).
6. See Tom Wishon, The Search for the Perfect Golf Club, pp. 13–18.
7. There are many good stories about the longest shot ever hit, including hitting drives into trucks or down roads, as in the movie Tin Cup. The official record for longest drive in a tournament is 515 yards by the remarkable Mike Austin, who was 64 years old at the time. Other official records tend to involve teeing off on the runway at an airport. However, without a doubt the longest drive ever hit was by cosmonaut Mikhail Tyurin on November 22, 2006. Tyurin stroked a 6-iron into space while strapped to the International Space Station. NASA submitted a low estimate for the shot (which they described as partially shanked) of slightly over a million miles. Russian estimates top a billion miles.
8. Not a Cinderella story in the Caddyshack fantasy sense. Entering the PGA Championship that year, John Daly was an unknown rookie struggling to keep his card. He was the ninth alternate, such a longshot to even get into the tournament that he did not play a practice round at the course, Crooked Stick (near Indianapolis). He shot 69 and 67 in the first two rounds and calmly outplayed the field over the weekend to win. A little-known fact is that a spectator, Tom Weaver, had died from a lightning strike in the first round at Crooked Stick. After his win, Daly quietly donated $30,000 of his $230,000 first prize to the Weaver family. Ironically, Daly’s career has followed a crooked stick, with highly publicized successes and failures on and off the course. He remains one of the most popular players in the game.
9. Both quotes are from Macrury, Golfers on Golf, pp. 21 and 12, respectively.
10. Mike Vitti, PGA Tour, personal communication.
Epigraph: See Don Wade, Talking on Tour, p. 368.
1. See Wade, Talking on Tour, p. 164. Later, Hogan was asked if he wanted to go to the driving range to watch Faldo hit balls. Hogan inquired as to whether Faldo used Hogan golf clubs. When told that Faldo did not, Hogan responded, “Then I think I’ll just sit here and finish my wine.”
2. As evidence that 27 is not enough putts, note that Tiger placed 17th out of 200 on putts made between 8 and 9 feet. It is unlikely that anybody is woefully deficient from 8 feet but highly accurate from 9 feet.
3. Each hole is counted only once. If the first putt is from 60 feet, then I compare the number of putts to the Tour average from 60 feet. I do not consider how long the second putt was. There is no distinction made between great lag putts and poor lags with long makes for the 2-putt. What is being measured is total putting performance. Also, instead of using the actual average number of putts for a given year, I use a number derived from the fitted curve to the number of putts. For ratings of other skills, I also use fitted curves in the computations.
4. There is some evidence along these lines for 2007. In the tournaments that Tiger played in, the average number of 3-putts was 0.623, and the average number of total putts was 29.1. In the tournaments that Tiger did not play in, those averages dropped to 0.609 and 28.6, respectively. There is more on this topic at the end of the chapter.
5. The number of shots of different types is not arbitrary. In 2008, 28,349 shots were taken from the fairway from 4–50 yards, 125,130 shots from the fairway from 50–200 yards, 17,063 shots from the fairway from 200–250 yards, 35,248 shots from the rough from 4–50 yards, and 39,600 shots from the rough from 50–200 yards. The weights are roughly proportional to the number of shots in each category.
6. From an interview with Laura Hill, PGATOUR.com.
7. See Douglas Fearing, Jason Acimovic, and Stephen Graves, “How to Catch a Tiger: Understanding Putting Performance on the PGA Tour,” pp. 1–49.
Epigraph: From Downs Macrury, Golfers on Golf, p. 61.
1. See my UMAP Module 725 in “A Mathematical Rating System” or my website, www.roanoke.edu/staff/minton/bynumbers.html. More details are given in the “Back Tee” section of chapter 12.
2. This large point differential is contrasted with NASCAR’s point system in Scott Berry, “Is Second Place the First Loser?”
3. The actual graph shown in figure 12.1 is that of .000039(x − 35)4 + .00009(x − 35)2, whereas the aging function found by Berry et al. is empirical and does not have a simple formula. The actual aging function graphed in Berry et al. does not appear to be symmetric about x = 35, although I tried to match the shape of their graph reasonably well.
4. Private communication with Scott Berry.
5. See John Zumerchik, Newton on the Tee, pp. 197–206.
6. See Jennifer Brown, “Quitters Never Win,” pp. 1–35.
7. The interpretation given by some in the media, that Tiger is so intimidating that he starts off each tournament with a de facto 2-shot lead, is not warranted. That might be what is happening, but Brown’s work was focused on the overall effect on the players when Tiger is winning every tournament in sight. Rich Beem, Zach Johnson, Trevor Immelman, Rocco Mediate, Y. E. Yang, and others have stood up nicely to Tiger in major championships.
