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Electronic copy available at: http://ssrn.com/abstract=1765964

Assessing Golfer Performance on the PGA TOUR

Mark Broadie

Graduate School of Business

Columbia University

mnb2@columbia.edu

Original version: April 27, 2010

This version: February 8, 2011

To appear in Interfaces

Abstract

The game of golf involves many diﬀerent types of shots: long tee shots (typically hit with a driver),

approach shots to greens, shots from the sand, putts on the green, and others. While it is easy

to determine the winner in a golf tournament by counting strokes, it is not easy to assess which

factors most contributed to the victory. In this paper we apply an analysis based on strokes gained

(previously termed shot value) to assess the performance of golfers in diﬀerent parts of the game

of golf. Strokes gained is a simple and intuitive measure of the contribution of each shot to a

golfer’s score. Strokes gained analysis is applied to extensive ShotLinkTM data in order to rank

PGA TOUR golfers in various skill categories and to quantify the factors that diﬀerentiate golfers

on the PGA TOUR. Long game shots (those starting over 100 yards from the hole) explain about

two-thirds of the variability in scores among golfers on the PGA TOUR. Tiger Woods is ranked

number one in total strokes gained and he is ranked at or near the top of PGA TOUR golfers in

each of the three main categories: long game, short game and putting. His dominance is a result of

excelling at all phases of the game, but his long game accounts for about two-thirds of his scoring

advantage relative to the ﬁeld. A similar approach is used to rank PGA TOUR courses in terms

of overall diﬃculty and diﬃculty in each part of the game. A preliminary analysis shows that the

recent change in the groove rule for irons by the United States Golf Association (USGA) has had

almost no impact on scores from the rough.

Acknowledgement: Thanks to the PGA TOUR for providing the ShotLinkTM data. Thanks

also to Kin Lo of the PGA TOUR, Richard Rendleman and Soonmin Ko for helpful discussions

and comments, Lou Lipnickey for extensive programming on the project, Alexsandra Guerra for

assistance with the data and the USGA for supporting the initial development of the Golfmetrics

software.

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Electronic copy available at: http://ssrn.com/abstract=1765964

1. Introduction

A golf score tells how well a golfer played overall, but does not reveal what factors contributed most

to that score. The goal of this paper is to analyze the play of PGA TOUR professional golfers in

order to understand and quantify the contributions of diﬀerent categories of golf shots (e.g., long

game, short game and putting) in determining a total golf score for an eighteen-hole round. This

performance attribution analysis is used to rank golfers in various skill categories. The relative

impact of each skill category on overall score is also examined.

Golf fans know that Tiger Woods is the best golfer of his generation, but it is often debated

whether his low scores are primarily due to superior putting, wedge play around the greens, driving,

or some other factor or combination of factors. Sweeney (2008) writes: “What really diﬀerentiates

Woods from everyone else is his ability to make more putts from the critical range of 10 to 25 feet.”

In June 2010, U.S. Open winner Geoﬀ Ogilvy said, “I think by now every player on tour is aware

that the biggest reason Tiger is the best is because he putts the best.” (Diaz, 2010). In spite of

these assertions, it is not clear whether putting is the most important factor contributing to Tiger’s

scoring advantage.

This paper shows that Tiger Woods’ scoring advantage in the years 2003-2010 was 3.20 strokes

per round better than an average tournament ﬁeld. Tiger is ranked at or near the top of PGA

TOUR golfers in all three categories (long game, short game and putting), and his dominance is a

result of excelling at all phases of the game. But Tiger’s long game accounts for 2.08 of the total

3.20 stroked gained per round, so about two-thirds of his scoring advantage comes from shots over

100 yards from the green. Tiger’s putting advantage versus the ﬁeld is 0.70 shots per round, while

his short game contributes 0.42 shots per round. Even though he is a phenomenal putter, his gain

from putting is less than the 1.01 strokes he gains from shots starting between 150 and 250 yards

from the hole, and comparable to the 0.70 strokes he gains from long tee shots.

Performance attribution analysis is diﬃcult using the current standard golf statistics, many of

which involve relatively crude counting measures. For example, the fairways hit statistic is the

count of the number of fairways hit on a long tee shot (i.e., on par-4 and par-5 holes) divided by

the number of tee shots. One problem with this statistic is that it doesn’t distinguish between

shots which barely miss the fairway, from shots that miss the fairway by a large amount and end

up behind trees, in water, or out of bounds. Many standard golf statistics have the drawback

that they mix several parts of the game together. For example, the sand save statistic counts the

number of times a golfer gets the ball in the hole in one or two shots from a greenside sand bunker

divided by the number of attempts. However, this statistic mixes together sand play with putting,

making it diﬃcult to isolate sand shot skill from putting skill. It is useful to have shot location

information in order to better measure driving skill, sand play and putting skill. The PGA TOUR

has collected this type of detailed data using their ShotLinkTM system since 2003.

In this paper, detailed shot data is used to assess and rank the performance of PGA TOUR

golfers in three main parts of the game: the long game (shots over 100 yards from the hole), the

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short game (shots under 100 yards from the hole, excluding putts), and putting (shots on the green).

The performance analysis is based on the concept of strokes gained (see Broadie 2008, where the

term shot value was used instead of strokes gained), which measures the quality of each shot based

on its starting and ending locations. As pointed out in Broadie and Ko (2009), the strokes gained

metric is related to the value function of a dynamic program. If, for example, a golfer hits a poor

sand shot followed by great putt, the sand shot will have a negative strokes gained value while the

putt’s strokes gained value will be positive. This approach allows each shot to be measured on its

own merits, which is not possible with the sand save statistic which combines both shots. Just as

golf scores are often compared to the benchmark of par, strokes gained represents the quality of a

shot relative to a benchmark deﬁned by the average performance of PGA TOUR golfers. Adding

strokes gained for shots in a given category gives a performance measure for that category and is

useful in understanding a golfer’s strengths and weaknesses and in comparing one golfer to another.

Strokes gained analysis is used to determine what separates the top golfers on the tour from others.

Since the publication of the landmark book by Cochran and Stobbs (1968), a large literature on

the scientiﬁc and statistical analysis of golf has developed. Recent surveys include Penner (2003),

Farrally et al. (2003) and Hurley (2010). Statistical analysis of amateur golfers was done in Riccio

(1990). An early attempt to quantify the value of a shot was given in Landsberger (1994). Several

papers have investigated which golf skill factors are most important in determining earnings in

professional tournaments. Examples include Davidson and Templin (1986), Shmanske (1992), Moy

and Liaw (1998), Berry (1999), Berry (2001), Nero (2001), Callan and Thomas (2007), Shmanske

(2008) and Puterman and Wittman (2009). Most of these studies were limited by the lack of

detailed shot information and had to rely on standard golf statistics (e.g., putting average, sand

save percent, fairways hit, etc.). The strokes gained approach used in this paper directly decomposes

a golfer’s score by the quality of each shot, and is an alternative to the regression analyses used in

many earlier studies.

Strokes gained analysis was introduced in Broadie (2008), primarily to determine which skills

most separate the play of professional and amateur golfers. Putting performance on the PGA

TOUR was investigated in Fearing et al. (2010), also using strokes gained analysis. In their study,

the putting benchmark was adjusted to account not only for the distance to the hole, but also the

diﬃculty of the green on each hole and the quality of putters in each tournament. Larkey (1994) and

Berry (2001) represent early eﬀorts to adjust tournament results for course diﬃculty and golfer skill

factors. More recently, Connolly and Rendleman (2008) employed a statistical model in order to

investigate golfer skill and streaky play on the PGA TOUR. The important idea in Larkey (1994),

Berry (2001), Connolly and Rendleman (2008) and Fearing et al. (2010), is that overall scores and

number of putts depend on golfer skill and on the diﬃculty of the course. Fewer putts are sunk

on bumpy greens and scores are higher on more diﬃcult courses, e.g., those with narrow fairways,

deep rough and many water hazards. However, discerning the diﬃculty of a course is problematic

when golfer skill is unknown: scores could be high because of less skill or a more diﬃcult course.

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The issue of disentangling golfer skill from course diﬃculty in golf scores also arises in creating golf

handicaps for amateur golfers. For issues related to golf handicapping, see Pollock (1974), Scheid

(1977) and Stroud and Riccio (1990).

This paper extends the analysis in Broadie (2008) in several ways. First, a benchmark function

representing the average strokes to complete a hole is estimated for PGA TOUR golfers. The

benchmark is interesting in itself, because it summarizes the skill of PGA TOUR golfers in vari-

ous shot categories. A component of estimating the benchmark is the automatic identiﬁcation of

recovery shots. An estimation procedure is used to simultaneously estimate the diﬃculty of each

course and round and to adjust the strokes gained results for the diﬃculty factors. In addition to

providing a better measure of golfer performance, this procedure allows courses to be ranked in

terms of overall diﬃculty and diﬃculty in each part of the game. Finally, the analysis is applied

to a database of more than eight million shots by PGA TOUR golfers, leading to many interesting

results, including the relative importance of the long game versus the short game.

In the next section the strokes gained concept is deﬁned and illustrated. The construction of

a benchmark function representing the average strokes to complete a hole for PGA TOUR golfers

is described in Section 3. In Section 4 the strokes gained approach is applied to rank PGA TOUR

golfers and analyze the factors that diﬀerentiate golfers on the PGA TOUR. PGA TOUR courses

are ranked as well. A preliminary analysis of the eﬀect of the USGA’s groove rule change is also

presented. Brief concluding remarks are given in Section 5.

2. Strokes gained

In this section the strokes gained concept is deﬁned, examples are given and a simple but important

additivity property of strokes gained is presented. The connection of strokes gained with dynamic

programming is mentioned at the end of the section.

Strokes gained is a simple and intuitive quantitative measure of the quality of a golf shot.

Suppose a function J(d, c) has been estimated, where drepresents the distance to the hole from

the current location (not the distance of the shot), crepresents the condition of the current ball

location (i.e., green, tee, fairway, rough, sand or recovery) and Jis the average number of strokes

a PGA TOUR golfer takes to ﬁnish the hole from the current location. For brevity, Jwill often

be referred to as the benchmark. Deﬁne the strokes gained of the ith shot on a hole that starts at

(di, ci) and ﬁnishes at (di+1, ci+1 ) to be

(1) gi=J(di, ci)−J(di+1, ci+1 )−1.

Strokes gained represents the decrease in the average number of strokes to ﬁnish the hole from

the beginning of the shot to the end of the shot, minus one to account for the stroke taken. For

example, suppose the average number of shots to complete the hole is 2.6 from a position in the

fairway forty yards from the hole. If the golfer hits the shot to one foot from the hole, where the

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average number of shots to complete the hole is 1.0, then equation 1 attributes a gain of 0.6 strokes

to the shot: it reduced the average number of shots to complete the hole by 1.6 and it took one

shot to do so, for a gain of 0.6. In general, a positive giindicates that a shot is better than a PGA

TOUR golfer’s average shot, while a negative giindicates that a shot is worse than average.

The units of strokes gained are strokes, e.g., a strokes gained value of −0.1 means the shot is 0.1

strokes worse than the benchmark. Because the units are the same for diﬀerent types of shots, e.g.,

long shots and putts, the strokes gained metric oﬀers a consistent method for evaluating diﬀerent

aspects of the game of golf. The strokes gained approach solves the problem of incommensurable

measures in standard golf statistics that was pointed out in Larkey and Smith (1999).

To give an example of strokes gained, suppose that PGA TOUR golfer A plays a long par-3

that takes the PGA TOUR ﬁeld an average of 3.2 strokes to complete the hole. Golfer A’s tee shot

ﬁnished on the green, leaving a 16-foot putt for birdie. From 16 feet, the PGA TOUR ﬁeld takes an

average of 1.8 putts to ﬁnish the hole. The PGA TOUR ﬁeld will one-putt about 20% of the time,

two-putt about 80% of the time, and rarely three-putt from 16 feet (1.8 = 20%(1)+80%(2)+0%(3)).

The ball started in a spot where the benchmark is 3.2 and ﬁnished at a position where the benchmark

is 1.8, so the strokes gained for the shot is 3.2−1.8−1 = +0.4. Golfer A left his birdie putt one

inch short. His ball started in a spot where the benchmark is 1.8 and ﬁnished in a spot where the

benchmark is one (the average number of shots to ﬁnish the hole for a tap-in is one), for a strokes

gained value of: 1.8−1−1 = −0.2. Golfer A’s missed putt represents a loss of 0.2 shots relative to

the benchmark, because he reduced the average number of strokes to complete the hole by 0.8 but

he used one putt to do so. Because a PGA TOUR golfer only sinks 20% of 16-footers, missing this

putt doesn’t cost a full shot: it really only costs 0.2 strokes. To complete the example, golfer A

tapped-in for par. The strokes gained equation (1) gives a value of zero for this putt, because

he reduced the benchmark from one to zero using one shot. This makes sense, because sinking a

one-inch putt neither gains nor loses shots relative to the benchmark.

The strokes gained metric has a simple but important additivity property: the strokes gained of

a group of shots is the sum of the strokes gained of the individual shots. Suppose a golfer takes n

shots on a hole. Then the total strokes gained for the nshots is:

(2)

n

X

i=1

gi=

n

X

i=1

(J(di, ci)−J(di+1, ci+1 )−1) = J(d1, c1)−n

because of the telescoping sum and J(dn+1 , cn+1) = 0 for the last shot which ends in the hole. In

the previous example, golfer A’s score of n= 3 represents a net gain of 0.2 strokes compared to the

benchmark of J(c1, d1)=3.2 from the tee. Golfer A did this with a great tee shot (+0.4 strokes

gained), a disappointing putt (−0.2 strokes gained), and a tap-in (0 strokes gained), for a total

strokes gained of 0.2 for the hole, consistent with equation (2).

Let’s consider PGA TOUR golfer B playing the same par-3 hole. Golfer B’s tee shot missed

the green long and left. From this position in the rough, suppose the average number of shots to

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complete the hole (the benchmark) is 2.6. The strokes gained equation (1) gives: 3.2−2.6−1 = −0.4,

so golfer B lost 0.4 strokes compared to the PGA benchmark. Golfer B hit his second shot from the

rough to inside of four feet from the hole, where the benchmark score is 1.1 (a PGA TOUR golfer

sinks about 90% of these putts). Applying equation (1) gives: 2.6−1.1−1 = 0.5, so golfer B’s

second shot gained a half-stroke compared to the benchmark. Golfer B sunk the four-footer and

the strokes gained equation gives 1.1−0−1 = 0.1. Golfer B’s score of three also represents a

net gain of 0.2 strokes compared to the benchmark value of 3.2 from the tee. Golfer B did this

with a poor tee shot (−0.4 strokes gained), a good chip from the rough (0.5 strokes gained), and a

one-putt (0.1 strokes gained), for a total of 0.2 strokes gained for the hole.

Golfers A and B had the same score on the hole, but they did it in very diﬀerent ways. If this

was a representative example, we could see that golfer A has a great long game, while golfer B has

a great short game. Strokes gained allow us to compare golfer A’s game to golfer B’s, both in total

strokes gained (for the hole, round, or season) and in various categories (e.g., long game, short

game, and putting). This observation can be formalized by decomposing the total strokes gained

for a round into diﬀerent categories as follows:

(3)

m

X

i=1

gi=X

i∈L

gi+X

i∈S

gi+X

i∈P

gi

where mis the total number of shots in a round, Lis the set of indices corresponding to long

game shots, Sis the set of indices corresponding to short game shots, Pis the set of putts and

where {1,2, . . . , m}=L ∪ S ∪ P. In a similar way, the strokes gained for a given category can be

further decomposed into subcategories. For example, the total strokes gained of all long game shots

can be split into the sum of strokes gained for tee shots and approach shots from various distance

categories. Unlike fraction of greens hit, proximity to the hole, or other statistical measures, the

strokes gained approach provides a consistent way to quantify the value of shots in various categories

and subcategories.

The game of golf can be modeled as a dynamic program. The score on a hole depends on the

strategy and results of each of the shots on the hole. The optimal strategy from the tee depends

on all of the possible outcomes of the ﬁrst shot and the optimal strategy for the second shot, which

depends on all of the possible outcomes of the second shot and the optimal strategy for the third

shot, etc. The solution of a dynamic program involves starting from the last stage, in this case

the shot which ends in the hole, and working backwards to determine the optimal strategy. The

Bellman (1957) equation says:

(4) J(di, ci) = min

µE[J(di+1, ci+1 )+1|(di, ci, µ)]

where the expectation is taken over (di+1, ci+1), the random distance and condition of the end of

shot i, given its start at (di, ci) and the strategy µ(e.g., target and club) chosen by the golfer.

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For more detail, see Broadie and Ko (2009). This paper does not address the strategy choices of

golfers, but PGA TOUR golfers are among the best golfers in the world, so it is not unreasonable

to assume that they play optimal or nearly optimal strategies and the observed data can be used to

estimate J(di, ci) = E[J(di+1, ci+1 )+1|(di, ci, µ∗)], where µ∗represents an optimal strategy. Now

an individual shot can be measured by the diﬀerence in the left and righthand sides of the equation

for a particular outcome, i.e., by J(di, ci)−J(di+1, ci+1)−1, which is the strokes gained deﬁnition

given in equation (1). This dynamic program viewpoint provides the justiﬁcation for the strokes

gained deﬁnition.

3. PGA TOUR benchmark

The strokes gained computation is based on a benchmark function that gives the average number

of strokes for PGA TOUR golfer to complete a hole. The benchmark typically increases with

the distance to the hole and depends on the course condition at the location of the ball, i.e., tee,

fairway, rough, green, sand or recovery. Shots from the rough are more diﬃcult than shots from

the fairway, and the benchmark is larger as a consequence. There are situations, typically from

the rough, where a direct shot to the hole is impossible because the path is blocked by trees or

other obstacles. In this case a golfer may elect to play a recovery shot, i.e., a short shot that is hit

back to the fairway rather than directly toward the hole. Recovery shots are placed in their own

category in order to better estimate the diﬀerential eﬀects of fairway and rough. The estimation of

the benchmark function, the recovery shot identiﬁcation procedure and empirical results are given

in this section.

The results in this paper are based on the PGA TOUR’s extensive ShotLink database, which

includes all shots at PGA TOUR tournaments from 2003 to 2010. The data is collected by 250

volunteers at each tournament. The ShotLink database contains more than eight million shots

(about one million shots per year), with shot locations measured to within one inch on putts and

one foot on other shots. Further information on the ShotLink system is given in Deason (2006).

The ShotLink database does not include detailed shot information for the four major tournaments:

the Masters, U.S. Open, British Open and the PGA.

The benchmark function (the average number of shots to complete the hole) needs to be deﬁned

in terms of observable information recorded in the database. Not all shots from the fairway with

125 yards to the hole are equal in diﬃculty, since there are many other factors involved: the ball’s lie

might be perfect or in a divot, the golfer’s stance might be level or on a hill, the wind could be calm

or gusting, etc. All of these other factors aﬀect the diﬃculty of a shot and the average number

of shots to complete the hole. However, the benchmark can only be computed from observable

information, and the ShotLink database includes the most important of these factors: the distance

from the hole and the condition of the ball (e.g., tee, fairway, green, sand, or rough). The benchmark

function can be interpreted as an average over these other unobservable factors.

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The goal is to estimate the benchmark function J(d, c), where drepresents the distance to the

hole from the current location and crepresents the condition of the current location (i.e., green, tee,

fairway, rough, sand or recovery). Statistical and model-based approaches are two main ways to

accomplish this. Statistical procedures include simple interpolation, linear regression, splines, kernel

smoothing and other methods. In model-based approaches a parametric analytical or simulation

model is formulated and optimization is used to determine the model parameters that best ﬁt the

data. Both approaches attempt to ﬁnd a benchmark that is close to the data and appropriately

smooth to take into the noise in the data.

The large size of the database allows for accurate estimation of the benchmark, since in most

distance and condition categories, there are many shots available to estimate the average score

to complete the hole. Experimentation with several approaches yielded similar results. Piecewise

polynomial functions were used as the form of the benchmark, except for putts on the green.

For putts on the green, a model-based approach is used to ﬁt one-putt probabilities based on a

simpliﬁcation of the putting model presented in Broadie and Bansal (2008). This is combined with

a statistical model for three-putts to give an average score function for putts on the green. This

approach is used to smooth the somewhat limited data for long putt distances.

3.1. Tee shot benchmark

From the tee, a simple linear regression of average score (J) on the distance to the hole (d, measured

in yards) for PGA TOUR pros using 2003-2010 data gives: J= 2.38 + 0.0041d. The distance to

the hole dis measured along the fairway from the tee to the hole (i.e., the dogleg distance, not the

direct “as the crow ﬂies” distance). In this regression, the data are grouped into 20-yard distance

buckets and the R2of the regression is over 98%. The slope of the equation implies that every

additional 100 yards of hole distance adds 0.41 strokes to the average score of a PGA TOUR pro.

(This regression is similar to the result 2.35 + 0.0044dobtained in Cochran and Stobbs (1968),

based on a smaller set of data collected from a single British professional tournament in 1964.)

In spite of the high R2, a linear regression does not provide an adequate ﬁt to the data as shown

in Figure 1. In particular, the average score from the tee exhibits a jump between long par-3 holes

at 235 yards and short par-4 holes at 300 yards (there is little data between these distances). The

computations in the paper are based on a more accurate piecewise polynomial ﬁt to the data. The

results are given in Appendix A.

Broadie (2008) ﬁnds that the average score from the tee is 2.79 + 0.0066dfor golfers whose

18-hole average score is 90. The slope implies that every additional 100 yards of hole distance adds

0.66 strokes to the average score of 90-golfers, while for PGA TOUR pros it adds 0.41 strokes.

The USGA refers to this slope as the ability to overcome distance. For 90-golfers, going from 180

yards (par-3 distance) to 580 yards (par-5 distance) will increase their average score by about 2.6.

But the par goes up by 2, so 90-golfers do worse relative to par on par-5 holes compared to par-3

holes. Going from a hole of 180 yards (par-3 distance) to 580 yards (par-5 distance), pros will see

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2.5

3.0

3.5

4.0

4.5

5.0

100 200 300 400 500 600

Distance to hole

Average score

Figure 1:Average score from the tee for PGA TOUR golfers in 2003-2010. Distance to the hole is

measured along the fairway from the tee to the hole, not “as the crow ﬂies.”

an average score increase of 1.6. The par goes up by two, so the pros do better relative to par on

par-5 holes compared to par-3 holes. The main reason is the 290-yard average distance that the

pros drive the ball, compared to an average drive of about 210 yards for 90-golfers. (Of course, on

par-5 holes 90-golfers have more chances to ﬂub shots or hit into trouble.)

Using the Golfmetrics database of primarily amateur data, described in Broadie (2008) an

average score from the tee of 2.79 + 0.0066dis found for golfers whose 18-hole average score is 90.

The slope implies that every 100 yards of hole distance adds 0.66 strokes to the average score, while

for PGA TOUR pros it adds 0.41 strokes. The USGA refers to this slope as the ability to overcome

distance. For 90-golfers, going from 180 yards (par-3 distance) to 580 yards (par-5 distance) will

increase their average score by about 2.6. But the par goes up by 2, so 90-golfers do worse relative

to par on par-5 holes compared to par-3 holes. Going from a hole of 180 yards (par-3 distance) to

580 yards (par-5 distance), pros will see an average score increase of 1.6. The par goes up by two,

so the pros do better relative to par on par-5 holes compared to par-3 holes. The main reason is

the 290-yard average distance that the pros drive the ball, compared to an average drive of about

210 yards for 90-golfers. (Of course, on par-5 holes 90-golfers have more chances to ﬂub shots or

hit into trouble.)

3.2. Benchmark within 50 yards of the hole

In this subsection average strokes to complete the hole from the sand, rough and fairway are

compared on shots within 50 yards of the hole. It is often claimed that professional golfers are so

good from the sand that they would rather be in the sand than in the rough. Figure 2 illustrates

the data (bucketed in 5-yard increments) and the ﬁtted curves. The ﬁgure shows that when the

distance to the hole is less than 15 yards or greater than 34 yards, sand shots have larger average

strokes to complete the hole than shots from the rough from the same distance. In the range from

15 yards to 34 yards, sand shots are easier than shots from the rough, on average. Conditioned on

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the shot starting within 50 yards of the hole, the average initial distance to the hole for shots from

the sand and rough is 16 yards, just about the distance of equal diﬃculty for sand and rough shots.

The average score can be translated into an up-and-down fraction, i.e., the fraction of the time it

takes two or less shots to ﬁnish the hole. From 15 yards from the hole, pros get up-and-down 51% of

the time from the rough or sand and 69% of the time from the fairway. At 25 yards from the hole,

pros get up-and-down 42% of the time from the sand, 35% from the rough and 54% of the time

from the fairway. These are averages over all situations; note that the outcome for an individual

shot will depend on the ball’s lie, the contour of the green near the hole and other factors. However,

the distance from the hole and condition of the ball are primary factors in determining the average

number of shots to complete the hole.

2.0

2.2

2.4

2.6

2.8

3.0

0 1020304050

Distance to hole

Average strokes to complete hole

Sand

Rough

Fairway

Figure 2:Average strokes to complete the hole from the rough, sand and fairway for PGA TOUR

golfers in 2003-2010.

3.3. Putting benchmark

In this subsection the estimation of the benchmark function for putts is discussed. The benchmark

is ﬁt in three steps. First, a one-putt probability function is ﬁt, then a three-putt function is ﬁt,

and then these two are combined into a benchmark average putts-to-complete-the-hole function.

This approach is followed for several reasons. First, the data is sparse and noisy for long putts

(e.g., greater than 50 feet from the hole) and so smoothing is necessary. Second, the procedure

works well for ﬁtting smaller data sets and it is useful to have a consistent procedure for all sets of

data. Finally, golfers think in terms of one-putts and three-putts, so these models and results are

of independent interest.

The one-putt probability function is based on a simple physical model for putts. Putting skill

is modeled with two components: a random distance and a random direction, both independently

10

distributed normal random variables. The random direction of the putt with respect to the hole

is α, with α∼N(0, σ2

α), so angular putt errors have a standard deviation of σα. The putt rolls a

random distance lwith l∼N(d+t, (d+t)2σ2

d), where dis the initial distance to the hole and tis

the target distance beyond the hole (all measured in yards). The standard deviation of the distance

a putt rolls, (d+t)σd, is proportional to the intended target distance d+t. If t= 1/2 yards it

means the golfer aims to hit the putt 1.5 feet beyond the hole. For the putt to have a chance of

ﬁnishing in the hole, the angle must satisfy |α| ≤ αc= tan−1(r/d), where dis the distance to the

hole and ris the radius of the hole (2.125 inches). In addition, the distance the putt rolls, l, must

be at least d, otherwise the putt will not reach the hole. If the putt is hit too hard (even if hit

straight at the hole) and rolls a distance greater than d+h, it will also not result in a holeout.

A holeout occurs if the putt rolls a distance lsatisfying d≤l≤d+hand is hit with an angle

satisfying |α| ≤ αc.

This model is a generalization of the Gelman and Nolan (2002) model which only takes putt

direction into account. It is a simpliﬁcation of Broadie and Bansal (2008), which models distance,

direction and green reading errors, but is not analytically tractable and requires simulation to

evaluate. The holeout criterion is used for analytical tractability. A detailed physical model for

holeouts, i.e., the putt ﬁnishing in the hole, was developed in Holmes (1991) and used in Broadie

and Bansal (2008). This model has few parameters, has a physical interpretation, is analytically

tractable, and it ﬁts the data very well. The probability of a one-putt, p1(d;σα, σd, t, h) is:

P(|α| ≤ αc)P(d≤l≤d+h) = P(|α| ≤ αc)P−t

σd(d+t)≤Z≤h−t

σd(d+t)

=2Φ αc

σα−1)Φh−t

σd(d+t)−Φ−t

σd(d+t),

(5)

where Zis a standard normal random variable and Φ is the cumulative distribution of a standard

normal. Given a set of one-putt data by distance to the hole, an optimization model is solved to

ﬁnd the best-ﬁt parameters σαand σd. The model can be ﬁt very quickly because of equation (5)

and the readily available routines for computing Φ. (The parameters tand hwere ﬁxed at t= 1/2

and h= 2/3.)

The three-or-more putt probability function is estimated by ﬁtting the equation:

(6) p3(d;a0, a1, a2, a3) = 1

1 + ea0+a1d+a2d2+a3.

for the parameters a0, a1, a2and a3. This functional form was chosen because, of the many forms

tested, it ﬁt the data very well. Four (or more) putts are observed in the professional data, but are

so rare that the ﬁt is not aﬀected. The optimization model to ﬁnd the best-ﬁt parameters is quick

to solve.

11

0%

20%

40%

60%

80%

100%

2 3 4 5 6 8 10 15 20 30 40 60

One-putt probability

Length of putt (feet)

0%

10%

20%

30%

2 3 4 5 6 8 10 15 20 30 40 60

Three-putt probability

Length of putt (feet)

Figure 3:PGA TOUR putting results using 2003-2010 data. Left panel: one-putt probability. Right

panel: three-putt probability. Dots represent the data and the curves are the ﬁtted models.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2 3 4 5 6 8 10 15 20 30 40 60

Average number of putts

Length of putt (feet)

Figure 4:Average number of putts by initial distance to the hole for PGA TOUR golfers in 2003-2010.

Dots represent the data and the curve is the ﬁtted benchmark model.

The average number of putts to holeout benchmark function is now easy to compute using

(7) J(d) = p1(d;σα, σd, t, h) + 2 (1 −p1(d;σα, σd)−p3(d;a0, a1, a2, a3)) + 3p3(d;a0, a1, a2, a3),

where the condition of starting on the green is implicit. This approach of separately ﬁtting a

physical model for one-putts, a statistical model for three-putts, and combining to give an average

number of putts curve is simple, easy to calibrate, and most importantly, ﬁts the data very well.

This method has the advantage of being able to see one-putt, two-putt, and three-putt probability

functions as well as the average number of putts function. A diﬀerent approach is used in Fearing

et al. (2010), where a statistical model is used for the one-putt probability function and a gamma

distribution is ﬁt to the remaining distance of missed putts.

The left panel of Figure 3 illustrates the data and the ﬁtted one-putt probability curve. The

12

parameters of the ﬁtted curve are: σα= 1.46, σd= 0.057 (with t= 0.5 and h= 0.667). The model

probability is almost always within one standard error of the data. (The standard errors are too

small to be shown clearly on the graph.) PGA TOUR golfers one-putt 50% of the time from a

distance of eight feet. It is interesting to compare this result from 2009 data with earlier results.

Cochran and Stobbs (1968, Chap. 29) found that in 1964 pros one-putt 50% of the time from a

distance of seven feet, though their sample was fairly small. Soley (1977, Chap. 4) found that pros

sunk 50% of their putts from seven feet using data from the early 1960s on regular tournament

courses. He found the same result at the 1974 U.S. Open at Winged Foot, but closer to six feet

at the 1972 U.S. Open at Pebble Beach. Using hand-collected data from PGA tournaments in the

1980s, Pelz found that pros sunk 50% of their putts from about six or seven feet (Pelz (1989, p.38)

and Pelz (2000, p.7)). The increase in the 50% one-putt distance from six or seven feet to the

current eight feet could be due to better conditioned greens, better putting skill, or a combination.

By contrast, amateur golfers with an average 18-hole score of 90 (90-golfers), one-putt 50% of the

time from ﬁve feet.

The putting data yields many interesting results. For example, the average number of one-putts

for a single pro golfer from 22 feet or over in a four-round tournament is only 1.4. This is a much

smaller number than most people expect, perhaps because of the bias from watching television

highlights of putts that are made, while missed 30-footers rarely make the highlight reel. A pro

averages just over ﬁve putts per round from 22 feet or more and sinks slightly less than 7% of these

long putts.

The right panel of Figure 3 illustrates the data and the ﬁtted three-putt probability curve. The

parameters of the ﬁtted curve are: a0=−0.106, a1= 5.49, a2= 0.000563 and a3=−0.00398.

It is not until 40 feet that the three-putt probability for PGA TOUR golfers exceeds 10%. PGA

TOUR golfers average 0.55 three-putt greens per round, or about 2.2 per four-round tournament.

Amateur 90-golfers three-putt about 2.3 times per round, or four times more often than pros.

Figure 4 shows how the average number of putts increases with distance for PGA TOUR golfers.

PGA TOUR golfers average two putts from 33 feet, i.e., the fraction of one-putts equals the fraction

of three-putts. Amateur 90-golfers average two putts from 19 feet.

3.4. Recovery shots

A shot is called a recovery shot if the golfer’s shot to the hole is impeded by trees or other obstacles.

Even if the golfer decides to hit toward the hole through a small opening in trees, or attempts a hook

or slice around an obstacle, it is still considered a recovery shot because the golfer is recovering from

trouble. In this subsection we discuss this category of shots, their importance in the benchmark,

and how they are identiﬁed.

Suppose a golfer hits a long drive but ends up behind a tree and is forced to chip back out onto

the fairway for the second shot, i.e., the second shot is a recovery shot. If the benchmark does not

account for recovery shots, the strokes gained of the tee shot may be close to zero, but the second

13

shot will have a very negative strokes gained since it didn’t travel very far. This doesn’t make sense,

because the problem was caused by a poor tee shot, not a poor second shot. The strokes gained

equation can account for this situation by identifying the condition of the second shot as a recovery

shot, and the benchmark will have a larger average number of strokes to complete the hole than

from a comparable distance in the rough. In this example, a using separate benchmark for recovery

shots will give a negative strokes gained for the poor tee shot and a strokes gained of close to zero

for the second shot. The recovery label is important for correctly allocating strokes gained between

the two shots and important for estimating the penalty for being in the rough versus fairway. In

this example, the rough was not the direct cause of the increase in score—it was an obstructed

route to the hole.

The recovery condition is an important element in assessing the quality of a shot, but the

ShotLink database does not have an identiﬁer of this category of shot. The main reason is that

labeling a shot a recovery shot is a judgement call, unlike distance to the hole which is an observable

and objective quantity. The recovery shot condition needs to be inferred from existing information

in the data. Because the database contains millions of shots, a manual identiﬁcation procedure is

not feasible. The automatic recovery shot identiﬁcation procedure has two steps. The ﬁrst step

ﬁnds shots that travel an unusually short distance (e.g., travel less r1= 40% of the distance to the

hole) or are hit at a large angle relative to the hole (e.g., an absolute angle greater than r2= 15

degrees with respect to the ball-hole line). Shots are also screened to started at least r3= 30 yards

from the hole. The parameters are determined by visually inspecting a number of shots which

satisfy the criterion. The second step ﬁnds shots that start close, e.g., within r4= 3 yards, to

shots by other golfers labeled as recovery shots in step one. For example, suppose two golfers are

in nearly identical recovery shot positions obstructed by trees. The ﬁrst golfer chips back onto the

fairway while the second golfer attempts a big slice around the trees. The ﬁrst golfer’s shot would

be identiﬁed as a recovery shot by step one, but the second golfer’s shot might not, even though

the shot started in the same position and was signiﬁcantly aﬀected by trees. The second step of

the procedure allows the shot of the second golfer to be labeled as a recovery shot.

Figure 5 illustrates three shots labeled as recovery shots by this procedure.1This method of

inferring which shots are recovery shots works very well, but two types of errors will occur. Some

shots will be labeled as recovery shots that are not, and some shots that are recovery shots will not

be labeled as such. Given the current data and judgement involved, it is not possible to design an

error-proof procedure. However, the magnitudes of the two types of errors can be controlled by the

choice of the parameters. Recovery shot identiﬁcation is important when comparing the average

number of shots to complete a hole from the rough versus the fairway. Once recovery shots are

identiﬁed, benchmark functions are ﬁt to the data using piecewise polynomials. The results are

given in Appendix A. Figure 6 shows the average strokes to complete the hole for recovery shots

1Golf course images from Google Earth are used to display the shots. The ShotLink database contains shot

starting and ending position using (x, y) coordinates that were translated to latitude and longitude for plotting.

14

Figure 5:Recovery shot examples. Left panel: Corey Pavin, 6/11/2006, hole 3, Westchester Country

Club. The shot indicated is labeled a recovery shot because of the distance criterion. Middle panel: Tim

Clark, 6/26/2005, hole 15, Westchester Country Club. The shot indicated is labeled a recovery shot

because of the distance criterion. Right panel: Fred Couples, 6/11/2006, hole 15, Westchester Country

Club. The shot indicated would not be labeled a recovery shot by the distance or angle criteria, but it

is labeled a recovery shot because it is nearby to another golfer’s recovery shot (not shown). Arcs show

100-, 150- and 200-yard distances from the hole.

and shots from the rough and fairway.

4. PGA TOUR golfer rankings and results

In this section strokes gained analysis is used to rank PGA TOUR golfers in various skill categories

and subcategories. The strokes gained are ﬁrst adjusted by the diﬃculty course for that round

in order to produce more reliable comparisons between golfers. This section provides details of

the adjustment procedure, results and discussion of the rankings, and analyzes which skill factors

separate the best golfers on the tour from others.

4.1. Course-round diﬃculty adjustments

Some four-round PGA tournaments have winning scores of 30 under par while for others it may

only be 6 under par. The diﬀerence of six shots per round is due to two main factors. First, the

course at one tournament may be easier than the course at another tournament. The diﬀerence in

diﬃculty can be due to length of the course, width of the fairways, ﬁrmness of the greens, height

of the rough, severity of bunkers, and other factors. Second, from one round to the next, scores on

15

2.5

3.0

3.5

4.0

4.5

5.0

100 200 300 400

Distance to hole

Average strokes to complete hole

Recovery

Rough

Fairway

Figure 6:Average strokes to complete the hole for recovery shots and shots from the rough and fairway

for PGA TOUR golfers in 2003-2010. Most recovery shots are in the range between 150 and 300 yards

from the hole. In this range, the average number of strokes to complete the hole is 0.6 strokes greater

from a recovery position than from the fairway and 0.4 greater than from the rough.

the same course can change dramatically because of diﬀerent weather conditions, especially wind

which aﬀects the ﬂight of the ball. Of all of these factors, only the length of the course is directly

included in the benchmark average score. So a golfer who shoots 12 under par in the tournament

with a winning score of 30 under par is likely to have played relatively worse than a golfer with a

score of 4 under par where the winning score is 6 under par. In order to make a direct comparison

of two golfers who play in a diﬀerent set of tournaments, it is necessary to adjust scores and strokes

gained for the diﬃculty of the course for each round.

Let gij represent the total (18-hole) strokes gained for golfer iplaying on a course and round

indexed by j. In order to separate golfer skill from the course diﬃculty for that round, the strokes

gained, gij , is modeled as

(8) gij =µi+δj+ij

where µirepresents golfer i’s intrinsic skill (i.e., the golfer’s average strokes gained on a PGA TOUR

course of average diﬃculty), δjrepresents the intrinsic diﬃculty of the course-round j, and ij is a

random mean-zero error term.2The model is estimated using a standard iterative procedure (see,

e.g., Larkey (1994), Berry (2001) and Connolly and Rendleman (2008)).

2In the ShotLink data, when a tournament is played on multiple courses, data is collected only at one course.

There are examples in the data with two diﬀerent tournaments played on diﬀerent courses on the same days, and

these are represented by diﬀerent course-round indices j. The δjare estimated as random eﬀects.

16

4.2. Golfer strokes gained results and rankings

The main golfer results are given in Tables 1−3. Table 1 gives PGA TOUR golfer rankings based

on the entire 2003-2010 data which contains over eight million shots. Ranks are relative to the

299 golfers with 120 or more rounds in the data.3Tiger Woods’ total strokes gained per round is

3.20, which means that he gains, on average, 3.20 strokes per 18-hole round versus an average PGA

TOUR ﬁeld. That is, 3.20 represents the µfor Tiger Woods estimated from equation (8). Tiger is

ranked ﬁrst in this category, with second place occupied by Jim Furyk, who gains 2.12 strokes per

18-hole round versus an average PGA TOUR ﬁeld. The diﬀerence between the two is an enormous

1.08 strokes per round. Diﬀerences between lower ranks are much smaller: the average diﬀerence

is 0.08 strokes between ranks 2 and 10 and 0.01 strokes between ranks 95 and 105. Not only was

Tiger the best golfer between 2003 and 2010, but he was the best by a large margin.

The strokes gained approach gives direct insight into where Tiger Woods gained the 3.20 strokes

per round. Table 1 shows that 2.08 strokes came from the long game (rank 1), 0.42 strokes from

the short game (rank 16), and 0.70 strokes from putting (rank 3). Tiger dominates the competition

because he excels in every category, but his long game contributes 65% (2.08/3.20) to his total

strokes gained relative to an average ﬁeld. Many people have commented on Tiger’s superior

putting, and the strokes gained analysis is consistent with this observation: he is ranked of third

with a gain of 0.70 putts per round. However, his gain from putting is less than the 1.01 strokes

he gains between 150 and 250 yards from the hole, and comparable to his long tee shots, where he

gains 0.70 strokes per round versus the ﬁeld.

Table 1 shows average strokes gained for the top ten golfers, and the long game contributes

65% (1.20/1.84) to their total strokes gained relative to an average ﬁeld. The bottom ten golfers,

ranks 290-299, lose 71% (−0.97/−1.36) of their strokes in the long game. The top ten golfers in

total strokes gained are all ranked in the the top 70 in long game strokes gained, but four of these

golfers are not ranked in the top 100 in putting. The bottom ten golfers in total strokes gained are

all ranked worse than 200 in long game strokes gained. These results suggest that the long game

is the most important factor that diﬀerentiates golfers at the elite PGA TOUR level.

Table 2 focuses on the long game, and shows that Tiger is ranked in the top 10 in each of the

long game subcategories. Of his 2.08 long game strokes gained, 1.01 strokes are gained between

150 and 250 yards from the hole. Although Tiger is known for his occasional wild drives, in the

long tee shot category he is ranked 7 (out of 299), and he gains 0.70 strokes per round versus the

ﬁeld, the same as his gain from putting.

Table 3 gives short game strokes gained results during 2003-2010. Steve Stricker had the best

short game overall, while Mike Weir and Luke Donald had the best greenside sand games. Table 4

focuses on putting and shows that David Frost, Brad Faxon and Tiger Woods were the top three

3The rankings are based on strokes gained per round. An argument can be made that a better measure of skill is

strokes gained per stroke, however both approaches give very similar results. Strokes gained per round is used here

because the additivity property makes it easier to see how total strokes gained splits into long game, short game and

putting strokes gained.

17

Table 1:Total strokes gained per round, broken down into three categories: long game (shots over 100

yards from the hole), short game (shots under 100 yards from the hole, excluding putts) and putting

(shots on the green, not including the fringe). Ranks are out of the 299 PGA TOUR golfers with at

least 120 rounds during 2003-2010.

Rank Strokes gained

Golfer Total Long Short Putt Total Long Short Putt

Woods, Tiger 1 1 16 3 3.20 2.08 0.42 0.70

Furyk, Jim 2 10 10 14 2.12 1.13 0.47 0.52

Singh, Vijay 3 2 5 195 2.05 1.63 0.51 −0.09

Els, Ernie 4 4 15 153 1.86 1.40 0.44 0.01

Mickelson, Phil 5 12 12 95 1.72 1.11 0.47 0.15

Donald, Luke 6 65 7 9 1.55 0.46 0.50 0.58

Goosen, Retief 7 19 22 46 1.52 0.90 0.33 0.29

Garcia, Sergio 8 5 60 220 1.47 1.39 0.23 −0.15

Scott, Adam 9 7 53 201 1.46 1.33 0.24 −0.11

Harrington, Padraig 10 54 4 42 1.44 0.57 0.56 0.31

Average 1.84 1.20 0.42 0.22

Boros, Guy 290 283 292 91 −1.14 −0.87 −0.43 0.16

McGovern, Jim 291 293 158 197 −1.15 −1.05 −0.01 −0.09

Waite, Grant 292 279 120 282 −1.17 −0.79 0.07 −0.45

Begay III, Notah 293 265 194 286 −1.23 −0.67 −0.09 −0.48

Bolli, Justin 294 267 274 264 −1.27 −0.69 −0.25 −0.32

Veazey, Vance 295 294 246 178 −1.33 −1.10 −0.19 −0.04

McCallister, Blaine 296 262 273 294 −1.49 −0.64 −0.25 −0.60

Gossett, David 297 292 103 295 −1.49 −1.01 0.12 −0.61

Duval, David 298 297 219 143 −1.51 −1.41 −0.14 0.03

Perks, Craig 299 298 195 249 −1.79 −1.44 −0.09 −0.26

Average −1.36 −0.97 −0.12 −0.27

Notable golfers

Couples, Fred 29 28 37 209 1.00 0.84 0.28 −0.12

Villegas, Camilo 30 13 126 212 0.99 1.05 0.06 −0.13

Westwood, Lee 43 17 253 129 0.83 0.97 −0.20 0.06

Pavin, Corey 99 252 8 26 0.33 −0.57 0.48 0.42

Durant, Joe 117 9 267 299 0.20 1.14 −0.24 −0.70

O’Meara, Mark 236 284 89 62 −0.49 −0.87 0.15 0.24

Daly, John 238 138 254 272 −0.50 0.08 −0.21 −0.38

putters during 2003-2010. Sergio Garcia is ranked 220 in putting overall: 271 in short putts, 179 in

medium putts and 85 in long putts. It is clear that the shorter the putt, the more trouble he has.

Sergio’s total putting strokes gained is −0.15, so he loses 0.85 strokes per round to Tiger Woods

just from putting. Brad Faxon gains 0.71 strokes on the ﬁeld in putting but loses 0.96 in the long

game (see Table 2).

Table 5 shows strokes gained results for Tiger Woods by year. Tiger Woods was ranked ﬁrst in

total strokes gained in each year from 2003 to 2009. Tiger Woods had the worst year of his career

in 2010, with his total strokes gained per round decreasing by three compared with 2009. His game

faltered across the board, dropping 1.19 strokes in his long game, 0.89 in his short game, and 0.91

18

Table 2:Long game strokes gained per round, broken down into ﬁve categories: long tee shots (tee

shots starting over 250 yards from the hole), approach shots 100-150 yards from the hole, approach

shots 150-200 yards from the hole, approach shots 200-250 yards from the hole and shots over 250 yards

from the hole (excluding tee shots). For space reasons, recovery shots and sand shots greater than 100

yards from the hole are not reported (but are included in the total long game strokes gained). Ranks

are out of the 299 golfers with at least 120 rounds during 2003-2010.

Rank Strokes gained per round

Long Long 100−150−200−Long Long 100−150−200−

Golfer total tee 150 200 250 >250 total tee 150 200 250 >250

Woods, Tiger 1 7 8 1 1 1 2.08 0.70 0.20 0.66 0.35 0.14

Singh, Vijay 2 3 20 8 7 14 1.63 0.81 0.16 0.33 0.19 0.07

Allenby, Robert 3 14 4 6 2 47 1.59 0.61 0.25 0.38 0.26 0.05

Els, Ernie 4 16 14 2 15 26 1.40 0.55 0.18 0.41 0.16 0.06

Garcia, Sergio 5 15 13 13 4 17 1.39 0.55 0.18 0.31 0.23 0.07

Perry, Kenny 6 6 25 10 18 95 1.37 0.73 0.15 0.32 0.15 0.03

Scott, Adam 7 18 5 12 48 16 1.33 0.54 0.25 0.31 0.10 0.07

Weekley, Boo 8 2 58 84 25 113 1.19 0.83 0.09 0.09 0.13 0.02

Durant, Joe 9 11 43 22 40 136 1.14 0.67 0.11 0.24 0.11 0.01

Furyk, Jim 10 51 7 4 12 134 1.13 0.32 0.21 0.40 0.18 0.01

Average 1.40 0.61 0.17 0.35 0.19 0.06

Notable golfers

Couples, Fred 28 28 22 92 26 117 0.84 0.47 0.16 0.08 0.12 0.02

Daly, John 138 55 242 252 123 128 0.08 0.31 −0.08 −0.15 0.03 0.02

Faxon, Brad 289 297 79 190 223 279 −0.96 −0.82 0.07 −0.05 −0.05 −0.08

Duval, David 297 298 249 211 293 164 −1.41 −1.08 −0.09 −0.08 −0.17 −0.00

Table 3:Short game strokes gained per round, broken down into three distance categories: 0-20 yards

from the hole, 20-60 yards from the hole, 60-100 yards from the hole (excluding sand and recovery shots

and putts). Greenside sand shots within 50 yards of the hole (‘sand’) are given in a separate category.

For space reasons, 0-100 yard recovery shots and 50-100 yard sand shots are not reported (but are

included in the total short game strokes gained). Ranks are out of the 299 golfers with at least 120

rounds during 2003-2010.

Rank Strokes gained per round

Golfer Short 0-20 20-60 60-100 Sand Short 0-20 20-60 60-100 Sand

Stricker, Steve 1 7 1 1 59 0.69 0.19 0.22 0.17 0.08

Olazabal, Jose Maria 2 1 27 66 7 0.57 0.30 0.10 0.04 0.15

Riley, Chris 3 9 4 45 3 0.56 0.18 0.15 0.06 0.15

Harrington, Padraig 4 5 11 4 39 0.56 0.21 0.12 0.15 0.10

Singh, Vijay 5 14 9 53 4 0.51 0.15 0.12 0.05 0.15

Weir, Mike 6 53 15 40 1 0.51 0.09 0.11 0.06 0.21

Donald, Luke 7 3 84 49 2 0.50 0.23 0.04 0.06 0.17

Pavin, Corey 8 4 30 20 23 0.48 0.21 0.09 0.08 0.12

Imada, Ryuji 9 18 21 38 19 0.48 0.15 0.10 0.06 0.12

Furyk, Jim 10 6 17 13 66 0.47 0.20 0.11 0.09 0.07

Average 0.53 0.19 0.12 0.08 0.13

Notable golfers

Haas, Jay 11 11 143 10 8 0.47 0.17 0.01 0.11 0.14

Mickelson, Phil 12 15 6 42 20 0.47 0.15 0.13 0.06 0.12

Woods, Tiger 16 22 8 47 64 0.42 0.13 0.13 0.06 0.07

Garcia, Sergio 60 64 75 60 101 0.23 0.08 0.05 0.04 0.04

Westwood, Lee 253 260 162 88 286 -0.20 −0.10 −0.00 0.02 −0.14

Daly, John 254 290 182 179 90 -0.21 −0.17 −0.01 −0.01 0.05

19

Table 4:Putting strokes gained per round, broken down into three distance categories: short putts

(0-6 feet), medium length putts (7-21 feet) and long putts (22 feet and over). Ranks are out of the 299

golfers with at least 120 rounds during 2003-2010.

Rank Strokes gained per round

Golfer Putt 0-6 ft 7-21 ft 22+ ft Putt 0-6 ft 7-21 ft 22+ ft

Frost, David 1 83 1 1 0.72 0.08 0.42 0.22

Faxon, Brad 2 21 3 2 0.71 0.19 0.31 0.21

Woods, Tiger 3 11 4 3 0.70 0.21 0.31 0.19

Crane, Ben 4 1 10 24 0.67 0.29 0.27 0.11

Roberts, Loren 5 4 13 13 0.65 0.25 0.26 0.14

Baddeley, Aaron 6 9 9 7 0.64 0.22 0.27 0.15

Chalmers, Greg 7 2 14 37 0.62 0.27 0.26 0.09

Parnevik, Jesper 8 3 27 9 0.61 0.25 0.21 0.15

Donald, Luke 9 14 17 16 0.58 0.20 0.24 0.13

Cink, Stewart 10 28 7 22 0.58 0.17 0.29 0.12

Average 0.65 0.21 0.28 0.15

Notable golfers

Stricker, Steve 19 15 60 19 0.46 0.20 0.13 0.13

Pavin, Corey 26 97 23 17 0.42 0.06 0.22 0.13

Mickelson, Phil 95 68 139 102 0.15 0.09 0.02 0.04

Singh, Vijay 195 152 252 97 -0.09 0.01 −0.14 0.04

Couples, Fred 209 294 102 41 -0.12 −0.28 0.07 0.08

Garcia, Sergio 220 271 179 85 -0.15 −0.16 −0.03 0.04

Daly, John 272 261 272 247 -0.38 −0.12 −0.19 −0.07

20

Table 5:Results for Tiger Woods, by year. Ranks for individual years are out of approximately 220

golfers with at least 30 rounds during each year. An exception was made to show Tiger Woods in 2008,

even though he only played in three PGA TOUR events. Ranks for 2003-2010 are out of the 299 golfers

with at least 120 rounds.

Rank Strokes gained

Year Total Long Short Putt Total Long Short Putt

Tiger Woods 2010 48 28 160 91 0.71 0.83 −0.20 0.08

2009 1 1 4 2 3.70 2.02 0.70 0.99

2008 1 1 3 4 4.14 2.56 0.72 0.85

2007 1 1 24 2 3.68 2.47 0.41 0.80

2006 1 1 16 21 3.78 2.83 0.45 0.49

2005 1 1 98 5 2.82 2.03 0.09 0.70

2004 1 5 11 3 3.07 1.62 0.49 0.96

2003 1 2 3 16 3.71 2.44 0.72 0.55

2003-2010 1 1 16 3 3.20 2.08 0.42 0.70

Rank Strokes gained per round

Long Long 100−150−200−Long Long 100−150−200−

Year total tee 150 200 250 >250 total tee 150 200 250 >250

2010 28 123 29 2 44 16 0.83 −0.08 0.16 0.48 0.12 0.10

2009 1 18 25 1 1 2 2.02 0.53 0.16 0.79 0.43 0.15

2008 1 7 9 1 1 51 2.56 0.60 0.25 1.17 0.40 0.05

2007 1 4 1 1 4 1 2.47 0.81 0.38 0.83 0.30 0.17

2006 1 4 52 1 1 1 2.83 0.91 0.13 0.94 0.62 0.16

2005 1 1 6 16 28 3 2.03 1.09 0.29 0.35 0.14 0.15

2004 5 17 54 2 9 7 1.62 0.53 0.13 0.58 0.24 0.12

2003 2 6 38 2 1 3 2.44 0.87 0.14 0.59 0.59 0.15

2003-2010 1 7 8 1 1 1 2.08 0.70 0.20 0.66 0.35 0.14

Rank Strokes gained per round

Year Short 0-20 20-60 60-100 Sand Short 0-20 20-60 60-100 Sand

2010 160 169 72 135 173 −0.20 −0.10 0.04 −0.02 −0.11

2009 4 6 1 67 14 0.70 0.25 0.25 0.03 0.17

2008 3 12 1 29 144 0.72 0.23 0.42 0.08 −0.05

2007 24 77 22 22 85 0.41 0.06 0.12 0.10 0.03

2006 16 17 108 18 90 0.45 0.20 0.02 0.12 0.03

2005 98 143 150 67 51 0.09 −0.03 −0.03 0.05 0.08

2004 11 70 2 40 86 0.49 0.08 0.27 0.07 0.04

2003 3 3 34 79 7 0.72 0.41 0.10 0.03 0.17

2003-2010 16 22 8 47 64 0.42 0.13 0.13 0.06 0.07

Rank Strokes gained per round

Year Putt 0-6 ft 7-21 ft 22+ ft Putt 0-6 ft 7-21 ft 22+ ft

2010 91 58 98 150 0.08 0.11 0.03 −0.06

2009 2 1 40 1 0.99 0.47 0.20 0.31

2008 4 29 12 5 0.85 0.20 0.40 0.25

2007 2 62 3 4 0.80 0.10 0.44 0.26

2006 21 32 58 17 0.49 0.17 0.12 0.20

2005 5 27 15 10 0.70 0.19 0.31 0.20

2004 3 53 2 9 0.96 0.12 0.62 0.22

2003 16 7 32 61 0.55 0.26 0.23 0.07

2003-2010 3 11 4 3 0.70 0.21 0.31 0.19

21

Table 6:Total strokes gained per round for selected golfers, by year. Ranks for individual years are

out of approximately 220 golfers with at least 30 rounds during each year. Ranks for 2003-2010 are out

of the 299 golfers with at least 120 rounds.

Rank Strokes gained

Year Total Long Short Putt Total Long Short Putt

Jim Furyk 2010 3 26 2 22 2.03 0.90 0.64 0.49

2009 3 31 8 4 2.12 0.80 0.53 0.80

2008 6 17 61 28 1.62 0.98 0.20 0.44

2007 10 14 8 105 1.68 1.02 0.62 0.04

2006 2 3 17 3 2.94 1.69 0.44 0.81

2005 4 6 8 26 2.27 1.32 0.53 0.41

2004 22 33 100 16 1.50 0.83 0.06 0.60

2003 4 13 5 10 2.55 1.31 0.61 0.62

2003-2010 2 10 10 14 2.12 1.13 0.47 0.52

Vijay Singh 2010 30 3 33 196 1.05 1.42 0.31 −0.68

2009 70 28 61 186 0.40 0.83 0.18 −0.61

2008 4 4 4 177 1.80 1.55 0.63 −0.38

2007 9 6 19 107 1.75 1.25 0.47 0.03

2006 6 6 9 90 2.07 1.43 0.53 0.10

2005 2 5 3 63 2.58 1.77 0.61 0.20

2004 2 1 7 120 2.86 2.28 0.61 −0.03

2003 2 3 6 63 3.06 2.23 0.60 0.23

2003-2010 3 2 5 195 2.05 1.63 0.51 −0.09

Ernie Els 2010 7 16 37 28 1.75 1.05 0.28 0.42

2009 16 6 23 152 1.37 1.35 0.32 −0.30

2008 25 10 26 190 1.10 1.16 0.40 −0.46

2007 2 2 27 104 2.16 1.74 0.38 0.04

2006 8 15 3 96 1.94 1.11 0.75 0.07

2005 3 3 60 32 2.37 1.80 0.18 0.39

2004 3 4 5 79 2.48 1.63 0.69 0.16

2003 10 8 16 159 1.90 1.67 0.51 −0.29

2003-2010 4 4 15 153 1.86 1.40 0.44 0.01

Phil Mickelson 2010 10 10 15 118 1.49 1.16 0.39 −0.05

2009 19 23 13 119 1.29 0.92 0.42 −0.05

2008 1 5 8 50 2.25 1.40 0.57 0.27

2007 3 11 4 59 2.06 1.14 0.69 0.23

2006 5 4 32 66 2.13 1.58 0.34 0.20

2005 8 18 7 49 1.82 0.98 0.54 0.30

2004 10 8 25 123 1.79 1.44 0.39 −0.04

2003 46 94 30 54 0.87 0.19 0.39 0.29

2003-2010 5 12 12 95 1.72 1.11 0.47 0.15

Steve Stricker 2010 1 9 3 15 2.36 1.17 0.64 0.55

2009 2 9 1 56 2.23 1.18 0.75 0.30

2008 14 112 1 26 1.31 0.03 0.83 0.46

2007 5 25 3 25 1.97 0.82 0.73 0.41

2006 18 113 2 20 1.47 0.15 0.82 0.50

2005 129 213 2 8 −0.05 −1.38 0.65 0.68

2004 144 213 9 12 −0.22 −1.41 0.53 0.67

2003 141 188 11 95 −0.35 −0.91 0.53 0.03

2003-2010 22 160 1 19 1.13 −0.02 0.69 0.46

22

in his putting. His combined results for 2003-2010 shows he is the best golfer of his era because of

his all-round excellence in every category, with his long game contributing 65% (2.08/3.20) of his

total strokes gained versus the ﬁeld.

Table 6 shows strokes gained results for selected golfers by year. Steve Stricker was the comeback

player of the year in 2006 when his total strokes gained increased from −0.05 to 1.47, moving from

rank 129 to 18. The improvement was almost entirely due to a better long game, with a long game

strokes gained increase from −1.38 to 0.15. He was also the comeback player of the year in 2007

where his total strokes gained increased from 1.47 to 1.97, moving from rank 18 to 5.

4.3. Inﬂuence of skill factors on golf scores

Many people claim that the short game and putting are the most important determinants of golf

scores. For example, Pelz (1999, p.1) writes, “60% to 65% of all golf shots occur inside 100 yards

of the hole. More important, about 80% of the shots golfers lose to par occur inside 100 yards.”

Several academic studies have reached similar conclusions. In contrast, strokes gained analysis of

PGA TOUR data shows that the long game is the most important factor explaining the variability

in professional golf scores.

For a single golfer, the relative contribution of each skill category can be assessed directly by

comparing strokes gained by skill category. Across golfers the relative contributions can be assessed

using variance and correlation analysis. Equation 8 is used to estimate µi, the mean total strokes

gained of golfer iand also the mean strokes gained of long game shots (µL

i), short game shots (µS

i)

and putts (µP

i). Note that µi=µL

i+µS

i+µP

i, and all quantities represent 18-hole round averages

estimated using equation (8). For notational convenience we drop the golfer subscript i. Then

Var(µ) = Var(µL) + Var(µS) + Var(µP) + 2 Cov(µL, µS) + 2 Cov(µL, µP) + 2 Cov(µS, µP) (where

each of the terms represents the variance or covariance across golfers). A unique decomposition of

Var(µ) is complicated because of the covariance terms. However, the covariance terms are quite

small and V≡Var(µL) + Var(µS) + Var(µP)≈Var(µ).4So deﬁne the contributions of the long

game, short game and putting to total strokes gained by: Var(µL)/V , Var(µS)/V and Var(µP)/V ,

respectively. More variability in a strokes gained category means that golfers have more opportunity

to distinguish themselves as better or worse golfers. Using data from 2003-2010 for golfers with

at least 120 rounds, the contributions to total strokes gained are 72%, 11% and 17% for the long

game, short game and putting, respectively. By this measure, the long game explains more than

two-thirds of the variation in total strokes gained.

Correlation results across golfers are summarized in Table 7. When the three broad skill cate-

gories are further divided, approach and tee shots in the 150-200 yard range are seen to have the

highest correlation with total strokes gained, with a correlation of 74%. Table 7 shows that the

correlation of putts gained with long game strokes gained across golfers is −14% (with a standard

4Using data from 2003-2010 for golfers with at least 120 rounds gives: Var(µ) = 0.50, Var(µL) = 0.35, Var(µS) =

0.06, Var(µP) = 0.08, Cov(µL, µS) = 0.01, Cov(µL, µP) = −0.02 and Cov(µS, µP) = 0.03.

23

error of 7%). At the tournament professional level, these skill factors are nearly uncorrelated,

as illustrated in Figure 7. The slight negative correlation can be explained by survivorship bias:

golfers with a subpar long game need better than average putting (and/or short game) to survive

on the PGA TOUR.

Table 7:Correlation results using 2003-2010 data for all PGA TOUR golfers with 120 or more rounds.

Top panel: Total refers to the total strokes gained per 18-hole round. Long refers to the total strokes

gained per 18-hole round for shots over 100 yards from the hole. Short refers to the total strokes gained

per 18-hole round for shots under 100 yards from the hole excluding putts. Putt refers to the total

strokes gained for per 18-hole round for shots on the green. The bottom two rows give the average

number of shots and fractions of shots in each category. Bottom panel: correlations of subcategories

with total strokes gained. Standard errors (computed by with standard bootstrapping procedure) are

given in parentheses.

Total (se) Long (se) Short (se) Putt (se)

Total 100%

Long 79% (2%) 100%

Short 54% (4%) 6% (6%) 100%

Putt 41% (5%) −14% (7%) 39% (4%) 100%

Number of strokes 71.1 32.2 9.8 29.1

Fraction of strokes 45% 14% 41%

Long game

Long 100−150−200−Short game Putt

tee 150 200 250 >250 0-20 20-60 60-100 Sand 0-6 7-21 22+

54% 61% 74% 66% 53% 50% 37% 44% 33% 27% 37% 40%

(4%) (4%) (3%) (4%) (4%) (4%) (5%) (5%) (5%) (6%) (5%) (5%)

Number: 13.9 4.8 7.1 3.2 1.6 4.3 2.1 1.6 1.7 16.0 7.9 5.3

Fraction: 19.6% 6.7% 10.0% 4.5% 2.3% 6.0% 3.0% 2.1% 2.4% 22.4% 11.1% 7.4%

Correlation and variability should not equated with importance. If every professional golfer hit

every drive 320 yards in the middle of the fairway, then long tee shots would have zero correlation

with score and the variability in long tee strokes gained would be zero. In this example, the

golfers are not diﬀerentiating themselves with their long tee shots—they all happen to be equally

outstanding in this skill category. However, it is still important to be a good driver of the ball: a

golfer who doesn’t hit his drives 320 yards in the fairway in this example will not survive on the

tour for long.

4.4. Course diﬃculty factors

The estimation of course-round diﬃculty parameters, δjin equation (8), allows courses to be

ranked just as golfers were ranked. By using individual shot data, course diﬃculty be further

explained and broken down into diﬃculty of long game shots, short game shots, and putts. A

related question of the diﬃculty of winning a tournament is studied Connolly and Rendleman

(2010). For handicapping purposes, the USGA rates course diﬃculty for zero-handicap (scratch)

24

-1.0

-0.5

0.0

0.5

1.0

-2-10123

Long game strokes gained

Putting strokes gained

Tiger Woods

Mickelson

Els

Duval

Faxon Donald Furyk

Singh

Daly

Figure 7:Scatter chart of putts gained versus long game strokes gained using 2003-2010 data. Each data

point represents the results for a single golfer; a few golfers are indicated to illustrate. The regression

trendline shows a slight negative correlation between the two skill categories (the correlation is −14%

with a standard error of 7%).

and bogey golfers by tabulating hole distances, counting the number and severity of bunkers and

other hazards, etc. Their method does not use scores nor shot information. To the best of our

knowledge, this is the ﬁrst attempt at ranking courses using shot data and the ﬁrst to break down

course diﬃculty by shot categories.

Deﬁne the diﬃculty factors for each course to be the average value of −δjfor all rounds played

at that course. The negative sign is used so that the most diﬃcult courses are ranked at the top.

Table 8 shows the ten most diﬃcult and the ten easiest courses that hosted tournaments during

2003-2010 and had at least 12 rounds of data. The TPC Sawgrass course, host of the Players

Championship and famous for the island green on its 17th hole, is ranked as the most diﬃcult

course on the PGA TOUR. The strokes gained approach explicitly accounts for the length of the

course, so courses are rated as more diﬃcult because of trees, hazards, rough height, ﬁrmness and

contours of the greens, etc. The strokes gained approach enables us to see which part of the course

contributes most to its diﬃculty and allows courses to be ranked for diﬃculty in the long game,

short game and putting. For example, Westchester Country Club is rated as the most diﬃcult

course for the short game and putting and Harbour Town Golf Links is the most diﬃcult course in

the long game category.

4.5. Eﬀect of the groove rule change

The USGA, golf’s rule-making body in the United States, recently changed the rules regarding the

grooves in irons. The rules were changed because of the perception that equipment advances in

the past decade made shots from the rough easier: clubs with sharper grooves allow highly skilled

golfers to impart more spin on the ball from the rough and stop the ball closer to the hole. The

25

Table 8:Ranking of courses by diﬃculty factors. Ranks are out of the 45 courses that hosted PGA

TOUR tournaments and had at least 12 rounds of data during 2003-2010.

Rank Diﬃculty factors

Course Total Long Short Putt Total Long Short Putt

TPCSawgrass 1 2 3 9 2.41 1.72 0.47 0.23

WestchesterCC 2 19 1 1 1.70 0.16 0.84 0.70

HarbourTownGolfLinks 3 1 27 18 1.69 1.78 −0.12 0.03

MuirﬁeldVillageGC 4 7 2 10 1.68 0.89 0.64 0.15

BayHillClub 5 3 28 11 1.54 1.58 −0.12 0.09

PebbleBeachGolfLinks 6 6 35 3 1.32 0.91 −0.21 0.62

WestinInnisbrook-Copperhead 7 5 22 12 1.20 1.21 −0.10 0.09

PGANationalChampionCourse 8 4 33 39 0.94 1.38 −0.20 −0.24

QuailHollowClub 9 16 12 4 0.77 0.22 0.05 0.50

TorreyPinesSouthCourse 10 14 25 8 0.48 0.36 −0.11 0.24

Average 1.37 1.02 0.11 0.24

LaCanteraGC 36 36 32 27 −1.07 −0.80 −0.16 −0.11

TucsonNat’lGolf 37 39 24 14 −1.11 −1.07 −0.10 0.07

WarwickHillsG&CC 38 35 37 29 −1.16 −0.77 −0.26 −0.13

MagnoliaGC 39 33 38 38 −1.19 −0.69 −0.29 −0.21

ForestOaksCC 40 37 30 41 −1.33 −0.94 −0.14 −0.25

AtunyoteGolfClub 41 40 40 17 −1.42 −1.13 −0.32 0.03

TPCSummerlin 42 44 6 43 −1.60 −1.67 0.33 −0.26

TPCDeereRun 43 43 23 44 −1.75 −1.29 −0.10 −0.36

SedgeﬁeldCountryClub 44 45 18 19 −1.97 −1.93 −0.06 0.02

En-JoieGC 45 41 45 30 −1.99 −1.17 −0.69 −0.13

Average −1.46 −1.15 −0.18 −0.13

purpose of the new rule is to “roll back” these equipment advances, so that shots from the rough

will have less spin and the rough more of a penalty compared to the fairway. The rules were put

into place on the PGA TOUR at the start of the 2010 season.

To estimate the eﬀect of the groove rule change, benchmark functions representing the average

strokes to complete a hole are estimated for the fairway and rough for each year. Recovery shots (as

described in Section 3.4) are excluded, so the rough benchmark functions are not biased by these

shots. Deﬁne the rough penalty to be the diﬀerence in the average strokes to complete the hole

between the rough and fairway at comparable distances to the hole. For example, from 120 yards

in the fairway the average number of strokes to complete the hole is 2.85 and from the rough it is

3.08. The penalty for being in the rough at 120 yards from the hole is an increase of 0.23 strokes.

Since the rough penalty varies slightly by distance, results are given for the average rough penalty

between 50 and 150 yards from the hole, where the rule is designed to have the maximum impact.

Figure 8 shows a decline in the rough penalty from 2003-2009. Surprisingly, the ﬁgure shows a

large drop from 2008 to 2009, prior to when the groove rule change went into eﬀect. Figure 8 shows

that the rough penalty was unchanged at 0.20 in 2009 and 2010 (with standard errors of 0.004).

Tests of ball spin indicate a measurable impact of the rule change, so it is puzzling that there has

been little impact on scores. Diﬀerences in the height, thickness and moisture of the rough and the

26

ﬁrmness of the greens will inﬂuence the results and these factors should, if possible, be incorporated

in the analysis. Another possible explanation is that the golfers adapted their swings and strategy

in order to minimize the impact of the change in ball spin. These issues are left for future research.

2.75

2.80

2.85

2.90

20102009200820072006200520042003

Year

Avg strokes, fairway

2.95

3.00

3.05

3.10

20102009200820072006200520042003

Year

Avg strokes, rough

0.18

0.20

0.22

0.24

0.26

20102009200820072006200520042003

Year

Rough penalty (50-150 yds)

Figure 8:Upper charts: average strokes to complete the hole from the fairway and rough. Lower chart:

rough penalty (the diﬀerence between the rough and fairway values). All three charts show results by

year for shots starting between 50 and 150 yards from the hole. Two standard error bars are shown in

each chart (standard errors were computed with a standard bootstrapping procedure).

5. Concluding remarks

The availability of detailed golf shot data makes it possible to create golf measures that allow

consistent comparisons between diﬀerent parts of the game. Using the starting and ending locations

of each shot, strokes gained gives the number of strokes a golfer gains or loses relative to an average

PGA TOUR tournament ﬁeld. Analysis of over eight million shots on the PGA TOUR in 2003-

2010 shows that the long game (deﬁned as shots starting over 100 yards from the hole) accounts

for more than two-thirds of the scoring diﬀerences between PGA TOUR golfers. In the 2003-2010

data, Tiger Woods led in total strokes gained, with a gain of 3.20 strokes per 18-hole round. He

gained 2.08 strokes (65% of the total) in the long game. A preliminary analysis of the impact of

the new groove rule for irons that went into eﬀect on the PGA TOUR in 2010 showed, somewhat

27

surprisingly, that there has been almost no impact of the rule on scores.

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30

A. Appendix

This appendix summarizes the benchmark average strokes-to-complete-the hole functions from tee,

fairway, rough, sand and recovery positions.

Table 9:Average number of strokes to complete the hole for PGA TOUR golfers from various starting

positions. Distance to the hole is measured in yards. Values are estimated using over eight million shots

during 2003-2010.

Distance Tee Fairway Rough Sand Recovery

10 2.18 2.34 2.43 3.45

20 2.40 2.59 2.53 3.51

30 2.52 2.70 2.66 3.57

40 2.60 2.78 2.82 3.71

50 2.66 2.87 2.92 3.79

60 2.70 2.91 3.15 3.83

70 2.72 2.93 3.21 3.84

80 2.75 2.96 3.24 3.84

90 2.77 2.99 3.24 3.82

100 2.92 2.80 3.02 3.23 3.80

120 2.99 2.85 3.08 3.21 3.78

140 2.97 2.91 3.15 3.22 3.80

160 2.99 2.98 3.23 3.28 3.81

180 3.05 3.08 3.31 3.40 3.82

200 3.12 3.19 3.42 3.55 3.87

220 3.17 3.32 3.53 3.70 3.92

240 3.25 3.45 3.64 3.84 3.97

260 3.45 3.58 3.74 3.93 4.03

280 3.65 3.69 3.83 4.00 4.10

300 3.71 3.78 3.90 4.04 4.20

320 3.79 3.84 3.95 4.12 4.31

340 3.86 3.88 4.02 4.26 4.44

360 3.92 3.95 4.11 4.41 4.56

380 3.96 4.03 4.21 4.55 4.66

400 3.99 4.11 4.30 4.69 4.75

420 4.02 4.19 4.40 4.83 4.84

440 4.08 4.27 4.49 4.97 4.94

460 4.17 4.34 4.58 5.11 5.03

480 4.28 4.42 4.68 5.25 5.13

500 4.41 4.50 4.77 5.40 5.22

520 4.54 4.58 4.87 5.54 5.32

540 4.65 4.66 4.96 5.68 5.41

560 4.74 4.74 5.06 5.82 5.51

580 4.79 4.82 5.15 5.96 5.60

600 4.82 4.89 5.25 6.10 5.70

31