T U M
Working Paper
The ISOPAR Method
Michael Stöckl, Peter F. Lamb, & Martin Lames
Fakultät für Sport- und Gesundheitswissenschaen Lehrstuhl für Trainingswissenscha und Sportinformatik Georg-Brauchle-Ring 62 80992, München [email protected]
First version: June 27, 2011 This version: June 12, 2012 Abstract The ISOPAR method is a method for characterizing the difficulty of golf holes and allows the per- formance of shots to be analyzed. The method is based on the ball locations provided by ShotLink™and the subsequent number of shots required to hole out from each respective location. ISOPAR values are calculated which represent the number of shots the field would require to hole out. These ISOPAR values can, a) be visualized on an ISOPAR map and, b) lead to a new performance indicator called Shot ality, which is the difference between the ISOPAR values of the starting position and finishing position, respectively. The Shot ality score can also be used to determine how many shots were saved per shot, or per type of shot, with respect to the performance of the field.
1 Introduction
In performance analysis, characteristics of a process which describe how an outcome was achieved are used to assess the performance itself (Hughes & Bartle, 2002) and are referred to as performance indicators. Classical performance analysis techniques in golf have focused on classes of golf shots
(James, 2007), such as driving distance, approach shot accuracy and puing average (James & Rees,
2008). Measures like greens in regulation, average pus per green and driving distance are intended to describe players’ abilities to perform certain types of shots, yet these abilities are not actually assessed.
For example, the beginning position of a pu is the result of the approach shot to the green. So a good puing average describes not only puing ability but also all previous shots on the hole – it is a composite measure. Therefore, if independent measures for different types of golf shots existed then strengths and weaknesses of a player’s game could be assessed (Ketzscher & Ringrose, 2002). Currently, golf performance analysis lacks performance indicators which reflect the influence one shot has on the next. For example, on each hole there is a chain of events which starts on the tee and ends once the ball is holed. Each shot represents an event and the final position of shot n determines the starting position for shot n+1. A model preserving the playing characteristics of the environment (for example, physical contours, playing conditions, etc.) and the stroke sequence is more suitable than simply an analysis of frequencies of discrete events.
2 Background
Cochran and Stobbs (1968) had the idea to manually collect shot data (ball locations) and to analyze per- formance based on these data. They wanted to measure the performance of professional golfers in dif- ferent parts of the game, figure out which of these parts is most important, and research in which parts
1 of the game the leading golfers are beer than the rest of the field. In the context of this study Cochran and Stobbs developed a model for calculating probabilities and the average number of remaining shots for holing out from certain (ranges of) distances. At the time of their study the lack of modern tech- nology prevented them from collecting more data and enhancing their approach. Landsberger (1994) built on the work of Cochran and Stobbs by refining the approach. Landsberger’s Golf Stroke Value
System (GSVS) provided a starting point for more recent work on establishing independent measures of performance.
Recent projects have emerged which have looked to further advance the shot value idea (Broadie,
2011; Fearing, Acimovic, & Graves, 2011; Minton, 2011)¹. Broadie (2008, 2011) developed statistical models to calculate probabilities of holing out and derives benchmarks as average number of remaining shots from the probabilities. One model provides benchmarks for holing out on the green based on the distance to the hole and another model computes benchmarks for holing out off the green, which additionally includes a classification of the ball location. Using these benchmarks Broadie has demon- strated a more valid method for describing the performance of individual shots, called strokes gained.
Strokes gained can be used to explain the contribution of each shot to the total score. Based on the shot value idea of Broadie (2008), Fearing et al. (2011) came up with a similar approach which is limited to the green. They applied various regression models to achieve the probability of making a pu and a prediction of the distance remaining aer a missed pu. In addition to the distance to the hole used by
Broadie, Fearing et al. (2011) consider the strength of the field and the difficulty of the green. From this they illustrate the use of these benchmarks to assess performance to individual shots using the same shot value idea as Broadie (2011). The PGA TOUR uses this approach as a measure for individual shots, which is called Strokes Gained - Puing. Both approaches provide very sophisticated models of puing performance with respect to the distance from the hole.
In the absence of independent measures of individual shot performance, several studies (Clark III, 2004; James, 2007; James & Rees, 2008; Scheid, 1990) have looked at the temporal variance of consecutive golf scores – both hole scores and round scores. Analyses of round scores showed very low correlations between scores of consecutive rounds when considered with respect to external influences on perfor- mance (i.e. weather conditions and course setup). Analyses of hole scores also showed low correlations
¹see PGA TOUR Academic Data Program page, available at: http://www.pgatour.com/stats/academicdata/ for de- tailed explanations of these projects.
2 between successive holes, again considering external influences like hole par and difficulty. Aside from the obvious fact that good players tend to shoot good scores and poor players tend to shoot poor scores, these results suggest that performance in golf is not subject to “streakiness”. In other words, the nature of the performance of individual shots which make up hole and round scores seems not to be well un- derstood. In summary, consecutive round scores do not depend on one another, and consecutive hole scores do not depend on one another. However, individual shots played on the same hole present a different scenario; these shots make up a continuous chain of events so that the finishing position of shot n represents the starting position for shot n + 1. Although shots on the same hole are related, one would expect the same lack of “streakiness” that has been demonstrated in the literature. This means that although a well played shot tends to set up an advantage on the ensuing shot compared to a poorly played one, a well played shot will not likely predict the performance of the ensuing shot.
3 The ISOPAR method
3.1 Framework
The previous described approaches of Broadie (2011) and Fearing et al. (2011), on which first measures for individual shots are based, are statistical and developed to predict performance and to compare per- formance to benchmarks of expected performance. Our approach is different from that and is specifi- cally aimed at characterizing the performance of the participating golfers, rather than predicting it. The framework of the ISOPAR project comes from a systems perspective and has been empirically applied to many levels of analysis of human movement and performance (e.g. Davids, Glazier, Araujo, & Bartle,
2003; Kelso, 1995; Mayer-Kress, Liu, & Newell, 2006). The central concept is that neurobiological sys- tems behave as complex systems and theories from physical sciences, e.g. dynamical systems theory, are appropriate for understanding and modeling human performance. Accordingly, golf performance is an emergent property of self-organizing dynamics and the confluence of constraints influencing the golfer (Newell, 1986). We have subsequently applied this perspective to golf performance on the PGA
TOUR measured by ShotLink™. The underlying assumption is that each shot a player faces, represents a new set of constraints and the player must adapt to the constraints associated with the shot, which can be divided up into three main categories: environment, organism, task (Newell, 1986). The constraints
3 do not necessarily prescribe one particular response (e.g. shot type), instead they guide the response selection of the golfer by excluding certain responses (Kugler, Kelso, & Turvey, 1980). Stöckl and Lames
(2011) have demonstrated the ISOPAR method for visualizing constraints in puing. A player’s puing performance is determined by a combination of environmental constraints (e.g. gradients of the green, distance to the hole, green speed, weather conditions), organism constraints (e.g. psychological influ- ences on the player, player’s green reading ability, player’s ability to perform pus) and task constraints
(hiing a golf ball with a club so that it rolls into the hole). The idea of visualizing the confluence of constraints off the green, by which the performance of players is determined, can be extended to entire holes to illustrate difficulty on a hole represented by the number of remaining shots – since each shot is part of a player’s shot sequence. Off the green, a player’s performance is also guided by the interaction of environmental, organism, and task constraints, however, their details may differ. For example, envi- ronmental constraints are the hole design (straight hole compared to a doglegged fairway),ball lie (e.g. fairway, rough, sand), line to the green (e.g. are there trees/bushes/other objects blocking the line to the green), or weather conditions (e.g. wind, rain); organism constraints can be psychological influences on the player, player’s decision making of the tactics for holing out with as few shots as possible, picking the ‘right’ club, or player’s ability to performing a planned stroke; task constraints are similar to those for puing, in that the player hits the ball with a club with the intention of the ball finishing close to the hole. However, the degrees of freedom involved with the swings used from off the green introduce a wider range of task specific constraints (e.g. achieving clean contact, club path into the ball, etc.). The coordination paern of the swing must be more adaptable for off-green shots because of the increased degrees of freedom involved with the movement itself as well as the increased variability in the results of the swing.
3.2 The concept
Here we present two analogies to help explain the following methods. In meteorology, lines of equal barometric pressure are ploed on geographical maps. These maps are called isobar maps and the lines are isobar lines. The term isobar (iso - meaning equal and bar - meaning pressure) is used appropriately as the isobar map shows lines of equal pressure. Small diameter, closed lines represent minima and maxima by which, areas of low-pressure and high-pressure can be identified. Densely packed isobar
4 lines indicate a steep gradient of air pressure. Meteorologists can therefore make weather predictions using isobar maps. Our second analogy is to contour maps used in geography to show elevation. Similar to isobar lines, lines of equal elevation are ploed on geographical maps. Here, densely packed lines represent steep ascents and descents. In both analogies, lines that are relatively close together represent steep changes in pressure or elevation, respectively. Likewise, lines that are relatively widely spaced represent areas of lile change in pressure or elevation.
For golf, we have developed the ISOPAR method for calculating a gradient of difficulty for a golf hole. The output can then be ploed on a map of the golf hole to visualize the difficulty of certain areas.
We call these maps ISOPAR maps and a detailed explanation of how they are calculated is provided below.
3.3 Development and testing
The ISOPAR method was originally developed for visualizing difficulty on the green based on the per- formance of the field on the green. Since we have the opportunity to use the ShotLink™database we can also calculate ISOPAR values and maps for entire holes. The calculation of ISOPAR values for entire holes is based on the same algorithm which will be described for greens in this section. To reduce com- putational complexity ISOPAR values are only calculated in a non-convex area in which ball locations were. In this section the development and testing of the method is described for this application. The three-dimensional spatial coordinates (x,y,z) of the green gives the first of three sets of triplets,
(xg,yg,zg), where g represents the number of measuring points. When available, this set of triplets can be used for ploing the physical contour of the green.
For each ball position, (x,y), the corresponding number of strokes, z, required for the player to hole out are used in the calculation. This gives our second of three sets of triplets (xp,yp,zp). For example, if a player took four shots on a hole, that player contributed four data points to our dataset: the x,y coordinates from the location of the first shot with a corresponding z value of 4 and the x,y coordinates from the second shot and a corresponding z value of 3 and so on.
5 3.3.1 Computing ISOPAR values and maps
Before explaining the details of the algorithm for computing the ISOPAR values and maps, a rough overview of the steps involved in calculating an ISOPAR map for a green is given (see Stöckl, Lamb, &
Lames, 2011):
1. Assign a grid to the green (Figure 1).
2. Calculate the ISOPAR value of every grid point subject to all measuring points with a modified
application of the exponential smoothing algorithm.
3. Compute a surface out of the ISOPAR values of the grid points using a smoothing spline interpo-
lation (Fahrmeir, Kneib, & Lang, 2009) to finely remove rough edges.
4. Calculate the ISOPAR map which consists of ISOPAR lines.
The following explains the steps for computing ISOPAR values and maps of greens in detail which we use for the calculation of entire holes as well. All computations were performed in MATLAB (The
Mathworks, Inc.).
Assign grid to green: A grid with a specified mesh size is assigned to the green (Figure 1). The ISOPAR values are computed at the grid nodes. For positions which lie between grid nodes the ISOPAR values must be estimated. Therefore, a grid with an extremely small mesh size represents the data very well, while a very large mesh size does not. However, there is a trade-off between representational power and computational intensity. A mesh size which optimizes this trade-off should be used.
Exponential smoothing algorithm: From Step 1, coordinates (xi j,yi j) were assigned to the grid nodes.
The corresponding zi j values which represent the ISOPAR values were then calculated; this gives the final set of triplets, (xi j,yi j,zi j),i = 1,...,m, j = 1,...,n. The algorithm used here is a well known smoothing algorithm; however, our application of the algorithm differs slightly from most applications. Typical applications of the exponential smoothing algorithm are in time-series analyses and based on pairs (xk,yk),k = 1,...,t, from which the value yt+1 at time xt+1 is computed. The modified application of this algorithm for calculating ISOPAR values is
6
j (x ,y ) ij ij
Ball Location Hole Grid Node Line Grid Node−Hole 60° Line Used Data
i
Figure 1: The mesh grid shown on the green. Green line represents the edge of the green. (xi j,yi j) represents coordinates for a grid point, blue dots represent ball positions, and red dots represent ball ◦ positions which are used for calculating the ISOPAR value at (xi j,yi j). The black, solid lines form a 60 angle which marks the boundary within which ball locations are used in the calculation.
based on the measuring points (xp,yp,zp), p = 1,...,q (q = number of sample points). The ISOPAR values zi j are computed based on these triplets. To use the exponential smoothing algorithm, which is based on pairs, we transformed the triplets into two-dimensional pairs, respectively. Since it does not make sense to include ball locations which are on the opposite side of the hole from the viewpoint of golf, we introduced a constraint for the usage of ball locations in order to calculate the ISOPAR value at a grid node. We empirically decided that ball locations which are considered for computing an ISOPAR value need to be in an area of 60 degrees le and 60 degrees right from the straight line between the pin location and the respective grid node
(the red data points in Figure 1). The transformation for every grid node was achieved by ordering the
7 (a) 6th hole (b) 18th hole
Figure 2: The ISOPAR maps for (a) the 6th hole at Bay Hill in the fourth round of the 2009 tournament and (b) the 18th hole in the fourth round of the 2008 tournament. The green line represents the edge of the green, the flag position is shown as a black dot. iso2.0 is shown in magenta. measuring points in ascending order (the nearest point first) with respect to the Euclidean distance
√ 2 2 di jp = (xi j − xp) + (yi j − yp) (1)
to the measured ball positions. This allowed the triplets from above to be wrien as pairs (di jp,zp). With the pairs sorted as described, we could use the exponential smoothing algorithm to calculate the ISOPAR values. In these pairings, (di jr,zr) represents the ball position with the shortest distance to the respective grid node and (di j1,z1) represents the ball position with the largest distance to the grid node. The exponential smoothing is calculated by
r−2 k r−1 zi j = α ∑(1 − α) zr−k + (1 − α) z1, (2) k=0 where 0 ≤ α ≤ 1 is the smoothing parameter (Hamilton, 1994).
The ISOPAR lines are calculated from the ISOPAR values (Figure 2). The ISOPAR lines, similar to the isobar lines used in our meteorological analogy, are the lines of intersection between planes which are parallel to the x,y plane in certain intervals and the surface which is calculated with the triplets
(xi j,yi j,zi j). The result is a contour map which empirically characterizes how many strokes “the field”
8 (a) (b)
Figure 3: Example of a portion of the grid surface (a) without smoothing and (b) with smoothing spline interpolation. took from each position on the green. Each line on the contour map is one of these lines of intersection, thus we argue that the ISOPAR lines give a visual representation of the difficulty of any shot on the green.
Smoothing spline interpolation: Because of the space between the grid nodes, the grid surface must be smoothed. Figure 3 shows the difference between the raw surface and the smoothed surface using a cubic smoothing spline interpolation (Fahrmeir et al., 2009).
n m ∫∫ 2 2 2 min β ∑ ∑(zi j − f (vi j)) + (1 − β)λ (D f (x,y)) dxdy (3) f i=1 j=1 where ∂ 2 ∂ 2 ∂ 2 D2 = + 2 + , ∂ 2x ∂x∂y ∂ 2y ( ) xi j vi j denotes the vector with entries , λ = 1 in our case and β is the smoothing parameter. When yi j β = 1, f is a natural spline interpolant – the cubic spline interpolant; when β = 0, f is a least square fit surface and as β → 1, the data remain relatively similar to the input.
Calculating the ISOPAR map: The ISOPAR lines were calculated in MATLAB. The ISOPAR lines are lines of intersection between the smoothed surface (calculated in the previous subsection) and planes which are parallel to the x,y-plane in certain intervals. For implementing the ISOPAR method we used intervals of 0.2, however, this value is not critical. The value for the interval should depend on the
9 objectives of and resources available to the user.
3.3.2 The performance indicator: Shot ality
Shot ality (SQ) is a post-hoc assessment of a shot taken. Similar to the shot value concept of Broadie
(2008), Shot ality is determined as the difference in ISOPAR value at the starting position (IPVbe f ore) and the ISOPAR value at the finishing position (IPVa fter) of the shot is calculated.
SQ = IPVbe f ore − IPVa fter (4)
Shot ality, as its name implies, represents the quality of a shot played. A shot of average performance, with respect to the data set (in this case the ShotLink™database), receives, by definition, a Shot ality score of 1 (proof shown below). A shot with a Shot ality higher than 1 is considered a well played shot and likewise, a shot with a Shot ality score of less than 1 is a poorly played shot.
Like the additivity property of the model of Broadie (2011), a unique property of Shot ality allows consecutive shots, which are performed in sequence (1,...,np) ending with the ball being holed, by a given player p to be weighted so that the sum of their Shot ality scores (SQ j) equals the ISOPAR value of the beginning position (IPV1) of the sequence:
np np−1 (4) − − ∑ SQ j = ∑ (IPVj IPVj+1) + IPVnp 0 j=1 j=1 − − − − = IPV1 IPV2 + IPV2 IPV3 + ... + IPVnp−1 IPVnp + IPVnp 0
= IPV1. (5)
− We have included 0 in the the final term IPVnp 0 to make clear that it represents the Shot ality of the final shot played on the hole (zero shots are required once the ball is holed).
Consider a hypothetical sequence of two pus on a green which starts from a position with an
ISOPAR value of 2.1. If the first pu missed, leaving a pu with an ISOPAR value of 1.1, the Shot
ality scores must be 1.0 for the first pu and 1.1 for the second, which adds up to the beginning
ISOPAR value of the sequence. If the first pu had been much worse, the holed second pu would necessarily have a higher value, the first lower, and then still add up to 2.1. If the second pu were missed, we now have a three shot sequence and these three Shot ality scores then add up to 2.1. This
10 concept applies to a sequence of shots of any length including the sequence of all shots played on a hole, as long as the final shot in the sequence results in the ball being holed. To follow this example, no maer the player’s score on the hole, the values of the Shot ality scores will add up to the ISOPAR value of the starting point of the sequence: the ISOPAR value at the tee (IPVTee). This leads us to another interesting property of Shot ality. In the ShotLink™database all tee shots recorded on the same hole
(and the same round) are assigned the same x,y coordinates – a single point. For this reason, we use the average score (Save) for the hole as the ISOPAR value at the tee
p ∑ S j j=1 IPV = S = , (6) Tee ave p
where S j are the hole scores for all p different players on the hole. Therefore, the sequence of all Shot ality scores for each player must add up to the average score for the hole. For example, another hypothetical golfer might score a birdie on a par 4 which has an average score of 3.92 which might involve a series of shots as follows: a good drive (SQ = 1.20), an slightly beer than average approach from that position (SQ = 1.05) and a very good pu (SQ = 1.67).
As mentioned above, the average Shot ality of all shots played on a hole (SQave) must be 1 and can now be shown by
p n j 1 · SQaveTotal = p ∑ ∑ SQi j=1 i=1 ∑ S j j=1 p (5) 1 · = p ∑ IPVTee j=1 ∑ S j j=1 1 · · = p p IPVTee ∑ S j j=1 p ∑ S j (6) 1 j=1 = · p · p p ∑ S j j=1 = 1, (7)
11 where p is the number of different players on the hole and n j is the number of shots played on the hole by each player.
Additionally, a new concept can be derived from Shot ality. Similar to strokes gained (Broadie,
2011; Fearing et al., 2011), already in use by the PGA TOUR, we assess the advantage gained relative to the average by a well played shot (or vice versa). Terminologically, we prefer Shots Saved instead of shots gained because a long pu made, saves instead of gains the player shots. Similar to the strokes gained concept (Broadie, 2008, 2011), Shots Saved is defined as
Shots Saved = SQ − SQave, (8)
where SQave denotes the average Shot ality of certain shot types (SQaveType), e.g. drives, or the average Shot ality of all shots (SQaveTotal).
4 Applying the ISOPAR method to ShotLink™data
While the methods of Fearing et al. (2011), Broadie (2011) and Minton (2011) can be used to make very good generalizations about the expected outcome of a shot based on its distance, the ISOPAR method is useful for answering a slightly different question. Given the factors which directly contribute to the performance of the field, how were certain shots performed with respect to the performance of the field?
4.1 Reading ISOPAR maps
In Stöckl et al. (2011) the concept of ISOPAR maps was originally described for on-green performance of amateur golfers. In this section we applied the ISOPAR method to on-green performance of PGA
TOUR golfers and extended the idea of calculating and visualizing difficulty to off-green performance of PGA TOUR golfers. According to Stöckl and Lames (2011) ISOPAR maps are suitable for identifying unique areas on the green. The iso-lines represent different levels of difficulty according to the number of remaining shots required to hole out. For example, if iso-lines were spread out evenly and circularly, we could conclude that all the constraints which influence performance were evenly distributed. Yet, we know
12 that many factors (e.g. distance to the hole, angle of approach, distance of the hole from the front of the green, speed and hardness of greens, etc.) directly influence performance, but they also indirectly influence performance. In other words, having a tree blocking the line to the pin constrains the kind of shot a player can play. This is an example of that factor directly affecting performance. An example of a factor indirectly affecting performance is a situation in which a player tries to play strategically, by aiming away from a hazardous area. The shot might result in a long remaining pu but has essentially eliminated the possibility of going in the hazard. The strength of the ISOPAR method is its capability of accounting for all the factors which influence performance. To illustrate this idea further, imagine all the players on a hole aim at a certain area of the green rather than the pin because the pin is close to a penalizing hazard. This would affect the distance-based statistical benchmarks and would make it look like everyone performed worse than expected. Accordingly, the benchmark could then be adjusted so that it is only representative of shots played on that specific hole. To take this example one step further though, imagine a situation in which long hiers give themselves such an advantage off the tee that they can play to the pin whereas the rest of the field still feel wise to play more conservatively.
This situation represents a non-linear relationship between players’ strategies. The same example is easy to think of on risk reward holes such as drivable par-4s and tricky par-5s. The ISOPAR method is capable of modeling these non-linearities in the data because it is a measure of performance rather than a predictor. The ISOPAR maps are capable of identifying how advantageous shots were (e.g. a drive that provides a good angle for the approach shot) as well as how hazardous hazards (e.g. rough, bunkers, trees, etc.) were.
In the folllowing subsections we will explain ISOPAR maps and how they can be interpreted. To read the ISOPAR maps we use the naming convention isoN to represent the ISOPAR line with value N.
4.1.1 On-green
The iso2.0 line is of importance, beyond iso2.0, 3-pus exist. Figure 4 shows the distribution of 1-, 2- and 3-pus on the 18th green at Bay Hill in 2008. The iso2.0 line is a result of the 3-pus shown in the figure.
Lorensen and Yamrom (1992), and later Penner (2002), modeled the difficulty of puing with differ- ent amounts of break and elevation change and from different distances. The authors showed that, not surprisingly, much more precision was required by the player as puing distance, break and elevation
13 Figure 4: The distribution of iso-lines, 1-, 2- and 3-pus on the 18th green in the 4th round of the Arnold Palmer Invitational in 2008. change increased. The ISOPAR maps visualize these factors as well as many other subtle factors which affect puing performance.
Since puing distance obviously increases outward from the hole, linearly and equally in all di- rections, the iso-lines should be circular on a flat green. However, since slopes are not symmetrically distributed across the green, the shape of the iso-lines can be useful in identifying easy or more difficult areas from which to pu. Useful characteristics of iso-lines are their a) circularity, b) density and c) their distance from the hole. If the map consists of circularly paerned iso-lines one can conclude that shot difficulty does not depend on the direction from which the stroke is taken. As the iso-lines become more elliptical certain areas of the green must be considered more favorable to pu from. The spread, or density, of iso-lines can be used to identify the severity of the gradient of difficulty on the green. A
14 Figure 5: The distribution of iso-lines on the 18th hole in the 4th round of the AT&T Pebble Beach in 2011. steep gradient is expected to coincide with undulated areas of the green but has not been empirically shown with ShotLink™data yet. The distance of the iso-lines from the hole can of course also be used to indicate difficulty of a pu. Reference values could be used as a comparison to provide context to the value of the iso-lines (e.g. Broadie, 2008; Cochran & Stobbs, 1968; Fearing et al., 2011; Tierney &
Coop, 1998).
4.1.2 Off-green
Figure 5 shows an ISOPAR map of the 18th hole in round four of the AT&T Pebble Beach in 2011.
The 18th hole at Pebble Beach is a unique par-5 because there is a tree in the middle of the fairway potentially blocking shots to the green. Furthermore, this hole is located directly on the coastline,
15 Figure 6: Zoomed in view of the ISOPAR map of the landing area in the fairway of 18th hole in round four at AT&T Pebble Beach in 2011. Red lines represent iso-lines which is the border on the le side.
Figure 6 is a zoomed in view of the ISOPAR map for the landing area of the drives. In the middle of the fairway is the tree, on the le side of the fairway is just a small strip of rough before the coastline starts, and on the right hand side of the fairway is a bunker. Looking at the iso-lines we notice that it was advantageous for the field to hit their drives on the fairway le of the tree where there was a local minimum of difficulty represented by the small, closed iso3.6 line. In contrast, the area behind the tree (the tree was blocking players’ line to the green) and close to or even in the bunker on the right hand side was much more difficult shown by the iso4.0 and iso4.2 lines. Hence, players whose drives ended up behind the tree or in the bunker had to take about half a shot more on this hole than players who were able to pass the tree with their drives or were able to keep their drives on the fairway to the le
16 of the tree. Among other constraints that influenced the players, this tree constrained the field’s play significantly.
4.2 Performance analysis
In this section we will show different applications of the performance indicators Shot ality and Shots
Saved in order to analyze performance.
4.2.1 Performance analysis based on ISOPAR maps on greens: Bay Hill in 2008 and 2009
To demonstrate the ISOPAR method the first time, we have used the ShotLink™data from the Arnold
Palmer Invitational presented by MasterCard in 2008 and 2009. In both years the tournament was won by Tiger Woods sinking dramatic pus on the final hole. These tournaments give us an opportunity to demonstrate performance analysis of the field using the ISOPAR method, as well as an analysis of the performance of Woods in both years.
In this section Shot ality and Shots Saved are used as performance indicators. The analyses in this section are based on ISOPAR values calculated for greens only. In order to analyze shots based on all shots, Shots Saved represents the difference between the Shot ality of any shot and Shot
ality of the average shot, which represents the field and has been shown to be exactly 1, and is called
Shots Savedtotal
Shots Savedtotal = SQ − 1. (9)
The Shots Savedtotal definition matches the definition of the strokes gained concept (Broadie, 2011;
Fearing et al., 2011). Shots Savedtotal represents the contribution of one shot by a player to that player’s total score with respect to the field’s performance².
Individual pus The ISOPAR method, because it is based on shot locations, can give Shot ality scores to individual shots. Table 1 shows the top-ten pus for the Arnold Palmer Invitational in 2008 and 2009, respectively. Notably, the pus with the highest Shot ality scores are not necessarily the longest pus. For example, in 2009 Daniel Chopra made a 31.2 foot pu on the 15th hole in the first round which had the highest Shot ality score of all pus in that year’s tournament, despite pus of
²In section 4.2.2 a slightly different application of Shots Saved will be introduced based only on shot types
17 more than double the length being holed by other players.
Table 1: Top-ten pus measured by Shot ality for the Arnold Palmer Invitational in 2008 and 2009.
2008 SQ Hole Round Distance () 1. Davis Love III 1.99 7 1 43.6 2. Bill Haas 1.99 2 1 61.1 3. D.A. Points 1.98 15 2 33.0 4. Charley Hoffman 1.98 3 1 36.8 5. Shaun Micheel 1.97 1 2 36.3 6. Mark Wilson 1.97 15 1 29.8 7. Tom Pernice Jr. 1.95 11 4 22.4 8. Brian Davis 1.95 6 2 23.0 9. Billy Mayfair 1.92 12 1 27.1 10. Kenneth Ferrie 1.91 2 1 41.0 42. Tiger Woods 1.83 18 4 24.2 n = 11,107
2009 SQ Hole Round Distance () 1. Daniel Chopra 2.12 15 1 31.2 2. J. J. Henry 2.11 8 1 55.7 3. D. J. Trahan 2.11 1 2 33.7 4. Zach Johnson 2.09 7 3 35.9 5. Ben Curtis 2.03 9 2 45.0 6. Fred Couples 2.01 11 2 37.9 7. Brian Gay 2.00 11 3 29.8 8. Aaron Baddeley 2.00 10 3 25.3 9. Heath Slocum 1.99 9 3 73.3 10. Jerry Kelly 1.95 18 4 38.1 40. Tiger Woods 1.84 13 1 16.4 203. Tiger Woods 1.69 18 4 15.9 n = 11,116
Tiger Woods’ winning pus in each year are shown in Table 1 in bold face (see also Figure 7 in
Appendix A). In 2008, the winning pu was the best pu by Woods of the week and the 42nd best pu out of over 11,000 pus in the entire tournament. In 2009, Woods’ winning pu was not his best of the week, his best was on the 13th hole in the first round,which was the 40th best pu of the week. His winning pu was the 203rd best pu of the week, again, out of just over 11,000 pus. This reveals exactly how well Tiger Woods performed on his final pu of the tournament, with the tournament on the line, two years in a row. One can, of course, argue that the winning pu was just one of 270 shots played in the tournament, and they all contributed equally to the outcome. However, we must
18 acknowledge, first that the preceding shots in the tournament were played sufficiently well so that
Woods had a chance to make a winning pu on the last hole; and second, the final pu is not like the rest because the consequences are known. In this sense we must appreciate the performance of Woods on these specific shots.
Shots Saved on and off the green Table 2 shows the top puing performers of the tournament and their off-green performance. On-green performance is calculated as above, however, calculating Shot
ality scores off the green is still under development (as of version from June 27, 2011; developments are introduced in the next section). Therefore, the off-green Shots Savedtotal can be calculated based on the average score of the field (hole, round or tournament) and the on-green score, which is already calculated. For example, in 2008, Tiger Woods’ score of 270 was 13.73 strokes beer than the average score of 283.73. Of the 13.73 stroke margin between his score and the field we calculated that he gained
1.13 on the green and, as a result, the remaining 12.60 strokes must have been from off the green.
In Table 2, the Shots Savedtotal performance indicator is introduced and shows a large discrepancy between Shots Savedtotal on the green and Shots Savedtotal off the green. At first glance it appears as though puing, because of the few Shots Savedtotal, is much less important than shots played off the green: this may in fact be the case but the topic requires some discussion first.
Shot ality and consequently, Shots Savedtotal, are independent measures of performance because the same metric is used for all shots. This means that any shot played can be directly compared to any other shot played. With that in mind, a speculative explanation for the discrepancy between on-green and off-green performance involves two factors: 1) There is a greater range of possible Shot ality scores for off-green shots compared to on-green shots. 2) a PGA TOUR player will typically take more off-green shots than on-green shots, so the number of elements in the off-green sum is greater than the elements in the on-green sum. Combined, these two factors may explain the discrepancy between
Shots Savedtotal on and off the green.
Anecdotally, one might notice in Table 2 that Brad Faxon is at the top of the Shots Savedtotal on the green list in 2009 (he was not in the field in the 2008 tournament). Each year on the PGA TOUR, no mat- ter how it is measured (number of pus, pus per GIR, or just how smooth the stroke looks to an expert eye), Brad Faxon is always among the best puers. In the 2009 tournament Faxon was the best puer and ranked fourth worst off the green. Clearly, Faxon was only able to make the cut in this particular
19 Table 2: Top-ten puers in the Arnold Palmer Invitational in 2008 and 2009.
2008 Shots Savedtotal Shots Savedtotal on the green off the green 1. Ken Duke 1.48 8.24 2. Tiger Woods 1.13 12.60 3. Hunter Mahan 0.79 8.94 4. Mark Wilson 0.47 -0.74 5. Carl Peersson 0.06 7.67 6. Woody Austin -0.46 6.19 7. Ian Poulter -0.47 1.20 8. Nick Watney -0.66 5.39 9. Frank Lickliter II -0.92 6.65 10. Joe Ogilvie -1.08 4.81 n = 71
2009 Shots Savedtotal Shots Savedtotal on the green off the green 1. Brad Faxon 1.32 -1.46 2. Lee Janzen 1.19 5.67 3. Lucas Glover 1.06 6.80 4. Daniel Chopra 0.58 8.28 5. Padraig Harrington 0.54 7.32 6. Zach Johnson 0.48 10.38 7. Tiger Woods -0.03 13.89 8. Ben Crane -0.06 7.92 9. Paul Goydos -0.34 3.20 10. Cliff Kresge -0.40 4.26 n = 73
tournament because of superior puing. As mentioned, Faxon is usually one of the best puers on the
PGA TOUR according to conventional statistics. These conventional statistics (i.e. pus per GIR) as we have already mentioned are a composite of previous shots played on the hole. If independent measures of performance, such as the ISOPAR method, had been available we may have noticed that Faxon was an even beer puer than previously thought. In Table 3, the leaders’ on- and off-green performance is shown. In both years, Woods performed, as the winner should, well on and off the green. He ranked 2nd in puing in 2008 and 7th in 2009. Off the green he ranked 9th in 2008 and in 2009, 5th. Combined, these performances on and off the green were good enough for him to win.
20 Table 3: Top-ten finishers in the Arnold Palmer Invitational in 2008 and 2009.
2008 Puing Shots Savedtotal Off-green Shots Savedtotal rank on the green rank off the green 1. Tiger Woods 2 1.13 9 12.60 2. Bart Bryant 11 -1.10 5 13.83 T3. Cliff Kresge 43 -4.70 2 15.42 T3. Vijay Singh 54 -6.26 1 16.98 T3. Sean O’Hair 16 -1.66 10 12.39 T6. Ken Duke 1 1.48 30 8.24 T6. Hunter Mahan 3 0.79 26 8.94 T8. Niclas Fasth 57 -6.82 3 14.55 T8. Alex Cejka 51 -5.33 7 13.06 T8. Carl Peersson 5 0.06 34 7.67 T8. Tom Pernice Jr. 55 -6.59 4 14.32 T8. Tom Lehman 30 -3.59 16 11.32 T8. Bubba Watson 34 -3.68 15 11.41 n = 71
2009 Puing Shots Savedtotal Off-green Shots Savedtotal rank on the green rank off the green 1. Tiger Woods 7 -0.03 5 13.89 2. Sean O’Hair 32 -2.58 1 15.44 3. Zach Johnson 6 0.48 21 10.38 T4. Pat Perez 15 -0.91 20 10.77 T4. John Senden 22 -1.76 11 11.62 T4. Sco Verplank 25 -2.04 10 11.90 T4. Nick Watney 27 -2.20 9 12.06 T8. Daniel Chopra 4 0.58 30 8.28 T8. Jason Gore 55 -6.24 2 15.10 T8. Kenny Perry 28 -2.35 15 11.21 n = 73
As exemplified by Vijay Singh, Niclas Fasth, Alex Cejka and Tom Pernice Jr. in 2008 and by Jason
Gore in 2009, it is possible to finish high in the tournament standings with relatively poor puing, if off-green performance is exceptional. The converse situation, in which poor off-green performance is balanced by excellent puing seems less profitable (for further context, see Table 9 in Appendix B which shows the boom ten players each year).
Using Shots Savedtotal, we were able to rank all the players in the field according to their on-green and off-green performance. The correlation (Spearman’s rank) between tournament rank and puing rank was ρ = .28 in 2008 and ρ = .44 in 2009. The correlation between tournament rank and off-green
21 rank in 2008 was ρ = .79 and ρ = .70 in 2009. These correlations are compelling evidence that off- green performance contributes to overall performance more than on-green performance. We should not discount the importance of puing, since it also is strongly correlated with overall performance.
We mentioned in the Shots Savedtotal on and off the green section that there is a discrepancy between the sum of Shots Savedtotal on the green and the sum of Shots Savedtotal off the green; and here show that off-green performance contributed more to overall performance than on-green performance. It should be noted that the discrepancy between on- and off-green Shots Savedtotal is not what implies the importance of off-green performance, rather the rankings in off-green performance. Those who were among the best off-green performance stood a beer chance of doing well in the tournament. Indeed, good off-green performance must be accompanied by on-green performance if one is to beat the best players in the world. These findings simply suggest that off-green performance is likely more important than previously thought.
4.2.2 Performance analysis based on ISOPAR maps of entire holes from 2011
In this subsection we use the performance indicator Shots Saved to analyze players’ performances with respect to different shot types during tournaments of the PGA TOUR in 2011. We calculated ISOPAR values and maps for 2,754 holes played in 153 rounds from the 38 PGA TOUR tournaments measured by ShotLink™.
A variation of Shots Saved, called Shots Savedtype, is introduced here to compare shots of the same type.
Shots Savedtype = SQ − SQaveType. (10)
Since shots are extracted from their original shot sequence in order to make this comparison, SQaveType does not necessarily equal 1. Shots Savedtype represents the quality of a shot in context of a certain shot type with respect to the field’s performance. We defined five different shot types: Drives, long approach shots, short approach shots, around the green shots, and pus. In the following, lists of the top ten golfers for each shot type are presented and compared to the most similar performance indicators currently used by the PGA TOUR.
22 Table 4: Top-ten puers in 2011 ranked by Shots Savedtype per round. Last column contains rank in PGA performance indicator Strokes Gained - Puing.
Rank Name Rounds Shots Savedtype Shots Savedtype Strokes Gained measured per Round total Rank (PGA) 1. Bryce Molder 71 0.773 54.863 3 2. Luke Donald 52 0.751 39.043 1 3. Charlie Wi 77 0.734 56.506 4 4. Steve Stricker 53 0.716 37.972 2 5. Kevin Na 68 0.591 40.159 8 6. Fredrik Jacobson 70 0.573 40.090 6 7. Brandt Snedeker 67 0.555 37.174 10 8. Greg Chalmers 79 0.553 43.668 5 9. Hunter Mahan 76 0.551 41.869 13 10. Jason Day 59 0.534 31.497 7 n = 202
Puing In table 4 the top ten golfers with respect to puing in 2011 are ranked by their Shots Savedtype values per round. In the last column of this table the golfers’ ranks in the PGA TOUR performance indicator Strokes Gained - Puing are listed. We calculated the Spearman rank correlation between the two puing rankings (ρ = .94), which shows a striking similarity in rankings. Comparing the ranks of the Shots Savedtype ranking and the Strokes Gained - Puing ranking of the top ten puing golfers in table 4 we can see that these players are also identified as the best puers by the Strokes Gained - Puing performance indicator. The small differences can be explained by the fact that the strokes gained method is based on benchmarks considering the distance to the hole, an indicator for the difficulty of the green, and the field strength in puing (Fearing et al., 2011) whereas the ISOPAR method implicitly considers all constraints which affect performance.
Driving Table 5 shows the top ten drivers in 2011 ranked by their Shots Savedtype values per round. We considered drives to be all tee shots taken on par-4s and par-5s. The players in table 5, we argue are popularly and anecdotally known as good drivers. We compared the Shots Savedtype ranking to the Total Driving performance indicator of the PGA TOUR which is intended to best account for a player’s driving ability. Many of the good drivers with respect to Shots Savedtype, like J.B. Holmes or Robert Garrigus, are not ranked well in the Total Driving performance indicator. The Spearman’s rank correlation between the Shots Savedtype ranking and the Total Driving ranking (ρ = .60) shows that
23 Table 5: Top-ten drivers in 2011 ranked by Shots Savedtype per round. Last two column contain ranks in PGA performance indicator Total Driving and Driving Distance.
Rank Name Rounds Shots Savedtype Shots Savedtype Total Driving Driving Distance measured per Round total Rank (PGA) Rank (PGA) 1. J. B. Holmes 47 0.991 46.588 90 1 2. Dustin Johnson 54 0.904 48.827 30 3 3. Gary Woodland 70 0.834 58.382 23 5 4. Robert Garrigus 69 0.787 54.317 94 4 5. Bubba Watson 59 0.780 46.003 35 2 6. Adam Sco 45 0.710 31.972 5 24 7. Jhonaan Vegas 70 0.595 41.644 84 8 8. Martin Laird 62 0.572 35.435 26 13 9. Bill Haas 72 0.566 40.745 76 48 10. Kyle Stanley 80 0.558 44.653 18 9 n = 202
these two rankings are not coupled tightly. Total Driving is a combination of two performance indicators for driving, the Driving Distance and the Driving Accuracy. The correlation between the Shots Savedtype ranks of the golfers and their Driving Distance rank (ρ = .74) is much stronger than the correlation between Shots Savedtype and Total Driving. Hence, Driving Accuracy, which is a binary measure of whether a drive ends up in the fairway, seems to over-influence the Total Driving performance indicator.
Approach shots Table 6 and Table 7 show approach shot rankings. We distinguished between long and short approach shots because we argue that players perform different types of swings for each of these two shot types. Generally, shots longer than 100 yds require full swings, the specific distances can be gaged by club selection. For approaches under 100 yds, players are almost always hiing some kind of wedge and are scaling the distance by adapting their swings. The ability to scale one’s swing according the shot (more common in short approaches), we argue, is a qualitatively different skill than performing full swings which rely less on swing modifications. ShotLink™also indicates whether a shot is a short or long approach. Approach shots were defined by ShotLink™as all shots taken from further than 30 feet from the edge of the green (except tee shots), which ended up on or around the green (within
30 feet). According to a benchmark used by the PGA TOUR an approach shot is a long approach shot if it is taken from further away than 100 yards. Alternatively, an approach shot is a short approach shot if it is taken from closer than 100 yards.
24 Table 6: Top-ten long approach shot players in 2011 ranked by Shots Savedtype per round. Last column contains rank in PGA performance indicator Approaches from >100 yards.
Rank Name Rounds Shots Savedtype Shots Savedtype >100 yards measured per Round total Rank (PGA) 1. Phil Mickelson 58 0.691 40.087 82 2. Rory Sabbatini 61 0.535 32.641 73 3. Bubba Watson 59 0.443 26.135 114 4. Luke Donald 52 0.380 19.741 10 T5. Sergio Garcia 45 0.378 17.031 93 T5. Kris Blank 83 0.378 31.390 10 7. Chris DiMarco 82 0.372 30.493 108 8. Jonathan Byrd 72 0.355 25.589 114 9. Robert Garrigus 69 0.338 23.355 8 10. Alex Cejka 47 0.309 14.543 4 n = 202
Table 6 shows the top ten long approach shot players with respect to Shots Savedtype per round in 2011. Most of the players ranked highly are again popularly known for their good long game or ball striking. We compared our ranking with the PGA TOUR performance indicator Approaches from >100 yards by computing Spearman’s rank correlation (ρ = .53). The two performance indicators are only correlated moderately. One reason for this is likely because the PGA TOUR performance indicator does not take into account the difficulty of the starting position of an approach shot - only how close to the hole the approach shot ends up. For example, an approach shot taken from 105 yards in the middle of the fairway which ends up 8 feet from the hole is assessed the same quality as an approach shot taken from 150 yards in the rough which ends up 8 feet from the hole according to the conventional performance indicator. In contrast, the ISOPAR method considers the difficulty of a shot and assesses the more difficult shot a higher Shot ality value and consequently a higher Shots Savedtype value. Because of this we argue that Shots Saved beer assesses the quality of the shot played.
Table 7 shows the top ten short approach shot players with respect to Shots Savedtype per round in 2011. Once again, we suggest that these golfers are well known as good short game players. We compared this ranking to the PGA performance indicator Approaches from <100 yards. These two per- formance indicators are also only correlated moderately (Spearman’s rank correlation ρ = .68), how- ever slightly stronger than Shots Savedtype and Approaches from >100 yards. The difference in these two rankings can be explained the same way as with the long approaches. The classical performance
25 Table 7: Top-ten short approach shot players in 2011 ranked by Shots Savedtype per round. Last column contains rank in PGA performance indicator Approaches from <100 yards.
Rank Name Rounds Shots Savedtype Shots Savedtype <100 yards measured per Round total Rank (PGA) 1. Nick Watney 55 0.227 12.495 1 2. Paul Goydos 72 0.218 15.693 5 3. Brian Gay 58 0.194 13.159 22 4. Steve Stricker 51 0.189 9.617 2 5. Stephen Ames 51 0.184 9.391 15 6. Camilo Villegas 42 0.182 7.638 28 7. Justin Rose 51 0.151 7.693 11 8. Luke Donald 41 0.149 6.125 9 9. Chris Kirk 67 0.141 9.444 19 10. Sco Piercy 51 0.136 6.955 3 n = 202
indicators do not take into account the difficulty of a shot and only focus on the outcome of the shot, the remaining distance to the hole, whereas the ISOPAR method accounts for a shot’s difficulty.
Furthermore, we can recognize that the top players’s Shots Savedtype per Round values are much smaller in this shot category than in all other shot categories which were studied. The Shots Savedtype per round values are smaller because short approach shots are played less frequently compared to the other shot types.
Around the green shots Table 8 shows the top ten around the green players ranked by Shots Savedtype per round in 2011. Around the green shots are defined as shots taken from an area within 30 feet around the green, a variable which is collected by ShotLink™. These golfers are also subjectively known for good short game performance. We compared Shots Savedtype around the green to the PGA TOUR’s Scrambling performance indicator although we admit it is not a completely valid comparison. The
Scrambling performance indicator represents how oen a player saves par when missing the green in regulation. So to do well in this statistic players could a) leave themselves easy around the green shots, b) play their around the green shots very well or c) make a lot of par saving pus. With any combination of these being possible it is difficult to make a meaningful analysis of performance using Scrambling.
The Shots Savedtype ranking is only moderately correlated (ρ = .54) with the PGA TOUR’s performance indicator Scrambling because this happens to be a category in which the performance of an individual
26 Table 8: Top-ten around the green shot players in 2011 ranked by Shots Savedtype per round. Last column contains rank in PGA performance indicator Scrambling.
Rank Name Rounds Shots Savedtype Shots Savedtype Scrambling measured per Round total Rank (PGA) 1. Chris Riley 68 0.465 31.639 20 2. Kevin Na 68 0.422 28.709 9 3. Bio Kim 58 0.418 24.261 142 4. Charles Howell III 91 0.380 34.577 5 T5. Justin Rose 63 0.337 21.260 95 T5. Brian Gay 74 0.337 24.912 7 7. Webb Simpson 83 0.327 27.150 16 8. Marc Leishman 80 0.311 24.852 71 9. Rod Pampling 60 0.308 18.500 10 10. Alex Cejka 47 0.290 13.625 111 n = 202
shot is lost.
An interesting example to illustrate this point is that of Bio Kim in 2011. Kim was ranked third according to Shots Savedtype but 142nd in Scrambling. Since Kim performed roughly average in puing
th in 2011 (Shots Savedtype rank = 99 ), Kim likely played many good shots around the green but was unable to make the ensuing par pu. This hurts his Scrambling rank but his ability to play ‘around the green’ shots is accurately reflected in the Shots Savedtype performance indicator.
5 Final remarks
The results presented in this working paper are specific to PGA TOUR tournaments. Especially, the results in section 4.2.1, which were outcomes of an earlier state of the project, are specific to the Arnold
Palmer Invitational in 2008 and 2009. Furthermore, the application of the ISOPAR method relies on calculating ISOPAR values and maps for each hole in each round based on all shots (all pus in section
4.2.1) taken by the participating players. Sometimes there are areas on a hole or a green where there are few isolated ball locations. Since the ISOPAR method models the performance of the participating players, the performance indicators Shot ality and consequently Shots Saved may not assess the
‘real’ quality of performances of those shots. Finally, we welcome any feedback from readers to help us improve and find new uses for the ISOPAR
27 method.
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29 30 Appendices
A ShotLink™, Google Earth and ISOPAR maps
(a) 2008
Figure 7: ISOPAR map for the 18th hole at Bay Hill during the Arnold Palmer Invitational presented by MasterCard in 2008. Orange lines are shown at intervals of 0.2 ISOPAR value, ball positions are shown as yellow dots and the winning pu by Tiger Woods was taken from the red ‘X’, the hole is shown as a black dot.
31 (b) 2009
Figure 7: ISOPAR map for the 18th hole at Bay Hill during the Arnold Palmer Invitational presented by MasterCard in 2009. Orange lines are shown at intervals of 0.2 ISOPAR value, ball positions are shown as yellow dots and the winning pu by Tiger Woods was taken from the red ‘X’, the hole is shown as a black dot.
32 B Shots Saved for the lowest finishers
Table 9: Shots Saved on and off the green for the lowest ten finishers of the 2008 and 2009 Arnold Palmer Invitational.
2008 Shots Saved Shots Saved on the green off the green T62. George McNeill -4.67 2.40 T62. Davis Love III -3.66 1.39 T64. Paul Goydos -4.47 1.20 T64. Steve Elkington -3.00 -0.27 T64. Andrew Magee -2.97 -0.30 T64. Fred Couples -1.77 -1.50 T68. Robert Gamez -7.88 2.61 T68. Marc Turnesa -7.44 2.17 70. Steve Lowery -4.27 -4.00 71. Heath Slocum -9.39 -0.88 n = 71
2009 Shots Saved Shots Saved on the green off the green T64. Boo Weekley -11.36 8.22 T64. Luis Oosthuizen -1.10 -2.04 T66. Skip Kendall -6.20 2.06 T66. Richard Johnson -4.48 0.74 T66. Kevin Streelman -3.52 -0.62 T66. Aaron Baddeley -2.99 1.15 T70. Oliver Wilson -8.63 2.49 T70. Brian Davis -5.08 -1.06 72. Woody Austin -4.81 -2.33 73. Bart Bryant -4.49 -5.65 n = 73
33