INDUSTRIAL ENGINEERING AND
OPERATIONS RESEARCH
PRESENTS
IEOR MONDAY SEMINAR
The USGA Golf Handicapping
System: Is it Fair?
Steven Nahmias
Eugene Yano
Yano Accountancy
Corporation
ABSTRACT
The system adopted by the USGA (United States Golf Association) for computing
golfers’ handicaps is a fairly complex procedure based on the best 10 of the
most recent 20 rounds posted. The question is if this approach results in fair
matches among players of different abilities. To test the fairness of the
system, we have constructed hole-by-hole distributions for five hypothetical
golfers at Palo Alto Municipal Golf Course, and simulated a variety of
different match and metal play scenarios among them. Our results indicate that
the system tends to favor better players (lower handicappers), and steadier
players (those with lower variance in their scores). However, we also observed
some surprising results for formats other than one-on-one match or medal play.
LITERATURE
Most agree the game of golf was invented by the Scots in the
mid 17th century, even though it does bear some similarity to games
invented earlier in
In 1857 H.B. Farnies’ Golfer’s Manual suggested the following adjustments in matches between two players of different abilities:
Third-one: a shot every three holes
Half-one: a shot on every other hole.
One more: a stroke a hole
Two more: two strokes a hole
Seen in the light of today’s handicapping system, such adjustments would be considered crude. In later years, handicapping systems were based on committee decisions. Each player would receive a number of strokes based on what the handicapping committee at his club felt was appropriate. By 1881, many clubs adopted the procedure of averaging a player’s best three scores for the year, and subtracting the course rating (essentially the total par for the course) to compute the player’s course handicap. Poorer players complained that this system strongly favored better players. (Based on what we know today and the results of this study, these complaints would seem to be justified.) In recent times, it was recognized that handicaps need to be adjusted based on the difficulty of the golf course. That is, players should receive more strokes on more difficult courses. However, there was no way of measuring course difficulty other than the par and course rating. The only way to distinguish the difficulty of golf courses was by the total number of strokes corresponding to par. Thus, if a player had a 12 handicap on a par 70 course, they would receive 14 strokes on a par 72 course. This system did not adequately account for differences in course difficulty, however.
This issue was addressed by the USGA when they adopted the slope system in 1987. The slope is a number attached to each golf course to measure the course’s relative difficulty. It ranges from approximately a low of 100 for very easy courses, to as much as 145 for very difficult courses, with 113 being the “standard” value. (Why the USGA chose to make 113 the standard, and not 100, is unclear). The slope is used to determine a golfer’s course handicap. The key point behind the slope concept is that it makes larger adjustments for higher handicap players than for lower handicap players. The logic behind this is that higher handicappers tend to have more trouble (relatively speaking) on difficult courses than do lower handicappers.
Prior Literature
While there has been interest in golf handicapping issues in the prior literature, we have not encountered any work which addresses the same set of issues we address, nor which use the same methodology. One of the earliest studies of golf using OR techniques is due to Pollock (1974). Pollock assumed known probability distributions for each hole score, and assumed handicaps were computed using the system employed by the USGA prior to the introduction of slope. He uses several approximations to obtain analytical approximations for both medal and match play between two players.
Kupper et. al. (2001) focus on the fact that the USGA handicap system is based on order statistics. They show analytically in a match between two golfers with the same handicaps but different variances, the golfer with the lower variance is favored. Our results corroborate theirs, but the scope of our study is considerably wider.
Others have suggested modifications in the present handicap system to account for differences in golfers’ consistency (see Smith and Prockow (1981)). This is not the issue we address here. Our interest is in determining the fairness of the current USGA handicap system. Another issue considered in the literature is where should strokes be given in match play competitions between two players (Hall and Swartz (1981)). Again, this issue is outside of our current interest, since the USGA specifies that strokes be given in the order of the hole handicapping of a particular golf course. They showed analytically that the outcome of the match will depend upon which holes strokes are given (which is obvious to anyone who plays the game).
The USGA Handicapping System
In this section we describe the USGA handicapping system currently
in use today (refer to the USGA Handicap Manual available online at http://www.usga.org/) .
After each round of golf, players are expected to post their scores (adjusted
scores actually – we will address this issue later) on a computer. These scores
are sent to one of several computers located regionally throughout the
It is important to distinguish the differences between the terminology used prior to 1987 with that used today. Prior to 1987, every golfer would have a handicap. This was an integer which reflected the player’s ability, the lower the better. The player’s handicap is the number of strokes the player would receive in any sanctioned competition (except for those where players are expected to play “scratch” or with no handicap, as do professionals). Today, a player does not carry a fixed handicap, but carries an index. The index is computed monthly and truncated to one decimal (for example, 15.4). The index is then converted to a course handicap based on the course slope. Hence, the index is a number associated with each player, and the handicap is a number associated with each player at a specific course. The handicap is still an integer.
The procedure used by the USGA for computing the index is fairly complex. Let’s suppose that a player with a 14 course handicap has just completed a round of golf with a score of 90. Let’s further suppose that in that round the player scored one 8 and one 9 during the round (the remaining scores being 7 or less). Before posting, the player computes an adjusted score. A 14 handicapper is not allowed to post more than 7 on a single hole. In this case, the player must adjust his or her score by 3 strokes ((8-7) + (9-7)), and post an 87. This score is then converted to a differential. Two numbers are required to convert this posting to a differential. One is the course rating and the other is the course slope. The course rating is a number close to the total par for the course, and is a relative measure of the length of the course. For example, a typical par 72 course might have a course rating of 69.5 if it is relatively short, or 73.4 if it is relatively long.
Let’s continue with the example. Suppose that our 14 handicapper has just posted an 87 at a course with course rating 70.4 and slope 128. The resulting differential is
(87 – 70.4)(128/113) = 18.8 (rounded to the nearest tenth).
Continuing with this example, suppose that the last 20 differentials posted by our 14 handicapper were:
16.3, 12.5, 14.0, 23.2, 11.8,
14.9, 18.3, 19.1, 12.8, 16.5, 17.4, 14.6,
13.3, 22.4, 13.9, 12.7, 19.2, 17.4, 15.3, 12.6.
The player’s index is based on the best 10 of the past 20 postings, shown in this example in boldface italic. These 10 are averaged to obtain 13.31 and then multiplied by 0.96 to obtain 12.78 and finally truncated (not rounded) to one decimal to yield the monthly index of 12.7. As new scores are posted, they replace older scores, so that the index represents a type of moving average, which is recomputed on a monthly basis.
The Rationale
What, one may ask, is the rationale for this complicated
calculation? One begins to understand the rationale for this system by becoming
familiar with the history of handicapping, and by better understanding the
purpose of handicapping from the USGA point of view.
A natural question is why doesn’t the USGA simply base
handicaps on average scores? First, actual scores must be replaced by
differentials to take slope and course rating into account. Golfers know that
two courses that may have the same par rating of 72 could be very different in
terms of difficulty. As noted above, the course rating is a measure of course
length, while the slope is a measure of difficulty. Note that these are not
necessarily the same thing. A long flat course with few hazards could have a
high course rating and a low slope, while a short hilly course with many
hazards could have a low course rating with a high slope. Hence, an 85 on
Spyglass Hill in
The second question is why not simply average the past 20 differentials rather than take the best 10 of the last 20? The reason for this seems to be historical. Recall that one of the very early handicapping systems was to only consider the player’s best three scores over the previous year. The idea has evolved that a person’s handicap should not reflect their average performance, but should reflect their good performance. The system was designed to be sure that a player is not be rewarded for an unusually poor score. Players know that very high scores are “tossed out” and will not figure into their index computation.
The rationale for the 0.96 adjustment is somewhat less clear. Note that any adjustment by a fixed multiplier will have a greater effect on high handicappers. If the average of ones best 10 differentials is 39.89, then one receives a (0.96*39.89) = 38.29 which comes to a 38.2 index, while if the average of the best 10 differentials is 3.26, the index is 3.1. Hence, the adjustment results in a 1.7 stroke penalty for the high handicapper and a 0.2 stroke penalty for the low handicapper. It is interesting to note that at one time this multiplier was 0.85, thus resulting in much larger penalties for higher handicap players. This was changed in recognition of the fact that an adjustment of this size provided too much advantage to better players. Presumably, the multiplier is included to provide a little more help for lower handicappers when competing against higher handicappers.
Is it Fair?
Now that we understand what the USGA handicapping system is, the question we wish to address is: Is this system fair? In the case of one-on-one matches, a fair system results in a 50/50 probability of either player winning. One must recognize that this ideal can never be exactly achieved, since course handicaps must be integers. And there are many different competitive formats. As we will see, the results vary by the type of format.
Based on the way handicaps are determined, we hypothesize the following:
1) Better players are favored over weaker players.
2) Steadier players are favored over more erratic players.
3) Different types of players are favored in different formats.
The rationale for 1) is that better players tend to have less variation in their scores from one round to the next. This means that using the best 10 order statistics to determine index should favor the better player. Also, we saw how the 0.96 multiplier penalizes higher handicappers more than lower handicaps. Hypothesis 2) comes from the fact that the system is based on order statistics. Steadier players have lower variance in their scores, and thus will be penalized less by the use of order statistics.
How to test these hypotheses is not obvious. One might think that using actual scores would be the best approach. However, as anyone familiar with the game knows, golfers are not consistent. Furthermore, current indexes may not reflect a player’s current ability level. Players that are improving will score better than their handicap, while players that are getting worse or who haven’t played in awhile, are likely to score worse than their handicaps.
For these reasons, we decided that a better way approach is
to simulate matches played by hypothetical golfers. For each hypothetical
golfer, we construct a probability distribution of scores on each hole. To be
sure that these distributions are reasonable, we assume that they correspond to
an actual course familiar to one of us (Nahmias.) We
will assume that all matches are played at Palo Alto Municipal Golf Course
(PAGC), a public links course located in
The score card for PAGC appears in Figure 1, and the slope and course rating information in Figure 2. Note that the course is a typical par 72 layout. As with most par 72 courses, each nine has two par 3’s and two par 5’s, with the rest of the holes being par 4’s. In order to simplify the analysis, we assume that all matches are played from the same set of tees (whites) and all matches are among men. (Matches between men and women are complicated by the different hole handicaps, and different slopes and course ratings.) Note that the score card also indicates hole handicapping. Each hole is given a number from 1 to 18. The purpose of hole handicapping is to rank holes according to their difficulty relative to par. The fourth hole, for example, has handicap 1, meaning it is considered the most difficult hole relative to its par of 4, while the tenth hole has handicap 18, meaning that it is considered the least difficult hole relative to its par of 4. Hole handicapping is important in determining on which holes players give strokes in match play and skins formats.
Note from Figure 2, that course ratings and slopes depend on which tees are being played, and the gender of the player. It is interesting to note that the red tees are considered more difficult for women relative to par than are the blue tees for men. (Women that compete in “men only” tournaments must do so as “men”. That is, they post scores from men’s tees and use slope and course ratings as if they were men. Some women actually maintain two sets of indices: one as if they were men and one as women.)
In order to test our hypothesis, we constructed five hypothetical golfers. These golfers may be classified as follows:
1) Scratch player. This is a player whose index is close to zero and scores near par.
2) Low handicapper. We define a low handicapper as a player with a single digit index.
3) Consistent bogie golfer. A bogie golfer scores close to one over par per hole on average. Bogie golfers have indices in the 14 – 18 range. A consistent bogie golfer is one that has relatively low variance. This might be an older player that was once a low handicapper, but has lost distance with age.
4) Inconsistent bogie golfer. This player also has an index in the 14 – 18 range, but has much higher variance. This might be someone who can hit the ball a long way, but has a tendency to “spray” the ball resulting in frequent penalties.
5) High handicapper. High handicappers have indices in the 20+ range. Our high handicapper has an index greater than 30.
While these five golfer types don’t represent everyone, they provide a good test of handicapping system. For each player type, we constructed hypothetical hole-by-hole probability distributions assuming play on the Palo Alto Municipal Golf Course. They appear in Figures 3 – 7. Note that Figures 3 – 7 include the mean and variance of each hole score. Assuming independence of hole scores, the mean and variance of the score per round are the sum of the means and variances of hole scores. Note that variance tends to increase as the player’s mean score increases, and that the means for players 3 and 4 are almost the same, but player 4 has a significantly higher variance.
Establishing the Index
The first step in any comparison is to determine the index for each player. To do so, we simulated 5,000 rounds for each player. After each set of 20 rounds, we determined the player’s index using the formulas described earlier, and then averaged the resulting 250 index calculations. The resulting indexes for each player are:
Player 1 (scratch player): 1.9. (CH = 2)
Player 2 (low handicapper): 6.9 (CH = 7)
Player 3 (consistent bogie golfer): 15.1 (CH = 15)
Player 4 (inconsistent bogie golfer): 14.9 (CH = 15)
Player 5 (high handicapper): 34.2 (CH = 35).
The course handicaps (CH) correspond to the slope from the
white tees at
CHi = Indexi * (115/113) rounded to the nearest integer. We should note that in order to avoid integer effects, we adjusted the probability distributions so that indexes and course handicaps were close. This can make a significant difference, especially for the scratch player. (For example, if the calculation of CH came to 1.4, this player would have to play with a handicap of 1, while if it came to 1.5, this player would play with a handicap of 2, which is a 100% difference in course handicaps based on only a 7% difference in indices. Rounding effects are less significant for higher handicap players.)
Medal Play
Once each player’s course handicap is established, we can then simulate matches among the players. Simulations were done in Microsoft Access. The first comparisons we consider are medal (stroke play) competitions among two players at a time. Again, we assumed 5,000 rounds played for each type of match considered. First, we consider all possible pairs of players. Given five different player types, this means that there are 10 different matches between different player types. Table 1 considers all 10 combinations assuming that each player plays at 100% of their course handicap. We have organized the matches so that player A is always a lower handicap than player B (except in the case of player 3 versus player 4 as they have the same course handicap). The key point to note here is that in every case, player A wins more than 50% of the time. That is, the lower handicapper wins more often than the higher handicapper. The largest difference occurs between players 1 and 5. Playing at 100% of their handicaps, the scratch player will win approximately 73% of the time against the high handicapper. Note also that player 4, the inconsistent bogie golfer has the next worse record after the high handicapper. These results are consistent with our hypotheses that the lower handicap player has an advantage over the higher handicap player, and that the steady player has an advantage over the inconsistent player.
The next question we address is what percentage of handicap differences provide the fairest matches in this format. These results are presented in Table 2. Due to integer effects, we could not get the percentage of wins to be exactly 50%, but in almost all cases it comes quite close. Note that in the case of player 3 versus player 4 there is no percentage of handicap differences that equates the two players, since they both have the same course handicaps. The results of this table should be interpreted as follows: Consider a match between player 1 and player 4, for example. Normally in such a match, player 1 would spot player 4 thirteen strokes (the difference in course handicaps). However in this case, player 1 would win nearly 70% of the time. However, if player 1 spots player 4 a total of 13(1.17) = 15 strokes, the match would be fair.
Match Play
We observed similar results in the case of match play. For those not familiar with golf, match play works as follows. The match is played on a hole-by-hole basis. Each hole is either won by a player or tied. The match ends when one player closes out the other. For example, if A is plus 3 with two holes to play, the match is over since B has no chance of winning. This would be referred to as a 3 and 2 win (3 up with 2 holes to play). If the match is tied after 18 holes, most competitions require a sudden death playoff. That is, the players continue the match until a player wins a hole. Another important point distinguishes match play from stroke play. In match play one must determine on which holes strokes are given. This is the purpose of hole handicapping. As an example, suppose that player 1 is playing a match against player 2. The difference in their course handicaps is 5 strokes. On what holes should these strokes be given? They are given on the holes having handicaps 1 through 5. From the score card in Figure 1, we see that the five strokes must be given on holes 4, 5, 6, 14, and 17.
We simulated match play format between two players. At 100% of the handicap differences, the better player, A, had an advantage over B as we hypothesized. The results of match play at 100% of handicap differentials are given in Table 3. Note that even though player A still wins the majority of matches, the percentage of time that A wins is not the same in match play as in stroke play. In most cases, the percentage of times that player A wins is less in match play than in stroke play. This suggests that the USGA handicapping system provides fairer competitions in match play than in stroke play. Table 4 gives the percentage handicap differences that are fairest for match play.
Skins
Another competitive format which has become popular in recent years is called skins. In order for a player to win a skin, he must defeat all his competitors on that hole. Note that skins is really just an extension of match play to more than two players. Hole scores are “net” scores. That is, the number of strokes scored minus the number of handicap strokes. In most skins games, strokes are determined the same way that they are determined in match play. Handicaps are spun off the lowest handicap player. Thus, if players 2, 3, 4, and 5 are playing a skins game, player 2 plays scratch (no strokes), player 3 gets 8 strokes, player 4 gets 8 strokes, and player 5 gets 28 strokes (which would correspond to 2 strokes on the first 10 handicap holes and one stroke on the remaining 8 holes).
There are two types of skins formats: with and without carryovers. A carryover occurs when no one wins a skin on a hole. In that case, the next hole is worth double. If that hole is tied as well, the next hole is worth triple, etc. If there are no carryovers, each skin has equal value. Our results indicated no significant difference between the two types of formats.
Given that we have shown that the lower handicap player has the advantage in every format considered thus far, we were surprised by the results for the skins format. We assume that the skins game is played among all the players, 1, 2, 3, 4, and 5. Suppose that all players are given 100% of their course handicaps. We observed the following results for skins games among the five players with and without carryovers.
Percentage of skins won – individual Skins
Player: 1 2
3 4 5
Percent: 13.6
25.3 20.0 20.8 20.3
Percentage of skins won – carryover skins
Player: 1 2
3 4 5
Percent 13.3
25.4 20.2 21.0
20.1
The results are surprising. The scratch player has the worst performance, while the low handicapper has the best performance. The three higher handicap players win roughly 20% of the time each. It is not clear why player 2 has an advantage in this format.
We considered the following issue: what fixed percentage of individual handicaps gives the fairest matches in skins? We could not find a fixed percentage that yielded 20% for each player. The closest we were able to come to this ideal was at 85% of course handicap differential. The results appear below.
Percentage of skins won – individual skins (85%)
Player: 1 2
3 4 5
Percent: 18.6
25.2 19.0 20.0
17.2
Percentage of
skins won – carryover skins (85%)
Player: 1
2 3
4 5
Percent 18.5
25.0 18.8 19.9
17.8
Note that in this case, player 2 still wins most often, but now player 5 wins least often.
Tournament Play
A tournament generally involves a larger group of players. In most tournament settings, attempts are made to separate high and low handicappers into flights, so that scratch players are not competing with 30 handicappers. However, there are cases where there is only a single flight, and all players compete head on. Based on our match play results, one might think that the low handicappers would have the advantage in this kind of format, but the simulations did not bear this out.
In order to simulate a tournament situation, assume that there are 20 players competing in a medal net score format. That is, the individual score is the gross score minus the course handicap for each player. We assume that the 20 players are four player 1’s, four player 2’s, . . . , and four player 5’s. We looked at both medal play (most actual tournaments) and skins formats. In each case considered, we determine the percentage of times the tournament winner was from category 1, 2, 3, 4, or 5. If two players tie, they are each awarded one half win; if three tie they are awarded one third win each, etc. We tested tournament play at a variety of course handicaps percentages. The surprising (and most telling) was that the fairest handicap percentage in the tournament setting was 100%. The observed results for medal play were:
Results for Medal Play Tournaments
Player Type Index Course Handicap % Wins
1 1.9 2 20.7
2 6.9 7 17.4
3 15.1 15 20.2
4 14.9 15 21.5
5 34.3 35 20.2
Our comment that these results are the most telling speaks to the issue of the reasoning behind the USGA formula for computing handicaps. Perhaps the methodology is aimed at making tournament competition fair. There is one surprising result here. Player 2 seems to have a disadvantage in this format, while player 2 had significant advantage in the 5 person skin game. We’re not sure why this occurs.
One might think that 100% of course handicaps would also provide a fair match in the case of tournament skins. Tournament skins means that our 20 players compete for skins on each hole. As with the 5 person skin format, in order to win a skin, a player must now beat all the players in the tournament on that hole. The results here are rather surprising. At 100% of course handicaps we observed:
Results for Tournament Skins
Player Type Index Course Handicap % Wins
1 1.9 2 8.0
2 6.9 7 11.6
3 15.1 15 12.1
4 14.9 15 26.5
5 34.3 35 41.8
In this format, we see that the high handicappers have a huge advantage. In addition, player type 4 (high variance bogie golfer) does significantly better than the steady bogie golfer as well as the low handicappers. Why is this? The reason is that player types 4 and 5 have the highest variance. With five players of type 4 and five players of type 5, it becomes much more likely that one of these players will have a good hole and win a skin.
The fairest overall adjustment in the skins tournament seems to occur at 80% of course handicaps. Having each player at 80% of his or her course handicap yields the following results for the skins tournament:
Tournament Skins at 80% of Course Handicap
Player Type Index # Strokes (80% of CH) % Wins
1 1.9 2 16.2
2 6.9 6 15.4
3 15.1 12 13.3
4 14.9 12 29.9
5 34.3 28 25.2
Note that a fixed percentage reduction affects the higher handicappers more than the lower handicappers. Even so, players of type 4 and 5 still seem to have a significant advantage, although this is the closest we could come to a fair system.
References
Hall, C. and Swartz, C., (1981). “The
Effect of Handicap Stroke Location on Best-Ball Golf Scores,” Mathematical Modelling,
2, pp. 161-167 (1981).
Henderson, R.W. (2001). Ball,
Bat, and Bishop: The Origin of Ball Games.
Kupper, L.L, Hearne, L. B.,
Martin, S. L., and
Pollock, S. M. (1974). “A Model for Evaluating Golf Handicapping,” Operations Research, 22, pp. 1040 – 1050.
Smith,
Appendix. Figures and Player Distributions.
Figure 1. Scorecard for
|
Rating and Slope |
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Rating |
Slope |
|
Black |
72.4 |
118 |
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Blue |
71.1 |
117 |
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White (Men) |
69.5 |
115 |
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White (Women) |
75.0 |
125 |
|
Red (Men) |
66.6 |
109 |
|
Red (Women) |
71.8 |
118 |
Figure 2. Course Rating and Slope Information for PAGC
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Player
1. Scratch Handicapper |
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Hole # |
Handicap |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
Mean |
Variance |
|
1 |
11 |
|
0.03 |
0.1 |
0.7 |
0.1 |
0.07 |
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|
|
5.08 |
0.5936 |
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2 |
7 |
|
0.08 |
0.72 |
0.16 |
0.03 |
0.01 |
|
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|
4.17 |
0.4211 |
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3 |
17 |
0.15 |
0.75 |
0.07 |
0.03 |
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2.98 |
0.3396 |
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4 |
1 |
|
0.08 |
0.72 |
0.11 |
0.07 |
0.02 |
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4.23 |
0.5971 |
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5 |
3 |
|
0.1 |
0.71 |
0.16 |
0.03 |
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4.12 |
0.3656 |
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6 |
5 |
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0.11 |
0.73 |
0.14 |
0.02 |
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4.07 |
0.3251 |
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7 |
9 |
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0.12 |
0.72 |
0.12 |
0.04 |
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4.08 |
0.3936 |
|
8 |
13 |
0.08 |
0.74 |
0.14 |
0.04 |
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3.14 |
0.3604 |
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9 |
15 |
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0.05 |
0.13 |
0.68 |
0.1 |
0.04 |
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4.95 |
0.5875 |
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10 |
16 |
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0.25 |
0.66 |
0.07 |
0.02 |
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3.86 |
0.3804 |
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11 |
4 |
0.11 |
0.71 |
0.13 |
0.05 |
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3.12 |
0.4256 |
|
12 |
14 |
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0.02 |
0.12 |
0.66 |
0.15 |
0.05 |
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5.09 |
0.5419 |
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13 |
12 |
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0.08 |
0.71 |
0.15 |
0.06 |
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4.19 |
0.4339 |
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14 |
8 |
0.09 |
0.75 |
0.13 |
0.03 |
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3.1 |
0.33 |
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15 |
18 |
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0.02 |
0.14 |
0.73 |
0.08 |
0.03 |
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4.96 |
0.4184 |
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16 |
10 |
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0.13 |
0.7 |
0.12 |
0.05 |
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4.09 |
0.4419 |
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17 |
2 |
|
0.08 |
0.73 |
0.1 |
0.07 |
0.02 |
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4.22 |
0.5916 |
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18 |
6 |
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0.1 |
0.64 |
0.2 |
0.06 |
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4.22 |
0.4916 |
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