A two parameter golf course rating system
D. KNUTH
Director of Handicapping,
US Golf Association
Abstract
Research by the United States Golf Association concluded that the primary technic error in handicapping was in its single parameter course rating system  the basis for the calculation of handicaps (also called the Standard Scratch Score in some countries). The major problem identified was that one parameter rating crated R "portability error" for nonexpert players. Depending on the difficulty and placement of obstacles, two courses with a difference in Course Rating of only three or four strokes could produce score differences of twelve strokes or more for average players. This portability error gave average and higher handicap players a major advantage over lower handicap players when competing at easier courses. The converse was also true. More than ten years of research, data collection and field testing produced a two parameter system. More than 10,000 courses in America were rated for both scratch and "bogey" golfers to produce both a USGA Course and Slope Rating. The "Slope" System provides that through charts, golfers receive additional handicap strokes when competing at difficult or "high Slope" courses and fewer strokes at "low Slope" courses. The relative increase or decrease in handicap strokes is proportional to the magnitude of players' handicaps.
Key Words: Handicap, Portability, Slope, Course rating, Scratch, Bogey golfer.
1. Introduction
The United States Golf Association appointed eight mathematicians to its Handicap Research Team (HRT) in 1979 to study the reliability of the USGA Handicap System, and in particular, to determine whether handicaps developed at one course were equitable for use at other courses. The author was selected to the HRT based on previous research in quantifying course obstacles for scratch course rating.
Thousands of scores from a broad range of golfers and courses were analyzed. After two years, the HRT concluded that basing handicaps only on a standard of Course Rating (Standard Scratch Score) created a "portability error" in the handicaps of average golfers when competing away from their home course. Despite the existence of an accurate course rating at different courses, in matches between players from different courses, the average player from the more difficult course consistently won because handicaps based only on one parameterthe expected score of a scratch golferare not portable.
2. Portability error
The player of scratch standard represents less than onehalf of one percent of amateur golfers. Approximately fifty percent of all male players in the USA have handicaps of 14 to 20 strokes, and the national average handicap of all men is 17. The average player cannot overcome distance and obstacles as well as scratch players. For example, obstacles such as a pond extending from the teeing ground 180 yards out to the fairway, are insignificant to scratch players, but such hazards are very significant for higher handicap players. The effect is that difference in score between dissimilar courses is much greater than the difference in course rating. Portability error can be quantified to be the observed difference in score between two courses by a group of players after subtracting the difference in course ratings.
The HRT found that the magnitude of the portability error depended on the size of the player's handicap, that it averaged three strokes for average players and in extreme cases amounted to 12 strokes.
Portability error is made more serious in the United States as shown from a USGA survey conducted in 1980. That survey found that on the average, golfers had played only fourteen of their last 20 rounds on their home course. Golfers who do not play away still are affected by portability when guests come to play at their courses.
Another USGA study has shown that less than 50 percent of American golf courses fit the mold of an average course, one for which the course rating provides the right number of handicap strokes for all players, and portability affects them only when they are visited by players from more challenging courses. In comparison to an average course, the majority of other courses are substantially more or less difficult for less skilled players than they are for scratch players. The high USGA Handicap of average golfers, the propensity of golfers to play their games at different courses, and the prominence of the difficulty (obstacle) factor all add up to a conclusion that portability was a very important problem for the American golfer. In view of this, a refinement to the USGA Handicap System was warranted. That refinement was to rate courses for the average or "bogey" golfer in addition to the scratch golfer.
3. Slope Rating
The USGA coined the term which numerically describes the difference in course rating difficulty between bogey and scratch players as the "Slope Rating," which is a second dimension in handicapping. Slope Rating is the slope of a regession line of total score versus USGA Handicap for a particular golf course. The Yintercept is the USGA Course Rating which is the better half score average of scratch golfers. The slope of the scores line of an average course has been observed to be 1.13 and USGA Slope Rating is referenced as 113 to deal in whole numbers.
A scratch golfer (zero USGA Handicap) will score proportionally fewer strokes than a 10, 20, and 30 handicapper. The slope of the line through their scores indicates the difficulty of the course, and the slope is steeper for more difficult courses and less steep for a less difficult course. In effect, the Slope System provides a separate USGA Course Rating for each handicap. It becomes readily possible to adjust scores for handicapping purposes which in turn minimizes the portability error.
In contrast, the preHRT USGA Course Rating System, which was based only on the scratch golfer, was a onenumber system which compared course differences for experts only and was based primarily on a yardage factor. It used one number as a description of golfers who play on their home course and on any other golf course. This worked adequately for the scratch golfer or for golfers who developed their handicaps on courses of equal Slope Ratings, but was unable to describe the ability of golfers from other golf courses if their courses were not of average difficulty.
The current USGA Course Rating System is based on a model which predicts the average of the better half scores of the 288 competitors in the U.S. Amateur Championship. The prediction is made by trained raters who measure a course electronically and evaluate each hole for the severity of the obstacles listed on the following page.
A model bogey golfer is one with a USGA Handicap Index of 17.5 to 22.4 strokes. A bogey golfer can hit tee shots an average of 200 yards and can reach a 370 yard hole in two shots. A "Bogey Rating" is determined by evaluating the obstacle factors from the standpoint of the bogey golfer, and then by applying a linear yardage equation based on ability to overcome distance. The result is equivalent to the average of the better half of a bogey golfer's scores under normal playing conditions.
A "USGA Slope Rating" is determined by multiplying the difference between the Bogey Rating and the USGA Course Rating by 5.381. This constant will produce Slope Ratings of 113 when the differential between the Bogey Rating and Course Rating is 21.0 (The expected difference in score due to "bonus for excellence"). A Slope Rating of 113 also is the empirically derived average value on standard American golf courses.
The yardage rating equation for scratch men is length (yards) divided by 220 plus 40.9 (bx18). For bogey golfers the values are 160 and 50.7, respectively. Following are the obstacles factors considered by rating teams in determining the amount of adjustment of the Yardage Rating in arriving at the USGA Course and Bogey Rating along with the weighting factor applied to each obstacle value rated on a scale of 0 to 10.
 Weight 
Obstacle  Scratch  Bogey 
Topography  .10  .12 
Fairway  .11  .09 
Recoverability and Rough  .14  .15 
Out of bounds  .10  .09 
Water Hazards  .14  .14 
Trees  .09  .14 
Bunkers  .07  .10 
Green Target  .09  .06 
Green Surface  .11  .08 
Psychological  .05  .03 
The 18hole sum of scratch weighted obstacle values is multiplied by .11 and 4.9 is subtracted to determine the summed scratch obstacle stroke value. The corresponding factors for the bogey rating are .26 and 11.5, respectively.
In addition, rating teams consider factors which cause a course to play significantly longer or shorter than its measured length. These "effective playing length" factors are:
Roll  Unirrigated and thin fairways and downhill landing areas result in the ball rolling farther than the normal 25 yards. Irrigated and lush fairways and uphill landing areas result in the ball rolling less than 25 yards.
Elevation  Holes which are uphill from tee to green play longer than those which are downhiI1 from tee to green.
Dogleg  Holes in which the fairway bends short of the normal drive zone will force the player to hit less than a full teeshot.
PrevaiIing Wind  Even though there may be as many holes with the wind as against, a constant wind, as on a seaside course, makes play more difficult.
Altitude Above Sea Level  The ball will carry a greater distance in high altitudes. The Yardage Rating of a course at an altitude of 2,000 feet or higher is adjusted downward to compensate.
4. The USGA Course Rating Model
The model used in developing the USGA Course Rating System was:
(F  R) (c) = a(1) + b(Y) = (s)
"F" is the matrix of rating values in which each of ten obstacles on each hole is rated on a defined scale of 0 to 10 by a rating team.
"R" is a reference matrix used to isolate the yardage terms based on parameters when yardage alone is judged to be an adequate explanation of hole difficulty.
"c" is the vector of coefficients (weights), "a(1)" and "b(Y)" are the yardage terms and "(s)" the vector of actual hole score averages.
The solution method used was the classic multiplication of both sides of an overdetermined system Ax = b by the transpose of A. This yields the well determined system AT A = AT b, and is equivalent to finding the least squares solution.
Holes from seven courses were rated from Amateur Championships and a similar number were rated for a team of 60 bogey golfers that provided data. A problem in determining optimum scratch coefficients was that because many holes in National championships are similar in length and obstacle difficulty, use of all hole data restricted convergence. By selecting 74 holes containing dissimilar variables, convergence was obtained with a goodness of fit of .989 for scratch golfers. The fit was .957 for bogey golfers.
An unbiased estimate of the holebyhole error of prediction is
stroke rms error in rating a golf course.
5. Application of Slope Rating
USGA Slope Ratings range from a high of 152 for Pine Valley Golf Club down to 55 for very short executive length golf courses. Approximately 50% of golf courses rated by the USGA have Slope Ratings between 100 and 120.
After the USGA Slope Ratings have been determined by a rating team, each golfer's handicap is converted to a decimal USGA Handicap Index, which is the handicap that would be developed by always playing an average Slope course (113) such as "Perfect Valley." For each round played, a golfer's Handicap Differential is computed according to the previous procedure (score minus the scratch rating), but modified for the USGA Slope Rating of the course ((Score minus scratch rating)x 113/Slope) . Handicap Differentials from high slope play will be reduced proportionately, while low slope differentials will be increased. The end result is that all USGA Handicaps are comparable, no matter where the players develop them.
Before the introduction of Slope, players with the ability of a 15 handicap might have developed a 20 USGA Handicap at Pebble Beach or only a 10 USGA Handicap at the easy "Open Flats." Through the Slope System, however, the USGA Handicap Index of each of these players would be a 15.0. The USGA Handicap Index travels with the golfer wherever the golfer plays.
6. Course Handicap Table
The final step of this system is to determine how many strokes each player requires when playing a golf course (including the home course). When golfers play on a course with a high USGA Slope Rating, the need for strokes increases as the difference in USGA Handicap Index increases.
Golfers determine how many strokes they receive by referring to a Course Handicap Table. Courses post the Table for each set of tees (championship tees normally have a higher Slope than the standard tees). Players check the Table before beginning play by locating their Index on the Table and reading the corresponding Course Handicap column. The Table values are determined by multiplying a player's Handicap Index by the Slope Rating of the course played, dividing by 113 and rounding off.
7. Implementation
Slope was tested at six extreme courses in Northern California in 1981 and statewide in Colorado in 1983. The USGA trained more than 4,000 volunteers from all 50 states to rate courses. Since then, Slope has been implemented nationwide in America and is in use at 10,000 golf courses by more than four million golfers. The portability error has been reduced to a level within the noise of sampling errors.
8. Future Research: The Player Variability Problem
The HRT has shown that 80% of all bogey golfers fit the model within one stroke. However, two distinct types of golfers fall outside of these limits. These types have been labeled "Steady Eddy" and "Wild Willy." Steady Eddy represents 12% of all golfers and he is a very straight, but short ball striker who has an outstanding short game. When taken from a short course to a long course, his score increase8 greater than the model would show, thus he is underhandicapped at a high Slope course. Conversely, Wild Willy is a longhitter, but is inaccurate. Representing 8% of bogey golfers, this type can be over handicapped on a long open course, but underhandicapped on any very tight and punitive course.
The HRT is considering a solution of adopting a normal model handicap formula which would mesh a two dimensional handicap to the Slope System. The solution could result in a Steady Eddy receiving more strokes on a high Slope Rated course than a Wild Willy of equal Handicap Index would receive.
References
Knuth, Dean L. "Proposed Course Rating System," April 21, 1976
Handicap Research Team, "Handicap Report, Volume I, Issues and Findings," January, 1981
Handicap Research Team, "Handicap Report, Volume III, Survey," November, 1980
Handicap Research Team, "Handicap Report, Volume IV, Golf Course Rating Procedure," April, 1981
USGA, "USGA Handicap System Manual, Part III, January, 1987
