## Thursday, July 17, 2014

### Why Is “113” in the Slope Handicap System?

Introduction and Summary - The number “113” has a ubiquitous presence in the USGA Handicap System.   It is used as a multiplier in calculating a player’s handicap differential.  It is used as a divisor in determining a player’s handicap.  It is not clear, however, how this seemingly strange number came to be such an integral part of the Slope System.  Three possible reasons are examined here.  First, a case can be made that “113” stems from the theoretical underpinnings of the Slope System.  A slope of 1.13 is assumed by the USGA to be the slope of the line relating a player’s average score with his handicap.  The USGA’s Course Rating Model, however, actually measures the slope of the line relating the average of the better half of a player’s scores and his handicap.  The slope of this line is 1.04.  To determine a course’s Slope Rating, the USGA takes the result of the Course Rating Model and multiplies it by 1.13/1.04.    Therefore, theory may explain where “113” came from, but not why it was chosen over “104”.
Second, the USGA has stated “113” is used since it is the empirically derived average Slope Rating on a standard American golf course.  Since the Slope Rating is essentially a statistical index, there can be no empirically derived average.  Any average Slope Rating would depend upon the USGA’s choice of a reference value.  The USGA’s choice of “113” determined the reference value, not the other way around.
Third, the USGA selected “113” for bureaucratic reasons.  This is the most likely explanation.  The USGA could have chosen any number for the reference Slope Rating—“100” would have been the ideal choice—without altering the handicaps of any player.  The number “113” was probably chosen because it gives the illusion of precision and adds mystique to the Slope System.  In the end, however, the selection of “113” unnecessarily complicated the Slope System but did not affect estimates of Indices or Handicaps.  Each of the possible reasons behind “113” is examined in turn.
1. The Number “113” Stems from Theoretical Underpinnings of the Slope System - The theory behind the USGA’s Slope system was first put forward in an article by Stroud and Riccio (hereinafter referred to as Stroud).   Stroud argued the “slope” of a course can be defined either of two ways.   First, the “slope” can be defined as the slope of average score versus a player’s Perfect Valley handicap where Perfect Valley is the reference course.  Second, the slope can be defined as the average of a player’s ten best differentials versus his handicap at the reference course.  The reference course would have a slope of 1.13 if a player’s average score is plotted.  The slope would be 1.04 if the average of a player’s ten best differentials is plotted.
This would seem to explain where “113” came from.  Unfortunately, the USGA’s Course Rating Model does not measure the slope of average scores versus handicap.  The USGA does not define either the Course Rating or the Bogey Rating with mathematical precision.  Course Rating, for example, is defined as the playing difficulty of a course for a scratch golfer based on distance and obstacle factors (i.e., the USGA Course Rating Model).  Knuth, however, writes that Course Rating Model attempts to predict the average of the better half of scores for the scratch and bogey player.  Using Knuth’s definitions of the Ratings, the slope actually measured is:

1)            Slope = (BDBP – BDSP)/Bogey Handicap
Where,
BDBP = Average of the ten best differentials of a bogey player
BDSP = Average of the ten best differentials of a scratch player
Bogey Handicap = Handicap of the Bogey Player at the Reference Course

By definition the slope of this line is 1.04 (1/.96).  The USGA could have chosen to go with "104" as the reference Slope Rating.  Instead, they chose to multiply both sides of the equation by 1.13/1.04.  This resulted in the USGA’s equation for the Slope Rating (Note: The Bogey Handicap has to be 20.16 for this to work.  This is different than the USGAS’s definition of the Bogey Handicap which is 20.0.  This small discrepancy, however, does not affect the workings of the Slope System):

2)            USGA Slope Rating = 5.381·(Bogey Rating – Course Rating)

The USGA has never published any evidence that the slope of the line plotting average score versus handicap is 1.13 at all courses.  For this to be true a crucial assumption has to be met: A player’s standard deviation of scores must increase at the same rate at all courses.  That is, on average the standard deviation of scores for any handicap level is not dependent on the Slope Rating.  The USGA adopted “113” even though it is based on this tenuous theory.  By adopting “113” instead of “104”, the USGA also increased the standard error of the estimate for the Slope Rating.  In summary, there is nothing in the theory of the Slope System that would make “113” a clear choice.

2. The number “113” was Chosen Since It Was the Average Value of a Standard American Course – Knuth has written that “113” was an empirically derived average Slope Rating.  Knuth does not document this finding.  An examination of how the Slope Rating is estimated shows that “113” was not an average.
The USGA used a multivariate regression model to estimate the Course and Bogey Ratings as a function of yardage and obstacle values.  The USGA’s Course Rating Model does not measure the Slope Rating directly, but rather the difference between the Bogey Rating and the Course Rating.  As an example, Table 1 shows the difference in Ratings for five courses.  The only way “113” can be the average Slope Rating is if the originator (i.e., the USGA) assigns the Slope Rating of “113” to a course that has the average difference between its Bogey and Course Rating (Course 3 in this example).  In essence, the Slope Rating of “113” was not empirically derived, but a conscious selection by the USGA.

Table 1
Difference Between Bogey and Course Rating

 Course Difference Between Bogey and Course Rating (Strokes) 1 19.0 2 20.0 3 21.0 4 22.0 5 23.0

3) “113” Selected for Bureaucratic Reasons – In the USGA’s Handicap System, any number (except zero) chosen for the Slope Rating of the reference course will yield identical results.    As an example, let’s compare the Stroud Slope Rating based on a reference course Slope Rating of "104" with the USGA Slope Rating of “113.”   Table 1 presents the Slope Ratings under the two methods for courses where the difference between the Bogey Rating and Course Rating is 26, 21, and 16 strokes.

Table 2
USGA and Stroud Slope Ratings

 Course Bogey Rating – Course Rating USGA Slope Rating Stroud Slope Rating 1 26 140 129 2 21 113 104 3 16 86 79

The two methods yield different estimates of the course Slope Rating, but identical estimates of a player’s index and handicap.   To see this, Table 3 presents the index of a player who has ten best differentials equal to DIFF on the Course 1.  His Index is the same under both Slope Ratings.

Table 3
Index of a Player with Ten best Differentials Equal to DIFF

 USGA Index Stroud Index Index =.96·Diff·(113/140) = .77· DIFF =.96·DIFF·(104/129) = .77·DIFF

Table 4 presents this player’s handicap at the three courses.

Table 4
Handicap of a Player with Ten Best Differentials Equal to DIFF

 Course USGA Handicap Stroud Handicap 1 =.96·DIFF(113/140)·(140/113) = .96·DIFF =.96·DIFF(104/129)·(129/104) = .96·DIFF 2 =.96·DIFF(113/140)·(113/113) =.77·DIFF =.96·DIFF(104/129)·(104/104) =.77·DIFF 3 =.96·DIFF(113/140)·(86/113) =.61·DIFF =.96·DIFF(104/129)·(79/104) =.61·DIFF

Handicaps are determined by the ratio of the difference between the Bogey Rating and the Course Rating at any course.  The Slope Rating chosen for the course of average difficulty has no effect on the efficacy of the Slope System.

So why did the USGA choose “113” as the reference Slope Rating?   When the USGA named the new handicap system the Slope System, it obviously needed a slope.  It is possible “113” was selected since it was easier and more understandable to talk of a player’s average score than the average of his ten best differentials.

A more sinister reason is USGA did not want a simple explanation of the Slope Rating and chose to make a player’s handicap a function of two rather meaningless numbers—the Slope Rating and “113.”  This added complexity and may have served to dampen criticism of the Slope System.  After all, it is difficult to criticize what you don’t understand.

Another possible reason for the USGA’s decision is that it wanted to give the illusion of precision.   The number “113” gives the impression that it was derived through exact science.  For example, if you hear that the attendance at a golf tournament is 25,000 it sounds like a rough estimate.  If the attendance is announced 25,301 it appears as an accurate estimate even though the method of counting is exactly the same.  The number “113” does not sound like it was drawn out of a hat, even though that is not far from the truth.

If the new handicap system had been named something like Handicap Adjustment for Course Difficulty there would have been no reason to introduce a slope.  Instead of the Slope Rating, there would have been something termed the Adjustment Rating.    A reference Adjustment Rating of “100” would have given intuitive meaning to the rating.  The Adjustment Rating would be the percentage of a player’s index he would receive as his course Handicap.  For example, if the Adjustment Rating was 120, a player’s Course Handicap would be 120 percent of his index.

Conclusion – There appears to be no strong theoretical or empirical basis for choosing “113” as the reference value for the Slope System.  There are possible bureaucratic reasons to explain the USGA’s decision to select “113.”  In the end, however, the decision unnecessarily complicated the Slope System but did not affect estimates of USGA Indices or Handicaps.

 Knuth, Dean, “A two parameter golf course rating system,” Science and Golf, The Proceedings of the World Scientific Congress of Golf, E & F.N. Spon, London, 1990, p. 143.
 The Slope Rating is essentially an index and any number can serve as the reference value.  The USGA Index, however, is not an index.  This is confusing and may explain why some countries have not adopted the term “Index.”  Australia for example, uses the term “Golf Australia Handicap” for what the USGA calls a player’s Index. Canada uses the term “handicap factor” instead of Index.
 Stroud, R.C., and L.L Riccio, “Mathematical underpinnings of the slope handicap,” Science and Golf, The Proceedings of the World Scientific Congress of Golf, E.FN. Spon, London, 1990, pp. 135-140.
 The USGA Handicap System 2012-2015, United States Golf Association, Far Hills, NJ, p. 11.
 Knuth, Dean, “A two parameter golf course rating system,” Science and Golf, The Proceedings of the World Scientific Congress of Golf, E.F. Son, London, 1990, p. 143.
 There is little evidence to suggest that “113” is the average Slope Rating in the United States.   A cursory observation of Slope Ratings listed by state golf associations indicates that the average Slope Rating is much higher than 113. Moreover, the reference Slope Rating for women is also 113.  What is the likelihood that the average Slope Rating is the same for both men and women?