To see how the “new” Slope Rating is related to the USGA’s
Slope Rating, it is necessary to review the origins of the USGA’s Slope
System. The theory behind the Slope
System was first put forward by R.C. Stroud and L.J. Riccio in an article
entitled

*Mathematical underpinning so the slope handicap system*.[1] Stroud starts with Figure 1 below that purports to show how the average unadjusted differential of players from Perfect Valley vary by course.[2]
Perfect Valley is
depicted as a course of moderate difficulty.
Stroud assigns Perfect Valley a slope of 1.13. The slope of 1.13 is due to three factors
embedded in the handicap system: 1) using only the best 10 of the last 20
differentials, 2) Equitable stroke control, and 3) the .96 bonus for excellence
factor. This gives the illusion that an average course would have a Slope
Rating of 1.13. This would not be
correct. Whatever course Stroud selected
as his reference course would have a Slope Rating of 1.13. For example, assume Stroud picked Panther
Mountain as his reference course. The
slope of the line relating Average Differential and Panther Mountain Handicap
would still be 1.13. The slope relating
the Average Score at Perfect Valley with Panther Mountain Handicap would be 0.92
(i.e., 1.13x1.13/1.39) and the slope at
Open Flats would be 0.74.[3]

What should a player’s
Standard Course Handicap (i.e., Perfect Valley Handicap or what the USGA now terms a player’s "Index") be at
other courses?[4]
For generality, let us name the course in question “Any Course.” The handicap formula tells us:

1) Standard Handicap(Any Course)
= ATBD(Any Course) x .96

Where,

ATBD(Any Course) = Average of Ten Best
Differentials at Any Course

The player’s Course
Handicap at Any Course can be written as,

2) Course Handicap(Any Course) =
ATBD(Any Course)/1.13

The ATBD(Any Course)
is related to his Standard Handicap by eq. 3,

3) ATBD(Any Course) x .96 =
Slope Rating(Any Course) x Standard Handicap

Where,

Slope Rating(Any Course) = Slope of the line
relating average differentials at Any Course
with Standard Course handicaps

Substituting eq. 3
into eq. 2, the Handicap(Any Course) becomes
4) Handicap(Any Course) = Slope
Rating(Any Course) x Standard Handicap/1.13

In summary, a
player’s handicap at Any Course is given by the product of his Standard Course
Handicap and Slope Rating(Any Course) divided by 1.13.

The Slope Rating,
however, has little intuitive meaning. A
more informative measure of a course is what is termed the “new” Slope Rating.
The new Slope Rating is simply the percentage of your Standard Handicap that is
allowed at a course. In essence,

5) Handicap(Any
Course) = New Slope Rating(Any Course) x Standard Handicap

Substituting from
eq. 4 into eq. 5, the New Slope Rating can be defined as:

6) New Slope Rating(Any Course)
= Slope Rating(Any Course)/1.13

If the New Slope
Rating of a course is 120, for example, a player gets to play at 120 percent of
his Standard Handicap.[5]

[1] Stroud,
R.C., Riccio, L.J., “Mathematical underpinnings of the slope handicap system,”

**Science and Golf**, E & F Spon, London, 1990, p. 136.
[2] Stroud actually plots average score versus handicap in
his paper. But he also claims (

*op.cit*., p. 140):*So the better half of average differentials (ATBD) divided by average score (for any handicap level) is 1.04/1.13 = 0.92.*

This
is not correct. A ten-handicap, for
example, would have an ATBD of 10.4 (ATBD= Handicap/.96). His average score would be the Course Rating
plus 11.3 (Handicap x 1.13). If the Course
Rating was 72.0, the ratio of ATBD to average score would be 10.4/83.3 =
0.12. What Stroud probably meant was
that the ratio of ATBD to Average Differential (Average Score – Course Rating)
is 0.92. Therefore, his figure has been
changed to reflect this correction. The figure also assumes the average
differential of a scratch player is zero.
This assumption is not correct, but is built into the Slope Handicap
System.

[3] The myth that 113 is the Slope Rating of an average
course was probably started by Dean Knuth, former Director of Handicapping for
the USGA, when he wrote:

“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 with whole numbers.”

*(**See*Knuth D., “A two parameter golf course rating system*,”***Science and Golf***, E*& F Spon, London, 1990, p. 143.)
Knuth’s explanation is either wrong or
possibly just not stated with precision.
The Slope Rating is the slope of the regression line of total score
versus USGA handicap

**for players from the reference course**(named Perfect Valley in USGA research papers). The slope of the regression line of score versus handicap for players at their home course will be 1.13 for all courses according to the USGA.
[4]
The handicap of a player at the Standard Course is measured in tenths and is
identical to what is now termed a player’s “index.” See “Simplifying the Slope Handicap
System,” www.ongolfhandicaps, 2/1/2013.

[5] The new Slope Rating has the same validity problems as
the old Slope Rating. Both the USGA’s old
Slope Rating and the New Slope Rating are based on the same two assumptions: 1)
There is linear relationship between average differentials at Any Course and
the handicaps at the Standard Course, and 2) Average differentials are 1.13
times a player’s handicaps. The validity
of these assumptions has not been tested in the peer reviewed literature.

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