Previous
posts (e.g., see

*The Mathematical Underpinnings of the "New" Slope Rating,*February 4,*2013) have documented the confusion at the USGA about the Slope Rating. This post takes another whack in the spirit of flogging a dead horse. Let’s start with documenting the USGA’s arguments as put forth by Dean Knuth, former USGA Director of Handicapping:**The USGA coined the term which numerically describes the difference in course rating difficulty between the bogey and scratch players as the “Slope Rating.”…Slope Rating is the slope of the regression line of total score versus USGA Handicap for a particular golf course. The Y-intercept is the USGA Course Rating which is the better half of score average of scratch golfers… The slope of the scores line of an average course has been observed to be 1.13 and the USGA Slope Rating is referenced as 113 to deal with whole numbers. A scratch golfer will score proportionally fewer strokes than a 10, 20, and 30 handicapper. The slope of the line through their score indicates the difficulty of the course, and the slope is steeper for more difficult courses and less steep for a less difficult course.*(Note: USGA Handicap refers to a player’s handicap before the introduction of the Slope System).

**[1]**
A minor error is that
Knuth uses the regression line of “total” score versus USGA Handicap. He probably meant to use “average” score as
shown in the work of Stroud and Riccio.[2] (The term “total” score has no obvious
meaning, so “average” score is substituted for the purposes of this post.) The larger error is the assertion that the
slope of the regression line of average score versus USGA Handicap will be the
Slope Rating. This is not the case. The slope of this regression line at any and
every course should be 1.13 if Stroud and Riccio are correct.

Why isn’t the slope of the regression line steeper at more
difficult courses as Knuth maintains? The
answer is straightforward. If a player
only plays his home course, his handicap is independent of the Slope
Rating. The relationship between average
score and USGA Handicap is only a function of the procedures for reducing an average
score to produce a handicap (i.e., taking only the ten best scores out of
twenty, equitable stroke control, and the 96 percent factor).

If Knuth was correct, a Slope Rating would have an absolute
value. Raters could simply use the
regression line relating average score and handicap in order to estimate the
Slope Rating. But the Slope Rating only
has relative value. The Slope Rating is
dependent on the Slope Rating of the reference course. What Knuth should have done is taken players
from the reference course and had them play the course in question. He should have then plotted the player’s
average score versus his handicap on the reference course. The Slope Rating would

**then**be the slope of the regression line relating average score versus the handicap of the player from the reference course.
Knuth compounds his error by arguing the Y-intercept of the
regression line is the Course Rating. The
Y-intercept, however, is the “average score” of a player with a scratch
handicap. The Course Rating however is
the average of a player’s ten best scores out of the last twenty. The Y-intercept is clearly not an unbiased estimate
of the Course Rating.

Several conclusions can be drawn from this critique of USGA
research:

1. The research papers appear to have been dashed off in
order to qualify for a conference at St. Andrews in July of 1990 (e.g., greek symbols were entered by hand). I suspect the authors were more concerned
with locking down their tee times on the Old Course than quality control. Why else would members of the USGA Research
Team (Knuth, Riccio, and Stroud) arrive at such disparate reasons for the
ubiquitous number 113 that pervades the USGA Handicap System?

2. The Conference editors (Cochran and Farrally) should have
been responsible for the quality of the papers presented. They overlooked this duty, probably for
monetary reasons. I suspect the USGA, a
major financial sponsor of the conference, could have submitted the menu from
Rusack’s Hotel and it would have been accepted.

Well that’s enough flogging for now. The next post will
examine the difficulty in empirically verifying Course and Slope Ratings.

[1]
Knuth, D, “A two parameter golf course rating system,”

**Science and Golf**, edited by A. J. Cochran and M. R. Farrally, E & FN Spon, 1990, pp. 142-143.
[2]
Stroud, R.C. and Riccio, L.J., “Mathematical underpinning of the slope handicap
system,”

**Science and Golf,**edited by A. J. Cochran and M. R. Farrally, E & FN Spon, 1990, pp. 135-140.