The USGA has long claimed that the slope of the line
relating average score to handicap is “1.13.”[1]
The USGA, however, has never presented any empirical evidence supporting its
claim. I suspect that is because no such
evidence exists. At the time of the
claim, the USGA had neither the computing power nor the data to make an
estimate of the slope. It is possible
that a small and unpublished USGA study found the slope to be around “1.13.” Such a limited study, however, could not
claim that “1.13” was a finding that applied to every course. It is more probable that the universal “1.13”
came from a theoretical calculation. To
make that calculation, two assumptions had to be made: 1) The standard
deviation of a player’s score is a linearly increasing function of handicap,
and 2) The standard deviation is, on average, the same for all players of the same handicap regardless
of course difficulty.
The necessity of these assumptions can be found through an
examination of the equations of the USGA
Handicap System.[2] Assume the standard deviation of a scratch
player is σ. The standard deviation of a
handicap player is postulated to be (1 + a·Handicap)·σ. In a normal distribution, the average of the
better half of differentials will be .8·σ below the average of all differentials. Therefore, the equation for a player’s
average score can be written as:
1) Average Score = Course Rating +
Average of Ten Best Differentials +.8·(1 + a·Handicap)·σ
Since a player’s handicap is equal to .96 times the average
of his ten best differentials, equation 1 can be rewritten as:
2) Average
Score = Course Rating + Handicap/.96 +.8·(1 + a·Handicap)·σ
The slope is the
average score of a handicap player minus the average score of a scratch player
divided by the handicap of the player:
3) Slope = (Average Score of Handicap
Player – Average Score of Scratch Player)/Handicap
= ((Course Rating +Handicap/.96 +.8·(1 +
a·Handicap)) – (Course
Rating +.8· σ))/Handicap
= (Handicap/.96 +.8·σ
+.8·a·Handicap·σ - .8·σ)/Handicap
= 1.04 +.8· a· σ
The 1.04 represents the part of the slope due to the USGA’s
Bonus for Excellence. Scheid has written
that the value of “a” is .05.[3] To have the slope of the line relating
average score versus handicap be 1.13, the standard deviation of the scores of
a scratch player must be 2.25. And so “1.13” was born in equation 4:
4) Slope = 1.04 + .8·2.25 ·.05 = 1.13
The slope of “1.13 depends on two assumptions that have
never been proven to be valid. The Slope
System, however, does not depend upon “1.13” being the actual slope. If the real slope was 1.05, then all Slope
Ratings would be off by 7.6 percent (1.13/1.05). Such a proportional error would not affect
the efficacy of the Slope System (i.e., handicaps and indices would remain the
same).
[1] Scheid contends that studies of large numbers
of golfers show that player means increase with USGA handicap at a slope near
1.13. Scheid does not cite any of these
studies, however. See Scheid, F. J., “On
the normality and independence of golf scores, with various applications,” Science and Golf, E & FN Spon,
London, 1990, p. 151.
[2] The USGA Handicap System 2012-2016,
United States Golf Association, 2012.
[3] Scheid, loc. cit.
