Tuesday, August 26, 2014

The Birth of "113"

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  It is possible that a small and unpublished USGA study found the slope to be around “1.13.”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.