Transcript PowerPoint Presentation - DESCRIPTIVE STATISTICS I

What is the number one
participation sport among
adults in Canada?
Is the RCGA Golf Handicap
System Equitable?
Background (1)
 In 1998, Statistics Canada surveyed approximately 10,000 Canadians
(aged 15 years and older) about the extent and nature of their
participation in sport during the previous 12 months.
 According to this survey, golf has replaced hockey as the number one
sport activity in Canada.
 Over 1.8 million Canadians (7.4% of the adult populatio n) reported
playing go lf on a regula r basis.
 Number of Canadians who golf 5,952,000 (Ipsos Reid, 2006).
 Percentage of Canadian men who golf (31.1).
 Percentage of Canadian women who golf (12.3).
Background (2)
 The Royal Cana dian Golf Association (RCGA) has served as the
nationa l governing body of amateur golf since 1895.
 The RCGA Handicap System is the RCGAÕsmethod of evaluating golf
skills so golfers of differing abilities can compete on an equitable basis.
Terminology
 Gross Score is the number of actual strokes taken for 18 holes (including penalty
strokes).
 Adjusted gross score = A playerÕ
s gross score adjusted under RCGA Handicap
System procedures for unfinished holes, conceded strokes and holes not played, or not
played under the rules of Rules of Golf, or Equitable Stroke Control
 Net Score = Gross Score Ğ Course Handicap
 Course Handicap (CH) = The number of strokes a player receives from a specific set
of tees at the course that is being played
 Handicap Factor (HF) = A measurement of the playerÕ
s potential scoring ability on a
course of standard playing difficulty
 Equitable Stroke Control (ESC) sets a maximum number of strokes that a golfer can
post on any hole depending on the playerÕ
s course handicap. The reduction of individual
hole scores must be in accordance with the chart below:
Cours e Handicap
0 or plus
1 through 18
19 through 32
33 and over
Maximum Hole Lim it
Bogey
Double Bogey
Triple Bogey
Quadruple Bogey
Some More Terminology
 Course Rating (CR) is an evaluatio n of the playing difficulty of a course
for scratch golfers under normal course and weather conditions. It is
expressed as strokes to one decimal place, and is based on yardag e
and othe r obstacles (e.g. Marine Drive Golf Club -- Blue Tees -- CR =
70.5)
 Slope Rating (SR) is an evaluation of the relative difficulty of a course
for players who are bogey golfers compared to the Course Rating.
Range 55 to 155. A course of standard playing difficulty has SR = 113.
(e.g. Marine Drive Golf Club -- Blue Tees -- SR = 126)
 Handicap Differential (HD) is the difference between a playerÕs
adjusted gross score and the RCGA Course Rating, multiplied by 113,
then divide d by the RCGA Slope Rating. HD is expressed as number of
strokes rounde d to one decimal place.
Calculation of RCGA Handicap
 A golferÕ
s handicap is based on the most recent 20 adjusted gross
scores that the golfer has submitted for handica p determination
purposes.
Step One
 Each adjusted gross score (AGS) is converted to a handicap differential
(HD)
Example
AGS
CR
Difference
HD
85
70.5
14.5
14.5 x 113 = 13.0
126
Calculation of RCGA Handicap
Step Two
 The golferÕs HandicapFactor (HF ) is calculate d as the mean of the
lowest 10 adjuste d gross scores multiplied by 0.96 with deletion of all
digits after the f irst decimal place.
Example
Mean o f lowest 10 AGS = 10.9
0.96 x 10.9 = 10.464
HF = 10.4
Calculation of RCGA Handicap
Step Three
 The golferÕs CourseHandicap (CH) is determined by multiplying the
golferÕsHF by SR and divided by 113. The resulting numb er is rounde d
to the neares t integer.
Example
Player ha s HF = 10.4
Will play from blue tees (CR=70.5, SR=126)
10.4 x 126 = 11.6
113
CH = 12
Rationale for the RCGA Handicap System
 The RCGA Handicap System assumes that the golfer will be honest, try
to make the best possible score on every hole, and submit every
acceptable score for peer review.
 The RCGAÕsprimary rationale for dropping the highest 10 of a golferÕs
20 most recent HDs wh en assigning handicaps i s that such ÒhighÓHDs
bear little relation to the golfe rÕsÒpotentialÓability.
The Competition
 ÒSteadyFreddieÓand ÒWild WayneÓare two golfers competing agains t
each othe r in an 18-hole net medal play match.
 The winne r of this competition is the golfer with the lower net score.
Example
Freddie shoots a gross score of 79 with a Course Handicap of 7
Wayne shoots a gross score of 88 with a Course Handicap of 17
WayneÕs net score of 71 beats FreddieÕsnet score of 72
What Does ÒEquitableÓ Me
an?
 We will consider the RCGA Handica p System equitable if both Freddie
and Wayne have the same chance of winning in a two-person, 18-hole
net medal play match.
Example
 Freddie and Wayne both tend to shoot about the same average
gross score for 18 holes.
 Freddie and Wayne also tend to shoot 18-hole gross scores that
are consistent with their nicknames.
 In such a situation, does Freddie have an advantage over
Wayne?
Statistical Developments (1)
 Gross scores, adjuste d gross scores, handicap factors, and course
handicaps are subje ct to random variation (refer to figures).
Histogram of gross score based on 1,114 rounds of golf played at 106 golf courses
100
90
80
Frequency
70
60
50
40
30
20
10
0
70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100101102103
Gross Score
Time Series Chart of 835 Gross Scores (1983-99)
100
98
96
94
92
Score
90
88
86
84
82
80
78
76
74
72
70
0
50
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850
Game Number
Statistical Developments (2)
 Suppose Steady Freddie and Wild Wayne are playing each other in a
net medal play competition on a golf course with course rating CR and
slope rating SR.
 Let SSF and SWW be independent rand om variables representing the
gross scores for each golfer.
 Let HSF and HWW be independent rand om variables representing the
course handicaps f or each golfer.
 Recall, Wayne wins the match if his net score SWW Ğ HWW is less than
FreddieÕs e
nt score SSF Ğ HSF
Statistical Developments (3)
 We can express the probability that Wayne defeats Freddie in a net
medal play match as follows:
Pr[Wayne defeats Freddie] = Pr[(SWW Ğ HWW) < (SSF Ğ HSF)]
 Using equa tions that relate HD to S and HF to H and performing some
algebra, we can rewrite the above formulas as
Pr[Wayne defeats Freddie] = Pr[(HDWW Ğ HD SF) < (HF WW Ğ HFSF)]
 What abo ut the behavio ur of the four random quantities HDWW, HDSF,
HF WW , and HFSF?
Conclusion
Question: Is the RCGA Golf Handicap System Equitable?
Answer: The current RCGA Handica p System is equitable i f Wayne and
Freddie tend to shoot about the same average score and are about
equally consiste nt when playing under the same conditions.
One small problem golf fansÉ in essentially all two-person golf
matches, the two golfers are not equally consistent and do not tend
to shoot the same average score.
Final Answ er: The current RCGA Handicap System is biased in favour of
the better golfer.
Recommendations
 The RCGA should not continue to ignore the imp act of varia bility in
golfersÕscores on its currently recommende d handicap system.
 A better system is one that provides a single numerical factor for each
golfer and is equitable despite the fact the two golfers are quite different
in their level of consistency.
 Use trimmed mean? (Throw out the two lowest and two highest
handicap differentials from the most recent 20).
 Revised measure of a golferÕsÒpotentialÓ? One standard deviation
below the mean?
What are ÒOrder StatisticsÓ?
 Given a sample of observations, the first order statistic is the
smallest observation in the sample, the second order statistic is
the next smallest, and so on.
 A typical situation of interestÉ . Assume that the observations in
question are independently selected values from a common
underlying distribution.
 The statistical properties of order statistics are different from
those of the original unordered observations
 In general, statistical methods involving order statistics are more
complicated than those based on the use of unordered data.