For those of you still here, I brought up a statistic I simply referred to as "short game" in my last post. What I've attempted to quantify in one number is how effective were the shots taken within 20 yards of the green and on the putting surface for a given round. It is essentially the complement to my previously described "shot efficiency" statistic. I played around with a number of different formulas over the past several years, but last year I settled on one that I think works well.
First, a few assumptions:
1. If we are striving for even par, then using PGA tour data as a gold standard upon which "perfect golf" should be measured is reasonable.
2. Putting itself is an adequate measure of total short game (both putting and chipping/pitch shots/bunker play) since the closer your short game shots get you to the hole, the fewer putts you should have.
The PGA Tour uses "strokes gained-putting" as their #1 putting efficiency stat. The basis for this is the work of Mark Broadie, a researcher at Columbia University, who determined the average number of putts for a PGA Tour pro from any given 1st-putt distance (see his paper here). For example, the average PGA pro takes 1.5 putts to get in from 8 feet; 1.87 putts from 20 feet; and 2.06 putts from 40 feet. Strokes gained-putting simply measures an individual player against the expected average for each of their putts; so if Phil Mickelson drains a 20 footer, he "gains" 0.87 putts against the average for that hole. The total gains (and losses) for each hole are added up to provide a score for the round, and a player's average per-round score is their strokes gained-putting number for the year. Last year Brandt Snedeker led the tour with 0.860 strokes gained-putting per round.
Now utilizing that statistic requires a precise measurement of the first putt distance for each hole; something that is not practical for the amateur golfer playing without the benefit of ShotLink technology. So what we need to determine is a way to estimate the average 1st putt distance for a given round; and that brings us to the complicated math.
Mark Broadie has done some other research that comes in handy here, including the average distance from the hole for PGA pros on shots from 20-60 yards and 100-150 yards. Using that, I determined a best-fit equation to describe the theoretical average 1st putt length:
P = (A/1.9259)^(1/1.2159)
where P = 1st putt length, and A = approach distance
Next, though, I needed to come up with an average approach shot distance (outside 20 yards) for a given round to plug in as "A" in the equation above. This required several steps:
1. First I calculated the average distance per shot for the round -- this is essentially the same as the shot efficiency, except I eliminated penalty shots since those ultimately do not impact the short game play. So avg distance per shot = course yardage / (score - putts - penalties).
2. Next I calculate the average rating per hole for the course; simply course rating / 18.
3. Then I calculate the percentage of the total distance that should be attributable to tee shots. Based on the USGA standard to determine course rating, that factor is 2.39 (in other words, the tee shot is worth 2.39 course rating points). So if the average rating per hole is 4.0 (i.e. a course with a rating of 72), then the percentage of the rating attributed to tee shots is 2.39/4 = 60%.
4. The inverse of that number (in the example, then, we are talking 40%) is the percentage of total distance that *should* be accounted for in approach shots to the green. I multiply that number by the average shot distance calculated in step 1 to get a theoretical average approach shot distance.
In sum:
A = (1-T)*(PISE)
where A = avg approach shot distance, T = avg tee shot distance, and PISE = penalty-independent shot efficiency
So now we have our dependent variable to plug into the first equation above, thus determining the average 1st putt length for a given round. One last step, though -- we need to compare that to the expected putts from that distance based on the PGA averages; rather than looking at a chart for each putting distance, I determined the equation for putts to hole from any distance (2-90 feet) as:
PP = 0.3759ln(P) + 0.6933
where PP = predicted putts, and P = avg 1st putt length
Confused yet? Let's look at an example. Last August, playing in the TOUR of Greater Boston Club Championship at Red Tail Golf Club, I shot 83 with 30 putts. The course rating is 71.9, played over 6698 yards. I had 1 penalty shot. So:
A = (1-(2.39/[71.9/18]))*(6698/[83-30-1])
A = (1-.598)*(128.8)
A = (0.402)*(167.45) = 51.78 yards per average approach shot
and
P = (51.78/1.9259)^(1/1.2159)
P = 14.96 feet average 1st putt distance
Based on the probable putts equation, the average tour pro will take about 1.704 putts to get in from that distance; over 18 holes that comes out to 30.672 putts/round. I took 30 putts, so my calculated strokes gained-putting for the round is +0.672.
Lastly, I wanted to put this on a similar scale to the shot efficiency data. So I arbitrarily set the expected putts value at 75% (since that's the "average"), and 5 putts saved as 100% and 5 putts lost as 50% -- essentially assigned letter grades of "C" "A" and "F" to those scores respectively. Using that scale, a score of +0.672 yields a short game score of 0.784.
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Extremely complicated, I know; but once you have the formulas plugged into an Excel file, all you need is 5 data points from the round -- yardage, course rating, score, putts, and penalty shots. And here's the thing: it works. The numbers match up well with my subjective impressions of how well I chipped and putted for most of the rounds that I've measured. My best short game round last year was in mid-August at Butter Brook, where I shot 85 but had just 27 putts -- 3.213 strokes gained-putting, for a short game score of 0.911. One of my worst was during the first round of the Asher Invitational at Fox Hopyard, a round in which I hit 69% of fairways and 50% of greens, but shot only 89 after 37 putts (including four 3-putts) for an abysmal 6.145 strokes lost-putting; short game score 0.443. Sometimes you just know when something is right, and this one feels that way.
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