Golf Erotica for Mathematicians

Math chalkboardI recently received an email from one of our golfers asking for an explanation of the “Odds” column on the Gross Score Report found at the bottom of the “Match Sign-Ups” page on this site. It dawned on me that others may have that same question. What follows is a copy of my response to the email.

If you have something important to do, like watch your grass grow or look for dust bunnies under your couch, skip everything in black below and go directly to the answer in red at the bottom of this diatribe. Either way, let me know if you have any other questions. Cheers.

What time is it, you ask? Let me tell you how to build a watch!

Years ago, the number crunchers at the U.S.G.A. analyzed (I presume) a couple million rounds of golf. They built a chart showing the probabilities that someone with a given handicap index would shoot a particular gross score on a golf course with a specified rating and slope. That chart was published in the USGA magazine and a variety of other places. When I saw it, I was enraptured. (Mathematicians and engineers like me are frequently aroused by statistical challenges. Yes, we’re a strange lot.)

When I saw the chart, I was driven to analyze the data in much greater detail. I called the USGA and spoke with someone in the appropriate department. I asked if they would share the raw data with me. The guy was aghast. “We don’t share our data with the public” he responded with arrogant contempt dripping from his voice. When I told him I wasn’t “the public” that I was in fact a USGA member, he was almost as impressed as if I had told him I was owned a pencil and knew how to use it. He wouldn’t budge. I’ve always been of the ilk that the only challenges worth pursuing are those that aren’t supposed to be possible. The guy just made me more determined than ever.

I set about the task of mapping all the data points from the chart that I thought necessary to accomplish the task. I then performed a nonlinear multiple regression analysis on all the manufactured data. I developed the following formula:
With that formula, I can produce a reasonably reliable estimate of the probability of a person shooting a given score.

Since I developed this tool, I have entered many tens of thousands of rounds of golf into my personal database. The formula has proven to be quite accurate when applied to my database, so the snooty dude at the USGA can go suck on his own database.

Needless to say, my formula – like the USGA predictor – is contingent upon a number of assumptions that suggest stasis, i.e., the golfer isn’t in the midst of a series of lessons, hasn’t had his legs cut off and the course is properly rated. Nonetheless, it provides an interesting metric that’s useful in a number of ways, some not terribly obvious. Think about using it as a tool for the handicap police to root out cheaters and sandbaggers.

So, in answer to your question … the “Odds” column reflects the probability that the golfer will shoot the score that was turned in. If the number appears in brackets, it means the number is actually the inverse of the probability, i.e. less than even odds.

Lateral Hazards – A Footnote

Santa golfingThe previous post (Camelback Golfers – Read This or Else; December 14, 2017), has generated far more “interest” than I had anticipated. Almost every response came to me in private rather than as a comment on this blog. After reviewing the responses, I can assure you that we not only have some good golfers in the group, we’ve also got some golfers that carry a sense of humor that can be used like a scalpel to surgically extract the essence of a situation and describe it in a fashion that make my sides hurt from laughter. I can’t share all the comments because they were sent in private. However, I will give the award for best humor to Dr. John Raines. Barbara must spend half her time doubled over in laughter.

I will share one comment and my response to it without mentioning any names (not even my own). One golfer said:

I assume you believe there are those that fudge?

Here’s my response.

“Three answers …

OFFICIAL ANSWER: I’ve had a number of [fill in tactful word for ‘complaints’] about those who (given the benefit of the doubt) don’t fully understand the rules surrounding lateral hazards. As a leader of the pack, I’m obligated to ‘educate’ new and old members alike. (Also, as Handicap Chair, I’m obliged to ‘educate’). Also, as the one with the media outlet (sounds more official than ‘blog’), I have the means to educate. Without exception, no one that complained mentioned any names.

NON-OFFICIAL ANSWER: Hell yes, there are those who fudge. If we were playing for Pokémon cards, most people wouldn’t care. However, money’s changing hands and ‘fudging’ shouldn’t impact that flow. Some people also point out that even if it doesn’t impact the money game, it tends to give the fudgers indefensibly low handicaps. They fear getting stuck with the vanity handicappers as a partner in a match or tournament.


When dealing with lateral hazards, there are frequently going to be judgment calls, especially on the question of “point of entry”. There is no section in the rule book concerning how to make a judgment. I think you learn that as a kid, maybe before. For example, I have seen many times where someone tees off on #8 Padre. The ball clears that water, but just barely. It lands in the grass and rolls back into the hazard. Does the golfer invoke the “two clubs from point of entry, no closer to the hole rule”? Or does he return to the “drop area” thus sacrificing twenty or thirty yards. It depends on whether the ball landed in the hazard or out of the hazard and rolled back in. Here’s the rub; the hazard ISN’T defined as the water. There’s usually a red line of demarcation on the other side of the pond. The ball often lands dry, but in the hazard. If so, back to the drop area. If it lands just past the red line, you can play much closer to the pin. Your call. The kicker is that from 175 yards, it’s difficult to pinpoint the precise landing spot, especially when comparing it to a red chalk line that has been largely eradicated from rain and the freshly grown grass. That call is further complicated by the fact that you’re bent over in a golfer’s curse pose vehemently bitching about the yardage on the cart GPS being wrong.

Nonetheless, it’s a judgment call. Do your best. As a suggestion in keeping with the season, consider the following:

You better watch out
You better not cry
Better not pout
I’m telling you why
Santa Claus is coming to town

He’s making a list
He’s checking it twice;
He’s gonna find out who’s naughty or nice
Santa Claus is coming to town

He sees you when you’re sleeping
He knows when you’re awake
He knows when you’ve been bad or good
So be good for goodness sake!

Going Native on Ambiente

High GrassWith a few new members and a few whose memories come into and out of focus from time-to-time, let’s review the rule associated with playing out of the “native grass” areas on the Ambiente course at Camelback.

Native grass areas are deemed “lateral hazards”. This means:

  1. YOU MAY NOT ground your club when addressing your ball. You MAY lightly touch the grass, but you MAY NOT do anything that alters the swing path such as taking practice swings that tear or uproot the grass or plants near the ball.
  1. YOU MAY NOT move any loose impediments in the hazard. You MAY NOT brush any rocks or pebbles aside. YOU MAY NOT pick up or move any twigs, pine needles, coyote droppings. You DO NOT get relief from “obstructions” if you’re in a hazard without incurring a penalty stroke. This includes things like the fire hydrant on the ninth hole.
  1. YOU DO NOT get free relief from standing water if you are IN the hazard. That’s sort of why it’s called a “hazard”. After the rains, many of the native grass areas become native rivers. No relief without penalty.
  1. YOU MAY NOT “build a stance”. You can place your feet firmly on the ground, but you may not uproot plants or kick big rocks around while taking your stance.

Relief from a lateral hazard is covered under Section 26 of the Rules of Golf. You have five options:

  1. Play the ball where it lies without penalty and subject to the prohibitions outlined above. Obviously, you have to find the ball to do this. No penalty.
  1. Stroke and distance. Return to the spot from which you hit the ball. You incur a one stroke penalty. If you hit the ball from the teeing ground, you are now hitting your third shot from the teeing ground.
  1. Drop a ball within two club lengths of the point where the ball crossed the margin of the hazard. You take a one stroke penalty.
  1. Drop a ball within two club lengths of a point on the opposite side of the hazard, but no closer to the hole than where the ball first crossed the margin of the hazard. You take a one stroke penalty.
  1. Drop a ball as far back as you wish on a line from the point of entry and the flagstick. You take a one stroke penalty.

Admittedly, it is called a “lateral hazard”, but this DOES NOT MEAN you can drop a ball laterally out of the hazard. You MUST drop within two club lengths of the point where the ball first crossed the margin of the hazard (assuming you’re taking relief as defined by #3 above).

What are the native grass areas? For most golfers, it’s pretty apparent that the arroyo area on the starboard side of the course is a “native grass” area. However, some golfers lose their clarity when they end up in a little “island” area of native grass. If it looks like native grass, you’re safe assuming it is a hazard. This includes places like (1) the tall grass between the cart path and the sand traps on the left side of the #3 fairway, (2) the grass areas running the entire length of the port side of pretty much every hole on the course, (3) the grass areas above and to the left of the traps on #18, (4) the tall grass area between the cart path and the #16 green, (5) the grassy area above the trap at the end of the dogleg on the #1 hole. These are just a few of the “native grass” areas. If it’s got flowers, it’s not fairway, it’s not rough, it’s native.

A couple of final comments on this topic: if you’re playing the Padre course and hit a ball into a native grass area that’s part of the Ambiente course, e.g., the area behind the twelfth green, IT IS NATIVE GRASS and deemed lateral hazard.

Consider this a “local rule” for The Jones Boyz Group. I don’t recall if Camelback has addressed this issue, so don’t claim it as an “official” local rule for the club without checking. There are areas on the course where the cart path runs through native grass areas. For example, on #3, #5, #6, #7, #8, #9 and other holes, there are sections of the cart path with native grass areas on both sides of the path. Technically, with a ball on the path or a ball adjacent to the path where the concrete interferes with your swing and the finish on your $100 club, you are NOT entitled to relief. However, by Executive Decree of the Tournament Committee, i.e., me, we will play with our own local rule. You MAY take relief from the cart path without penalty. However, you MAY NOT take relief out of the hazard. The free drop must be within a club length of the nearest relief from the cart path, but within the hazard.

We’ve got a great golf group and we should be proud that we have fostered a culture where we play by the rules. We don’t improve our lies. We don’t bump the ball. We’re pretty much “by the book”. Hopefully, this helps some of the newer members of the group to stay on the high road.

In the final analysis, the best way to avoid conflicts with these and other rules is to hit your shots into the fairway. I’m thinking about trying that approach. I’m always open to new things.

Why Play the Ball “Down”?

intheroughFrom time-to-time, competitors approach me before a match and suggest that because it rained the day before or the course was over-seeded a month ago or the mower blades need to be sharpened or the tides have been running higher than normal in Malaysia or their grandmothers have been ill or their balls don’t like bad lies or blah, blah, blah. In all but the rarest of circumstances, we elect to play the ball “down”. No winter rules. No improving the lie.

Why? Because the rules of the game say play it down!

However, I don’t opt to play it down solely so I can be a “good boy” and play by the rules. You know me better than that. The rules say “play it down” for reasons. And here they are – straight out of the USGA Handicap System Manual, Section 7-2.

  1. Such a Local Rule conflicts with the fundamental principle of playing the ball as it lies;
  2. Preferred lies is sometimes adopted under the guise of protecting the course when, in fact, the practical effect is just the opposite – it permits moving the ball to the best turf, from which divots are then taken to injure the course further;
  3. Preferred lies generally tends to lower scores and a Handicap Index, thus penalizing players in competition with players whose scores are made without preferred lies;
  4. Extended use or indiscriminate use of preferred lies will place players at a disadvantage when competing at a course where the ball must be played as it lies.

In a book entitled “How to Cheat in Golf – Confessions of the Handicap Committee Chairman”, the author (one of my favorites) dedicates Chapter Four to “The Biggest Cheat in Golf”. He makes it clear the biggest cheater is the golfer that takes steps of any kind that result in a unjustified lowering of his handicap index. Bumping your ball, improving your lie, and playing “winter rules” means that the biggest cheat in golf is you!

Let’s play it down.

The Pope and the Slippery Slope

Math ForumulasI recently noticed a posting on the bulletin board of the locker room at the Scottsdale club at which I play golf. Obviously posted by a higher handicap golfer, it heralded the claims of Dean Knuth, “The Pope of Slope”, that low handicap players retain a distinct advantage in head-to-head competition over higher handicap players. Although I have a lot of respect for the opinions of Dean Knuth, I took some issue with the article as printed in Golf Digest. Below is the original piece as taken from the bulletin board as well as my response. I then went into my database of golf matches and analyzed nearly one million head-to-head matches. I’ll present the empirical data upon request. What do you think?

Taken from Golf Digest June 2014 Issue

Another Reason to Ask for More Strokes

You would think that a golfer with a course handicap of 12 would have a decent chance of beating a scratch golfer, provided he was given his full 12 shots. But that golfer has only a 25 percent chance of winning, says Dean Knuth, former Director of the USGA’s Handicap Department. “The USGA set up its system to favor better players with a built-in bonus for excellence,” Knuth says. “It’s a philosophy that handicaps should be based on potential rather than average ability.” For every six strokes in handicap difference, the better player has a one-stroke advantage, Knuth says. So in a match between an 8 and a 14, the 8 handicapper has a 60 percent chance of winning. You might want to remember that before wagering.

But now (as Paul Harvey used to say), The Rest of the Story …

Dean Knuth is unquestionably one of the experts in the world of golf handicaps, however, he may be sugar-coating this one and allowing us middle to high handicappers to cry foul when there isn’t one.

The slight-of-word about handicaps being based on “potential” rather than “average” ability is a semantic dance with roots buried in the “USGA Handicap System”, but those roots are clouded and misunderstood. They are part and parcel of a circular argument that doesn’t make a lot of mathematical sense.

I suspect Mr. Knuth is alluding to the fact that for the purposes of handicap calculation, the differential for any given round is multiplied by 96% which the USGA calls its “Bonus for Excellence”, an “incentive for players to improve their golf game.” On the surface, it appears to favor the lower handicap players. But does it really?

I challenge you to go to the driving range and question one, ten or a thousand golfers as to why they’re practicing. I don’t believe any of them will respond they’re there because of the USGA’s “Bonus for Excellence”. They’re there because they lost two balls in a water hazard the previous day and they can’t afford to lose two more tomorrow. To suggest the “Bonus for Excellence” incents golfers to improve or for that matter that the USGA should concern itself with your level of motivation is delusion.

The “Bonus for Excellence” does tend to mitigate a statistical advantage that the higher handicap players have over the lower handicap players. Generally speaking, the higher the handicap, the greater the variability of scores. A scratch golfer’s net scores generally fall within a couple of strokes of par. A twenty handicap golfer’s net score will usually fall within four or five strokes of par – both over and UNDER. In other words, a twenty handicap golfer has a much higher probability of shooting a net 65 than a scratch golfer.

On any given day, in a head-to-head match, the scratch golfer may have a miniscule statistical edge over the high handicapper. However, it’s not the big edge Mr. Knuth suggests.

In a recent tournament at a nearby club, eighty-eight golfers competed where prizes were awarded to the best twenty-two net scores, i.e., 25% of the field. Slightly more than 20% of the field carried single-digit handicaps. If Mr. Knuth’s claims are valid, wouldn’t you expect the list of winners to carry more than 20% of the single-digit handicapper’s names? Alright, how about close to 20% of the winners come from the group of low handicappers? How would you explain only 10% of the winners being single-digit? The fact of the matter is that of the twenty-two prize winners, not a single one came from the single-digit list. With a closer look at the probabilities, this outcome was not unexpected.

No matter how you cut the math, here’s my conclusion. With all due respects to Mr. Knuth … If you come to me and ask for an extra stroke because your handicap is six strokes above mine, you’ll be talking to yourself. Play well.

Howard Jones

Handicap Committee Chair – Camelback Golf Club

Pay No Attention to that Man Behind the Curtain

Behind the CurtainI have a much greater understanding of one of the most famous quotes from the 1939 Hollywood spectacular, “The Wizard of Oz”. For I have indeed looked behind the curtain. As a result, rumors fly that I am no longer a member of Gainey Ranch Golf Club. I have received numerous calls and emails asking me if it is true. So allow me to put the rumors to rest. Continue reading “Pay No Attention to that Man Behind the Curtain”

Tales from your Handicap Committee

If you’re not a “math guy”, this may be of little interest. Unless that is, you’re interested in reading a simple “how to piece” on how to manipulate your handicap. Before I get started, let me point out that there are those – on and off the Handicap Committee – that don’t believe the “numbers” can reveal anything. What follows is an example of what the numbers can tell us. You be the judge. And yes, this is very real data for a very real golfer at Gainey. It may have been moved to slightly to mask the identity of this golfer, but it is real.

John Doe DifferentialsThis particular member plays a fair amount of his golf on courses other than his home course. The chart below shows his posted differentials for a substantial (read: statistically significant) period of time. The blue diamonds show his differentials when playing on Gainey Ranch G.C. The red squares are his posted differentials when playing at away courses.

A couple of things immediately jump out at you. Home differentials are clearly randomly distributed around roughly thirteen. The actual number isn’t the important thing; it’s the distribution that is significant. You can see they range from a low of around seven to a high of roughly twenty. If you analyze the distribution, it is what you would expect for someone in this golfer’s handicap range, i.e., standard deviation of around three strokes. Ninety-five percent of the differentials should (given the laws of statistics) fall between approximately seven and nineteen. What an amazing coincidence – they do!

Now look at the away scores. Their average is closer to sixteen, nearly three strokes higher than those recorded at Gainey. Does this fact imply handicap manipulation? Maybe – maybe not. It can certainly be argued that Gainey Ranch’s handicap doesn’t “travel well”. I’m confident there’s an element of truth to that. You can also make a case that a golfer isn’t as familiar with away courses and a lack of course knowledge results in higher scores. Perhaps this is true, but how many times can that excuse be used? After all, once you’ve played a particular course four or five times, you should have a pretty good level of familiarity and that excuse tends to evaporate into the morning mist.

For the sake of this discussion, let’s temporarily accept that a three stroke difference in the scoring average is acceptable. If that’s the case, shouldn’t the entire grouping of data points move up approximately three strokes? For this golfer, it does, but . . .

The “randomness” disappears! It’s as if the entire bottom half of the distribution is missing! Could it be that the scores weren’t posted “properly”? Hell yes. That’s one possible explanation. The standard deviation of away postings is approximately one stroke! This seems to be statistically “improbable” (for those of you not paying attention, this is called a gross understatement).

An average differential of sixteen with a standard deviation of one stroke means that ninety-five percent of this golfer’s away differentials fall within the range of fourteen and eighteen. This is well within the “you must be joking” range of statistical probabilities.

There are possible explanations. For example, it could be that by playing a more difficult course with which you have no familiarity whatsoever magically makes you a much steadier and more consistent golfer than you are at home. To me, that’s like saying the more you drink, the better you drive.

There are a couple of other explanations. I’ll sit back and see if any of you come up with them and post them as comments here. In the meantime, I continue to do battle with those who say numbers don’t prove anything and I’ll watch to see how this golfer performs in the upcoming Member/Member tournament. This golfer isn’t alone. He has company. I’d love to say, “If you don’t mind, I don’t mind”, but I do. It’s my job.