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BCrosby

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The Tom Paul Theorem
« on: April 17, 2002, 04:29:43 PM »

      We are all familiar with Fermat's Last Theorem.  Aren't we?  Then there's Godel's Incompleteness Theorem.  There always seems to be some joker at the bar who wants to give you his own take on it.    

      Now we have the Tom Paul Theorem.  

      The Tom Paul Theorem states that the design quality of a hole is a function of the  distribution of under par and over par scores on it.  The wider the distribution, the better the hole.

      I think the theorem is plausible and worth testing.  (More importantly, the Tom Paul Theorem (hereinafter the "TPT") may  -- at long last – identify great golf holes with something like mathematical certainty.  No more messy debates.  No more acrimony.  We can get past golf course architecture issues and focus on more important things like my son's terrible grades.)

      Since I have at hand lots of Masters scoring data, let's test the TPT on the par 4's at ANGC in 2002.  The following is based on cumulative scoring for this year's tournament.

      First, let's try to nail down the the scoring distribution for the holes.  This would be the ratio of non-pars (eagles, birdies, bogies and others) to pars (the "Non-Par Ratio") on each par 4.  I weighted each occurrence of an eagle as a 2 (since it's 2 under par and should be given extra weight) and each occurrence of an "other" as a 2 (which understates the aggregate "other" scoring, but it makes the math much easier).  Each par, birdie and bogey was weighted as 1.  On this formula I came up with the following:


                    non-pars/pars     Non-Par Ratio      

No. 1            88/179 -             .49            
No. 3            76/191 -             .39            
No. 5            96/174 -             .55            
No. 7            96/173 -             .55            
No. 9            123/151 -       .81            
No. 10            104/167 -       .62
No. 11             98/174 -        .56
No. 14            125/146-        .85
No. 17            103/162 -       .63
No. 18            115/158 -       .72


      The Non-Par Ratio is not the whole story, however.  It could be that some high Non-Par Ratios are due solely to lots of bogeys and "others" on a particular hole.  In other words, really hard holes would have high Non-Par Ratios.  Ditto for really easy holes.  TPT, however, seeks out holes where there are lots of under par AND over par scores.  It seeks holes with a wide distribution of scores.

      So let's take each par 4 at ANGC and find the ratio of under par scores to over par scores for that hole, again giving eagles and "others" the same weighting as above (the "Under-Over Ratio").  (Again, I am open to suggestions that this weighting my be inappropriate.)  The results are as follows:

                           Under-Over Ratios

            No. 1            .375                        No. 3            .8                        No. 5            .435                        No. 7            .30                        No. 9            .35                        No. 10            .23
            No. 11            .4
            No. 14            .38
            No. 17            .32
            No. 18            .15

      Obviously, if an equal number of birdies and bogies are scored on a hole, the Under-Over Ratio would be 1 for that hole.  Note that No. 3 at .8 is the hole that comes closest to having a 1.  A fantastically wide distribution relative to other holes at ANGC.  All the other holes had many more over par scores than under par scores, which is what you would expect.  

             Hang with me here.

      The final step is to multiply the Non-Par Ratio by the Under-Over Ratio and the holes with the highest resulting numbers should win the TPT derby.  

              Why?  The Non-Par Ratio tells us which holes had the most non-par scores (whether birdies or bogies) and the Under-Over Ratio tells us the distribution of those non-par scores.  We want to identify under the TPT those holes with both the highest non par scores and those with the broadest distribution of those non par scores.  Put differently, we want to weed out holes where scores are bunched around par, very easy holes (lots of birdies, few bogies) and very hard holes (lots of bogies, few birdies).

      Thus we multiply the Non-Par Ratio by the Over-Under Ratio (the "TPT Ratio")for each par 4 at AGNGC with the following result:

                               TPT Ratio

            No. 1            18.3                        No. 3            31.2                        No. 5            24                        No. 7            16                        No. 9            28                        No. 10            14.2
                          No. 11            22.4
            No. 14            32.3
            No. 17            20
            No. 18            10.8

      (Decimal points have been shifted to the right.  For reasons I don't recall, I rounded some numbers but not others. Sorry.)

      If the TPT is correct, the highest of the above numbers should be the best par 4's at ANGC.

      The TPT ranking comes out as follows:

   No. 14
   No. 3
   No. 9
   No. 5
   No. 17
   No. 1
   No. 7
   No. 10
   No. 18

     Amazingly, that's more or less where I would have rated the par 4's ANGC on a purely subjective basis.  I don't agree with the relatively high ranking of No. 9.  At a minimum I would rank it after No. 5.  Maybe lower.  But other than that I don't have much to quibble with.  Really amazing.  

    Congratulations Tom Paul.  Could this mean the Nobel Prize?  Tenure at Harvard?  A weekend at Myrtle Beach?  It's rare that someone throws out a theory on these boards that can actually be tested.  Rarer still that it passes the test.

     The results shocked me, frankly.  I did not expect the results to be so elegant, as my mathematician friends might say.  Move over Fermat.  Make way Godel.  There's a new kid in town.

Bob

P.S.  I am no mathematician.  I encourage comments and criticisms as to my methods and calculations.  I may have made fundamental mistakes that make the foregoing garbage.  Be gentle.  But it has been an interesting exercise.
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

ChrisB (Guest)

Re: The Tom Paul Theorem
« Reply #1 on: April 17, 2002, 04:59:30 PM »
So does that mean that, according to the TPT, the best holes in the world are short, reachable par 5's with a lot of trouble so that you're either making eagle/birdie or a big #?

How would #17 at the TPC rate?
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

TEPaul

Re: The Tom Paul Theorem
« Reply #2 on: April 17, 2002, 04:59:55 PM »
Bob:

You're no mathmetician? Man, you could sure fool me. I'm no  mathmetician at all and can barely figure out the math, but how did you figure out what to mulitiple against what? That's how obtuse I am about math!

If this works mathematically you're the one who deserves the Nobel prize. I'm assuming too that first prize is a weekend in Myrtle Beach for golf, second prize is the tenure at Harvard and third prize the Nobel Prize!

I don't know ANGC well at all but wouldn't you assume that #9 might fall into that position because of the razor thin margins for error on the green itself? I mean miss it wide on either side and you know what you're looking at and isn't it also probably the easiest green on the course to hit the green and have the shot come back off of? Or even chip back off the green or even putt one off of? Add to the severe back to front cant the opposite topographical configuration at the LZ and that might add up to some real intensity that some really good pros can handle with a birdie but the marginal ball strikers get into the over par category fast. In other words the margins for error across the entire score spectrum are real thin but the reward is still very much there, but only for the best of the best!

But it does look elegant and if the math does work, don't ask me how!
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

TEPaul

Re: The Tom Paul Theorem
« Reply #3 on: April 17, 2002, 05:12:12 PM »
ChrisB;

#17 TPC is a good point? A quandry! It certainly has gotten recognition and fame for some reason.

As to something like a short par 5 that real quality shots can eagle and birdie but marginal ones can result in a high score--probably that's it in effect--that would be ANGC's #13 at its best. That type of thing would also fall almost perfectly into the definition of so-called great shot values!

All this stuff is really based on some of the thoughts and writing and discussions I've had with Geoff Shackelford that the best holes offer the most interesting options and underlying those options themselves are high although occassionally mindbending degrees of TEMPTATION with  risk/rewards of real consequence!

It always made perfect commonsense to me so I thought logically a wide score spectrum should indicate a really good and interesting hole and architecture.

The other side of the spectrum would be the type of hole that golfers play the same way day after day with a narrow band of score. How could the latter be interesting and fun compared to the former?
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

corey miller

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #4 on: April 17, 2002, 06:32:31 PM »
Bcrosby- pretty interesting I must think about the best way to solve the problem.  What is wrong with determining what holes have the mean score closest to par yet with the highest standard deviation.  after converting all >triple bogeys to doubles.

the mean close to par would insure both high and low scores and  the std. dev would insure a wide distribution around the mean.

« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

BCrosby

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #5 on: April 17, 2002, 08:12:58 PM »
Tom -

No, you deserve the Nobel.  Or did you opt for the trip to Myrtle?  I'm just a tiller in the field.  

My guess is that the TPT won't give very clear answers for par 3's.  We'll see.  Except for par 3's with lots of water or other high penalty hazards near the green (like no. 17 at the TPC), I'll bet the range of values from one par 3 to another are very narrow (because the scoring ranges are so narrow) thus making the TPT less useful as a predictor of design value.

Chris B -

To make the TPT work, you would have to calculate it over par threes as a group, par fours as a group, etc.  You wouldn't want to compare the number for a par 3 to a par 4 just because of the inherent differences is scoring ranges.

As to par 5's, I haven't run any numbers (I will with the Masters data) but I suspect you are right.  You should find that short, risk/reward par 5's will win every time.  

But that's the way it should be, isn't it?  Aren't they they best par 5's?  Ross, MacK, Tillie, Thomas certainly thought so.  I think so.  If the TPT numbers confirm this, it's another indication that the TPT is a good tool.

Corey -

In your short post you have convinced me that you know much more math than I do.  Please continue to ponder the TPT and let us know what changes, improvements, etc. you think appropriate.  For example, I'm sure there is an easier way to do it, I just can't figure out what it is.  I also have doubts about how I did the weightings.

I do think it would be fun to develop the thing and actually use it in design debates when there is sufficient scoring data available.

Bob
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

Jeremy Glenn. (Guest)

Re: The Tom Paul Theorem
« Reply #6 on: April 17, 2002, 08:22:53 PM »
BCrosby,

Interesting... (?)

The trouble I see is that changing the par of the hole would change the hole's "design quality".

This, I don't agree with.

Might I suggest instead to use the hole's scoring average, rather than par?
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

BCrosby

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #7 on: April 17, 2002, 08:59:38 PM »
Jeremy -

I don't think you want to use the scoring average as your starting point becase then the ratio of under par scores to over par scores will - by definition - always be pretty close to 1.  And that leaves you without a tool to distinguish high birdie holes (i.e. easy holes) or high bogie holes (hard holes) from holes with a wide distribution of both birdies and bogies (i.e. the good holes under the TPT).

I do agree that ideally you don't want a system that turns on an arbitrary par value.  But I'm not sure that's a terrible thing.  Changes in par only affect long par 4's and short par 5's and courses only fudge par on these holes for tournament set ups.  You usually know the par for which the hole was designed and you would use that number in your calculations, all things being equal.  For example, if Hootie decided to call no. 13 at ANGC a par 4, we could still crunch the TPT numbers on that hole as a par 5 if you wanted.  The raw scoring data would be the same whatever par was assigned to the hole.

You are right, the system does assume that the par of a hole has some reaonable relationship to how it actually plays.  But except for fairly rare situations, I don't think that is too big a leap of faith.

Bob
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

TEPaul

Re: The Tom Paul Theorem
« Reply #8 on: April 17, 2002, 10:27:35 PM »
I'm not sure that in this context par is all that useful on certain holes and in certain situations and that fits into Shackelford's feeling about the Masters, and the potential drama on certain holes under certain architectural mixes and how even apparently minor changes can effect or minimize that drama by really skewing the sophisticated risk/reward decision making of certain touring pros.

Take some holes like ANGC's two back nine par 5s. What's the par on them and what difference does that really make? Well, they're short dangerous par 5s but to the pros in the Masters they really aren't "par" 5s and they look at them in relation to what others in contention are likely to do on them and calculate their strategies accordingly. And of course that can always change depending on many situations relating to overall tournament scoring calculations and certainly on the number of players in contention of the final nine as evidenced by some of the real tournament heroics like Nicklaus's eagle on #15 in '86, or some of the disasters of other contenders who opted on the side of some added risk due to their calculated needed results and also the reasoning of Hogan of why he didn't need to go in two because an eagle wasn't needed.

So why wouldn't just the raw spread be as interesting as anything else? Is it really necessary to consider a hole that produces numbers that falls a bit more on the birdie side an "easy" hole? I wouldn't think so if it also produces some numbers that fall into the serious "other" category even if occasionally!

In a way this is part of the general theme of what I call "high intensity" courses that are made up of a number of holes that are "high intensity". Maybe a better term would be "high anxiety". What I mean by that is many of these holes may not even be all that difficult to play or not as difficult as some might think but the point is they are the holes and courses that keep you concentrating because a lapse in judgement or a poorly thought out shot can deal you a serious "other" (a big number) very quickly. And so many of the holes that I'm thinking of this way are recognized as great architecture!

The "high intensity" courses in my area are Merion, Pine Valley and Huntingdon Valley, three courses that just happen to be the three primary stars in my area.

Even really good par 3s should often produce the same dramatic spreads, even if it may be on different sides of actual par overall. Mostly the median or mean would probably be on the high side of par though, but not necessarily so.

Par 3s like PV's #3 or #5, Merion's #3 or #17 or even #13, Huntingdon Valley's wonderful #5 or even #3. A very interesting one to look at too because of its extreme shortness but weather effects certainly would be Pebble's #7!

I've often suspected that par 3s generally and some par 3s specifically are nowhere near where they actually need to be in handicap allocation. I feel that if the USGA's handicap and GHIN system would go to "hole by hole" posting the raw data would be there to show where handicaps really need to be allocated in a more realistic sense and some par 3s would show that reality in no time with the raw hole by hole data.

These kinds of par 3s are where the good player shines most of the time and the marginal player makes a number other than par a lot of the time or a big number more of the time than on other holes that have a greater handicap allocation. The question of the frequency and extent of the "other" on any hole though, gets into something I call "spillage" in a handicapping context that happens through the application of "gross score" posting (single number 18 hole posting) into a match play format that could be completely corrected by hole by hole posting but that's another story for another time!

But handicapping, because of the way it may have evolved off of course rating and mostly being all about raw distance, obviously par 3s generally don't fall into that category or that handicap allocation area.

Bob, who knows you just might be able to mathmetize other interesting things in golf architecture like routings, superior routings or even the elusive "ideal" routing.

Did you see the movie "A Beautiful Mind" yet? You could apply to these kinds of things what Ron Howard did to John Forbes Nash's breakthrough realization on what became the "Nash Equilibrium" (mathmetized formulae) of "Game Theory",  of even social interrelationships and negotiating techniques.  

Remember Nash in the bar at Princeton with his four other friends when the six girls walked into the bar with one unbelievably beautiful girl and how that became the basis for his breakthrough thinking? All five of them fixated on the one beautiful girl and Nash realized that none of them would ever get lucky that way since there were too many of them and she would probably chose none of them because of that. The best way was for all of them to pair off on the five other girls and they would all get lucky. The only one who never would get anyone, most ironically, would be the beautiful girl!

You could do that with holes, courses and whole routings and mathmetize it! You could show why and how there are poor routings, good routings and great routings but also the extreme unlikelihood of the "ideal" routing simply because 18 "ideal" holes are so difficult to string together to be individually "ideal" as they might naturally negatively effect each other or compromise each other somehow!

Unfortunately, the simple truth of the extreme "subjectivity" in architecture might scotch all this mathematizing but if your mathematical formulae are "elegant" enough we might be able to scotch "subjectivity" in architecture too!

More unfortunately this all could lead to Pat Mucci's dream of total "formulaics" in golf architecture at which point I would instinctively disassociate myself from any of it and then you might have to construct some mathematical test for the quality of randomness!

I'll talk to Dr. Katz first before pursuing this any further.
« Last Edit: December 31, 1969, 07:00:04 PM by -1 »

TEPaul

Re: The Tom Paul Theorem
« Reply #9 on: April 18, 2002, 05:11:26 AM »
Maybe Corey Miller's idea of using a mean number as the base line is a better idea than using the par number (a mean number like 4.23 or 5.19 as exs). It would be the more realistic or the true scoring base line number, correct? Par is a useful base line number, I suppose, but it's just an arbitrary expectation nevertheless, correct? Wouldn't using par as an arbitrary number be a bit like the USGA using 109MPH as the base line to test the conformity of golf balls? In a test condition it works but how realistic was it?

But on second thought how would you really be able to accurately compare par 4s to each other? Same for the other par holes? I am obtuse mathematically!
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

BCrosby

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #10 on: April 18, 2002, 06:59:03 AM »
I think you have to start with par.  The point of the TPT is to establish a mathematical basis for comparing the playing values of a group of roughly similar holes.  You are not trying to come up with the inherent "value" of any one hole, but simply how its shot values stack up against holes that play more or less the same way, i.e., with similar pars.  That gives you a ranking.  If enough of those rankings come out close to the rankings that astute observers of golf design would predict, then you have a basis for thinking the TPT is on to something.  

If you start with the mean scores or average scores on each hole you get a meaningless apples to oranges comparison between holes.  More importantly, comparing the spread of birdies and bogies to the mean score doesn't shovel any explanatory coal because that spread of birdies and bogies is already built into the mean or average score for that hole.

Again, we are not trying to ascertain the difficulty (or the easiness) of a hole.  We are trying to identify those holes with the widest distribtution of both below par and above par scores.  If may turn out that some of those holes average below par scores or above par scores.  That's irrelevant.  The TPT issue is the distribution of scores.  The hypothesis is that it is the distribution of high and low scores that makes for a well designed hole and - at least at the ANGC - the theroy seems to do a good job of identifying well designed holes.

In fact, one of the central strengths of the TPT is that it provides an analytic framework for thinking about the design of golf holes that does not depend on resistance to scoring.  Birdie holes can be great holes.  For example, even though Doak worries about Pacific Dunes being too easy, I'll bet a crunch of the TPT numbers would yield a number of holes there with very high rankings.  Unfortunately, we probably won't have the raw scoring data to do that calculation until PD hosts some sort of major tournament.
 
Of course if the par assigned to a hole has no reasonable relationship to how that hole was designed, the TPT will break down.  But wierd par designations occur only rarely.  It only seems to happen when the USGA gets involved.   (Other than publishing crisp new rules books very year, somebody remind me why I am paying dues to those guys.)

Another reason to use par as the beginning point is that it allows you to compare holes of similar par across different golf courses.  You need lots of raw scoring data and a field of players of more or less similar skills.  That doesn't happen very often.  The Masters is a perfect test bed for the TPT.  The US Open would be too.  If the USGA provides similar scoring data for Bethpage, it would - in theory - be possible to do comparisons.  How does no. 14 at ANGC stack up against the best par 4 at Bethpage under the TPT?  You could get a ranking of the par 4's at the two courses and do a Ran Morrissette match, one version based on the TPT numbers and one based on Ran's always impeccable instincts.

Bob  
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

corey miller

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #11 on: April 18, 2002, 07:34:45 AM »
The reason i made any mention of par is it was my belief you wanted as wide a distribution of scores both above par and below par which again means you want a mean near par with a wide deviation.

The data(if you can e-mail me, though I am traveling for a few days)  will wind up being a normal distribution(bell-shaped) curve with the mean or high part of the curve pretty close to par.

Not sure how golf scores effect the two tails of the curve, as it is not possible to score a 0, and a one on these holes is improbable.  Also, you are skewing(for good reason I believe) the other tail by limiting scores to double-bogey.  In other words the right side tail(high scores) can actually have a rising slope or more >=double bogeys than bogeys.

Within one std dev 68% of scores will fall, and within two 95%.  so for great variability you want the std dev to be as wide as possible.

in my first tour of duty in school we had to do this stuff by hand, later just plop into a program amazing how my grade improved.....
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

Mike_Cirba

Re: The Tom Paul Theorem
« Reply #12 on: April 18, 2002, 07:41:25 AM »
Tom Paul,

Interesting your mention of Nash's equilibrium theory as relates to course architecture.

However, somewhat ironically, you probably have inadvertedly given credence to the Tom Fazio philosophy of design.

It would seem for equilibrium theory to apply to golf course architecture, in the way you suggest, two things would need to remain static;

1) The amount of land for the routing
2) The topography of that land as fit for golf usage

By way of contrast, a modern architect would try to change those static attributes.  If aquiring additional, more suitable (if much more spread out routing) land for golf wasn't a possibility, they would probably focus on changing number two.

Using your example, Tom Fazio might argue that Nash's barroom revelation was simply short-sighted.  Instead, he would seek to change the five other girls into "10's", as well, probably through massive reshaping, restructuring, and artifice (and silicone, or at least bunker-woll ;) ).  That way, he might argue, everyone wins.

Of course, the debate then becomes to what extent is the cost involved in the "makeover" inevitably good for the game, and more importantly, can the trained eye still detect that the beauty is surface only, without real depth?

With escalating costs, boilerplate design, and often diminished strategy and quirk, does everyone really win?    
« Last Edit: December 31, 1969, 07:00:04 PM by -1 »

Tom MacWood (Guest)

Re: The Tom Paul Theorem
« Reply #13 on: April 18, 2002, 07:48:01 AM »
16-Cypress Point, 17-St.Andrews and 13-ANGC might be the greatest par-3, par-4 and par-5 in golf - they are all 1/2 pars. #16 is a 3.5 and 17 & 13 are 4.5s. Is the par designation of these holes relevant?
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

BCrosby

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #14 on: April 18, 2002, 01:36:06 PM »
By popular demand, here are the Tom Paul Theorem Numbers for the par 3's and par 5's at ANGC.  The supporting math is available on request.

Par 3's in their order of finish, best to last:

No. 12   -  51.5
No. 16   -  45
No. 6     -  29
No. 4     -  21

Par 5's in their order of finish, best to last:

No. 13  -  67
No. 8    -  42
No. 15   -  29
No. 2     -  23

Pretty damn amazing.  No. 4 is lower than I might have guessed, but otherwise very close to my subjective ranking of the par 3's.  I have always thought that no. 8 was a better hole than no. 15.  Not as visually dramatic perhaps, but a better hole.  No. 13 comes out as the leader of the pack, as it should.

Tom Mac asks what happens with 1/2 par holes like 13.  The short answer is  - nothing.  It ranks at the top of the class the way any mathematical system worth its salt should rank it.

But Tom's question is a good one, so I shifted the numbers on 13 and 15 to treat them both as par 4.5's.  I adjusted the number of birdies, pars, bogies etc. to account for a par of 4.5.  (It's not easy, try it some time.) The effect of that shift is, of course, to shift the point from which you are going to measure the distribution of scoring more towards the lower scores.

The results are surprising.

No. 13 gets a slightly higher ranking at 78.

But No. 15 jumps to an astounding 147, a number far higher than any ever measured.  Almost double the TPT number for no. 13.  That happens, I think, because the distribution of scoring on 15 is skewed so heavily towards birdies that any change in par towards the birdie end of the range (i.e., pretending par is 4.5 rather than 5) has an extreme effect on the final number.  At least I think that is the explanation.  Another possible expanation is that the "other" category on no. 15 is severely underweighted, i.e, the "other" scores on that hole included a lot of 8's and 9's.  I don't know and don't have access to data that might answer that.

Remember that the point of the TPT is to find holes with a wide and relatively equal distribution of BOTH under par and over par scores.  The TPT penalizes holes with few birdies and lots of bogies; it also penalizes holes with few bogies and lots of birdies; and it also penalizes holes where the scores are lumped closely around par.

No. 15 didn't rank higher initially on the TPT because in the Masters it played as a relatively one dimensional birdie hole (V.J. Singh's play on Sunday notwithstanding).  

No. 13, on the other hand, had a much more broad and even distribution of under par and over par scores.  There were lots of birdies and lots of bogies.  The kind of hole that the TPT says is a well-designed hole.  Again, the theorem confirms my subjective judgment.

Gotta love it.

Bob

« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

TEPaul

Re: The Tom Paul Theorem
« Reply #15 on: April 18, 2002, 03:51:17 PM »
I always loved #13 ANGC for a lot of reasons--now I've even got the math to prove it! Thanks!

One of the great short "go/no go" holes in the world and even my own club has a "conceptual copy" of it we will now take extra care to bring out of decades of doldrums with tender loving care.
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

Tom MacWood (Guest)

Re: The Tom Paul Theorem
« Reply #16 on: April 18, 2002, 03:51:28 PM »
Bob
How did you come up with 67 for #13? I came up with 181.

4-eagles(x2) + 83-birds + 35-bogies + 10-others(x2) = 146
133-pars

146/133=1.1

4-eagles(x2) + 83-birds = 91
35-bogies + 10-others(x2) = 55

91/55=1.65

1.1 x 1.65 = 1.81 or 181

If you convert #13 to a par-4 like was done at 17-St.Andrews the numbers change dramatically.

4-birdies + 133-bogies + 45-others(x2) = 227
83-pars

227/83=2.73

4-birdies
133-bogies + 45-others(x2) = 223

4/223=.02

2.73 x .02 = .05 or 5

Although the strategic merit of the hole doesn't change, a change in the par designation from 5 to 4 has a dramatic effect on its number - the hole goes from by far the best hole on the course to by far the worst hole.
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

Jeff_McDowell

Re: The Tom Paul Theorem
« Reply #17 on: April 18, 2002, 04:08:45 PM »
Amazing stuff!!

This reminds of something Einstein said. Please accept my paraphase.

I understood the theory of relativity until the mathematicians got a hold of it.

I know you're trying to compare different golf holes. On a single hole analysis, I think of the TPT in terms of the bell shaped curve. The closer the scores resemble a bell-shaped curve, the less intersting.
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

George Pazin

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #18 on: April 18, 2002, 05:09:56 PM »
How tough would it be to find the raw data for the last 5-6 years to see if the numbers were changed by all the screwing around with the holes?
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »
Big drivers and hot balls are the product of golf course design that rewards the hit one far then hit one high strategy.  Shinny showed everyone how to take care of this whole technology dilemma. - Pat Brockwell, 6/24/04

Mike_Cirba

Re: The Tom Paul Theorem
« Reply #19 on: April 19, 2002, 07:01:36 AM »
Despite the solid analytical work done here by BCrosby, which certainly merits study and praise, I think we should not lose sight of the fact that we're only talking about the functional aspects of hole design here, and attempting to create a valid measure of that function.

I'm reminded of the scene in "Dead Poet's Society", where Robin Williams as the professor has the students open their book the first day and go to a section where a literary scholar has developed a mathematical formula to rate poetry.  After discussing it briefly, Williams has the students rip the section from their books, as he doesn't want his students limited in their thinking by such artificial constructs.

While the functional components of golf course architecture are fundamentally important, I contend that the aesthetic art, balance, flow, natural integration, and harmony of golf course architecture are at least as important.  Otherwise, we could probably play this game to targets on a driving range.    
« Last Edit: December 31, 1969, 07:00:04 PM by -1 »

BCrosby

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #20 on: April 19, 2002, 07:09:25 AM »
George -

I had the same thought.  If anyone has easy access to the scoring data at the Masters for past years, please email me a source and I'll do the comparisons.  

I am checking with the Atlanta Journal-Constitution (who is the source for my current info and who run a pretty full statistical slate every year on Masters scoring) but getting into thier archives is a huge pain.

Tom -

Your first calculation hits on a problem I struggled with.  It only comes up with the numbers for par 5's.  The problem is the Over-Under Ratio.  It is unique to par 5's that they tend to have more under par numbers than over par numbers, but that ratio gets real funky if it is expressed as a number larger than 1.  

My solution is to adopt a rule that the Under-Over Ratio must always be expressed at a fraction.  (We can discuss this off line if you want but I think it makes mathematical sense.  We are looking purely for a ratio that expresses the distribution of overs and unders.  We are totally agnostic about whether there are more birdies or more bogies.  All we want is a measure of the distribution of non-par scores.  A number larger than 1 distorts the utility of the ratio.  I am not a competent enough mathematician to know why, but I can give you lot's of examples of how non-sensical it gets.)  

The problem first hit me when I used your methodology (btw, that's the way I did it the first time too) on no. 15.  You come up with at TPT number for no. 15 of 912, which is crazy.  So I wondered if there was a fundamental flaw in the system.  

The fix, as noted, is to be sure you express the Under-Over Ratio as a fraction.  For par 5's, because there are more birdies than bogies, you make total bogies the numerator and total birdies the denominator and come up with the fraction. Everything then shakes out quite nicely.  To wit:

Hole        Non-Par Ratio   Over-Under Ratio   TPT Number

No. 13     146/133=1.09    55/91=.6              65

No. 15     184/96=1.92      32/152=.21           40

No. 8       101/170=.59      42/59=.71             41

Your analysis of no. 13 as a par four is fascinating.  Using the foregoing "fraction" rule for the Over-Under Ratio and then doing the rest of the calculation as you did, I come up with an even worse TPT number of .006 or .6.

But that is exactly what you would expect.  No. 13 is a great, great par 5 but a lousy par 4.  Why?  Because there would be extremely few birdies and lots of scores at par or higher.  The scoring would be extremely skewed towards higher scores. In short, No. 13 becomes just another US Open long, hard par 4.

It is in precisely this sense that the par of a hole matters. Yes par can be arbitrarily assigned.  Anyone can assign any par they want to a hole and the raw scores will be identical.

But if you can get over the metaphysical hump that perceptions matter (because they really and truly do matter), par is a big deal.  It is the framework from within which we make shot decisions.  It is the baseline against which we are tempted to take certain risks or avoid certain hazards.  That it is a second shot on a par 5 is not irrelevant to my go/no go decsion for a shot over the creek.

It is also my belief that something close to that philosophy of par is what informs the design decisions of the best architects, now living or dead.  And, as best I can figure, the TPT Number is helps identify the best of their holes.   At least I think it does when applied to ANGC.

Bob      
 

 

« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

Richard_Goodale

Re: The Tom Paul Theorem
« Reply #21 on: April 19, 2002, 07:13:00 AM »
Based on this and other threads, I suggest we rename this collective brainwave the "Harris-Paul-Crosby Conjecture."  If and when the Nobel Prize committe comes a calling, I for one, will recommend that they also include Ran Morrissett in the Rpize list for facilitating this meeting of the minds.

PS--I am sure that Tom Paul understands that "hole by hole" posting of scores will greatly accelerate the proof and ultimate accepteance of this theory/conjecture.
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

Tom MacWood (Guest)

Re: The Tom Paul Theorem
« Reply #22 on: April 19, 2002, 07:24:31 AM »
Bob
I do agree that the best holes provide the most variation in score - holes that have number of options of play. #13 for example can be played very boldly - with big rewards and big downsides. Or it can be played as a traditional par-5 or it could even be played ultra conservatively. What I don't understand is the notion that it goes from a great par-5 to a poor par-4. Surely there would be fewer birdies, but its all relative. The strategic options would remain the same, the great vaiations of score would remain the same. Under that rational, 17-St.Andrews and 16-Cypress Point are poor holes. For the formula to work I think it has to ignore par. But I also agree with MikeC that any formula is only part of the reason for a hole's greatness and that math can't possibly measure the many other important qualities.
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »

George Pazin

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #23 on: April 19, 2002, 07:32:49 AM »
As a recovering math geek, my gut feeling is that the problems with respect to par & numbers greater than one are related to the fact that Bob is, in a a sense, trying to perform a standard deviation function in a more roundabout fashion. As Corey Miller states, dig up the old stat formulas or get yourself a good stat program & try using that.

Rich seems to have everything else - surely he has a good stat program we can borrow. ;)
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »
Big drivers and hot balls are the product of golf course design that rewards the hit one far then hit one high strategy.  Shinny showed everyone how to take care of this whole technology dilemma. - Pat Brockwell, 6/24/04

BCrosby

  • Karma: +0/-0
Re: The Tom Paul Theorem
« Reply #24 on: April 19, 2002, 07:44:23 AM »
Tom -

I have to disagree with you.  Any par 4 that yields only 4 birdies after four days of play by the best golfers in the world is a lousy par 4.  It is either bady designed or has a par that is not reasonably related to its playing characteristics.  It is way too hard and, therefore, uninteresting to play or to watch others play.  It is a hole that only the USGA on steroids would want on a course.  It would be a bad hole.

Extrapolate that level of difficulty to golfers like you and me. Sheesh.

Why does par matter? Again, because the par of a hole sets the risk/reward bar for that hole. Standing in the middle of no. 13 fairway and being required to clear the creek with my second shot to score par is not the kind of "strategic choice" that defines great holes.  It is quite the opposite.

Bob    
« Last Edit: December 31, 1969, 07:00:00 PM by 1056376800 »