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Golf is about how many strokes it takes to make the ball 18 times. To keep this really simple, lets say that the stats you just provided are the stats for ever putt the good player and the bad player face. So to make the ball 18 times, the good putter will have to putt: 18/80% = 22.5 putts on average. the bad putter will have to putt: 18/50% = 36 putts on average. Now after the bumps, here's what happens: Good putter 18/60% = 30 putts on average Bad putter 18/37% = 48.6 putts on average. The good player increased by 7.5 putts to hole the ball 18 times. The bad putter increased by 12.6 putts to hole the ball 18 times. The bad putter was impacted more by the bumpy greens than the good player. The problem is too complicated to just compare make rates. You have to apply those rates.

Really simplified version of the spirit of this post. Lets say make rates fall 10% on average for the good putter and fall 9% on average for the bad putter. This is in agreement that the bumps narrow the range of make rates. Now apply this to the # of putts each player typically hits per round: 10% * 29.5 = 2.95 more putts for the good putter each round. 9% * 33.5 = 3.015 more putts for the bad putter each round. You can't just evaluate this problem through make rates. The make rates can narrow, and still hurt the bad putter more than the good putter in total score.

Also apologies for the poorly formatted tables.

Ok lets actually run through this in detail.I’m going to show that the OP may be correct when it comes to how bumpy greens effect individual putt make rates, but wrong about how it effects the putters performance overall. I’ve already discussed how I’m modeling the putts.You can see it here here. Make rates have been captured for professionals and amateurs on todays greens.I’ve previously referenced a paper by Mark Broadie that has a set of those make rates which will be used here. Distance (ft) Pro Make Rate Average Make Rate 2 99.87% 93.20% 3 95.40% 76.30% 4 84.80% 60.40% 5 73.00% 48.10% 6 62.30% 39.40% 7 53.80% 33.20% 8 46.80% 28.20% 9 41.00% 24.30% 10 36.40% 21.10% 15 22.00% 12.30% 20 14.60% 8.10% 25 10.30% 5.80% 30 7.60% 4.60% 40 4.70% 3.00% 50 3.30% 2.20% These are the make rates I set my model to match.After running each distance though my Monte Carlo model these are the make rates I get: Distance Pro Make Rate Average Make Rate Simulated Pro Make Rate Simulated Amateur Make Rate 2 99.87% 93.20% 99.88% 93.14% 3 95.40% 76.30% 95.35% 76.02% 4 84.80% 60.40% 84.89% 60.52% 5 73.00% 48.10% 73.65% 48.24% 6 62.30% 39.40% 62.92% 39.65% 7 53.80% 33.20% 53.52% 33.12% 8 46.80% 28.20% 46.35% 29.11% 9 41.00% 24.30% 41.33% 24.31% 10 36.40% 21.10% 35.49% 21.85% 15 22.00% 12.30% 23.25% 12.05% 20 14.60% 8.10% 14.96% 8.80% 25 10.30% 5.80% 10.16% 5.32% 30 7.60% 4.60% 7.45% 4.75% 40 4.70% 3.00% 4.62% 2.90% 50 3.30% 2.20% 3.18% 2.32% The models fit to real life in terms of make rates is reasonable.Now we add a ½ bump random bump in either direction to the distribution and the make rates become this. Distance Pro Make Rate Average Make Rate Simulated Pro Make Rate With Bump Simulated Amateur Make Rate With Bump Gap Change 2 99.87% 93.20% 98.16% 88.10% Widens 3 95.40% 76.30% 89.80% 73.37% Narrows 4 84.80% 60.40% 79.80% 58.59% Narrows 5 73.00% 48.10% 68.99% 47.54% Narrows 6 62.30% 39.40% 60.20% 38.71% Narrows 7 53.80% 33.20% 51.56% 32.27% Narrows 8 46.80% 28.20% 46.81% 28.31% Narrows 9 41.00% 24.30% 40.64% 23.93% Widens 10 36.40% 21.10% 34.87% 21.16% Narrows 15 22.00% 12.30% 21.48% 12.21% Narrows 20 14.60% 8.10% 14.63% 7.84% Widens 25 10.30% 5.80% 10.30% 5.67% Widens 30 7.60% 4.60% 7.59% 4.67% Narrows 40 4.70% 3.00% 4.37% 2.89% Narrows 50 3.30% 2.20% 3.30% 2.06% Widens Now, notice, most of the gaps in terms of make rate have narrowed, just as the OP said they would. Others have said that from 8 feet the better putter gets harmed more. I’m not ignoring that and I agree with you that from 8 ft the better putter very likely gets harmed more. And yes, at this point, it looks like the OP is correct, but not all shots have been examined yet. Now, data for frequency of first putt at a given distance was also published by Broadie: Distance (ft) Number of First Putts 2 1.15 3 0.46 4 0.86 5 0.86 6 0.86 7 0.89 8 0.89 9 0.89 10 1.76 15 2.62 20 1.75 25 1.25 30 1.47 40 1.28 50 0.93 Since we know typical first putting distance, we can apply both sets of make rates for both players before the bumps and after to see what effect they have on make rate. Distance Number of First Putts Pro Make Rate Amateur Make Rate Pro # of putts made Am # of putts made 2 1.15 0.9987 0.932 1.148505 1.0718 3 0.46 0.954 0.763 0.43884 0.35098 4 0.86 0.848 0.604 0.72928 0.51944 5 0.86 0.73 0.481 0.6278 0.41366 6 0.86 0.623 0.394 0.53578 0.33884 7 0.89 0.538 0.332 0.47882 0.29548 8 0.89 0.468 0.282 0.41652 0.25098 9 0.89 0.41 0.243 0.3649 0.21627 10 1.76 0.364 0.211 0.64064 0.37136 15 2.62 0.22 0.123 0.5764 0.32226 20 1.75 0.146 0.081 0.2555 0.14175 25 1.25 0.103 0.058 0.12875 0.0725 30 1.47 0.076 0.046 0.11172 0.06762 40 1.28 0.047 0.03 0.06016 0.0384 50 0.93 0.033 0.022 0.03069 0.02046 Total 17.92 Total 6.544305 4.4918 Distance Number of First Putts Pro Make % Bumped Amateur Make % Bumped Pro # of putts made Am # of putts made 2 1.15 0.9816 0.881 1.12884 1.01315 3 0.46 0.898 0.7337 0.41308 0.337502 4 0.86 0.798 0.5859 0.68628 0.503874 5 0.86 0.6899 0.4754 0.593314 0.408844 6 0.86 0.602 0.3871 0.51772 0.332906 7 0.89 0.5156 0.3227 0.458884 0.287203 8 0.89 0.4681 0.2831 0.416609 0.251959 9 0.89 0.4064 0.2393 0.361696 0.212977 10 1.76 0.3487 0.2116 0.613712 0.372416 15 2.62 0.2148 0.1221 0.562776 0.319902 20 1.75 0.1463 0.0784 0.256025 0.1372 25 1.25 0.103 0.0567 0.12875 0.070875 30 1.47 0.0759 0.0467 0.111573 0.068649 40 1.28 0.0437 0.0289 0.055936 0.036992 50 0.93 0.033 0.0206 0.03069 0.019158 Total 17.92 Total 6.335885 4.373607 And guess what, the OP is again correct that relatively speaking, the poor putter makes 0.1 fewer putts per round and the better putter makes 0.22 fewer putts per round. The gap on one putts does narrow. People have said I don’t understand the OP’s math.But I hope you see, I do understand it, and this math isn’t that much different directionally when it comes to make rates of longer putts. But for some reason they stop here, without examining what happens to the 12 and 14 other putts that still need to be holed. Let’s assume that half of these remaining putts are hit from 2 ft, and half from 3 ft.For all putts after the second, we will assume they are from 2 ft.Now I’m assuming both players putt from the same distance for their following putts, which is clearly a bad assumption. Since good putter has more skill they will be typically be closer for their following putts. So please remember this assumption I’m making here HURTS my case, but I’m doing it anyway. When you calculate what happens to the two players having them hole out from 2 and 3 ft after a miss, on a green without bumps you get this: No Bump With Bump Second putt Second putt Second putt distance (Ft) Pro # of putts Amateur # of putts pro make am make distance Pro # of putts Amateur # of putts pro make am make 2 5.688 6.714 5.680 6.258 2 5.792 6.773 5.685 5.967 3 5.688 6.714 5.426 5.123 3 5.792 6.773 5.201 4.969 total 0.269 2.048 total 0.697 2.610 Third putt Third putt Pro # of putts Amateur # of putts pro make am make Pro # of putts Amateur # of putts pro make am make 2 0.269 2.048 0.269 1.909 2 0.697 2.610 0.685 2.299 total 0.000 0.139 total 0.013 0.311 fourth putt fourth putt Pro # of putts Amateur # of putts pro make am make Pro # of putts Amateur # of putts pro make am make 2 0.015 0.299 0.015 0.265 2 0.015 0.299 0.015 0.265 total 0.000 0.034 total 0.000 0.034 All the data summarizes as follows No Bump Pro Amateur Pro Total Putts Am Total Putts one putt 36.4% 25.0% 6.544 4.492 two putt 61.7% 63.2% 22.213 22.761 three putt 1.5% 10.6% 0.806 5.726 four putt 0.0% 0.7% 0.001 0.519 Total 29.565 33.497 With bump Pro Amateur Pro Total Putts Am Total Putts one putt 35.2% 24.3% 6.336 4.374 two putt 60.5% 60.8% 21.774 21.873 three putt 3.8% 12.8% 2.054 6.897 four putt 0.1% 1.5% 0.050 1.094 Total 30.213 34.239 Difference 0.648 0.741 Notice, including the second and third putts makes the total putts per round go up MORE for the bad player than the good player.This is because the bad putter must hit MORE short putts than the good player, and the bumps are going to affect the short putts too. The bumps affect the bad player’s total strokes in aggregate more in the second and third putts, making the gap in TOTAL putts widen. Now you don't have to agree with my assumptions here. What I want people to see is that even if make rates do narrow in the poor putters favor from mid and long distances, the overall question of who is more effected requires considering every shot, not just the first one.When you do consider every putt, it’s perfectly reasonable to believe that the good putter ends up better off than the poor putter. In golf the lowest score wins. The lowest score includes all putts, not just the first putt.

I've been in a cart that rolled over when I was younger, but it probably wasn't the courses or the carts fault.

In very simple math terms, you are not defining the second variable of the equation. You are not considering the difference in miss rate due to variability, rather you're assuming that the “misses more” rate of player A will be equal or greater than the “misses more” rate of player B? This is the biggest factor and it’s not being addressed with your math. Trackman measured the launch direction for the average PGA tour player at +/- 0.5 degrees and the launch direction for the average amateur player at +/-0.8 degrees. We know that even if the A player and B player were equally skilled at playing break and pace, the A player's misses would be nearer to the hole than the B player. Thus it would take a larger bounce to move a typical B player's miss back to the hole than it would for player A. The relationship between the A player's miss and the B player's miss may stay fairly constant even as the magnitude of bounce direction increases. In other words, it’s likely going to be easier for player A’s ball to be bounced back into the hole than player B's ball. This is why you can’t just oversimplify the problem and assume away the complex aspects of how bounces affect putts. It’s just not a simple one-variable solution.

I think you are saying you believe the distribution at the hole will be like a uniform distribution, with the same probability at every point. Is that correct?

Another great post. Glad to see you recognize how complicated understanding the ball's path on the green is. Green reading, speed calculations, techniques for both bad and good putters, actual slopes, optimal speed for entering the hole, and then of course bumps. A wide spectrum of inputs. It's no wonder Mark Broadie thought he needed 14 variables to understand how a putt travels across the green. So needless to say, I'm confused how, if Mark Broadie, one of the most respected, if not the most respected golf statistician, felt he need to get to this level of complexity to figure out the impact of a large hole on putting, but my, much less sophisticated method is too complicated, and your extremely simplified opposite ends of the spectrum thought experiment has enough detail to understand the problem?

I agree it's complicated. What distribution do you think makes more sense to use than normal. Or do you only think you can understand the very complicated answer of where ball goes by trying to model each variable of putting independently. It's perfectly reasonable to think you need that much complexity to model the potential paths of balls on the green. I would probably agree that my model is likely over simplified.

If I had said, "statistically a top 10 putter that year that many think was the best putter of the era", would that have changed the likely hood of a coincidence that this player won on unusually bumpy greens? "Great putter wins on controversially bumpy greens." vs best putter probably would have been a better choice.

Thank you for saying that assuming a normal distribution is understandable. I think you bring up a good point about the skewedness of the distribution. Some players don't have symmetrical distributions. Now across a group of players this might get canceled out, but for individual players, it wouldn't. I've said before that the answer to this question is "it depends" and certainly at a singular level, a certain bad players could be expected to outperform good players on bumpy greens, and certain bad players could be expected to underperform on bumpy greens, all depending on the typical distribution of thier putts. I agree that better players would tend towards a tighter more normal distribution and the poorer putters would have a wider lower peaked curve. Thats how I tried to model it. Interestingly, Broadie didn't assume the player always aims at the hole and tried to optimize making the putt. He assumed the players were trying to putt optimally for total score, including 3 putts. With slopes, this could mean just what you said, the player doesn't aim at the hole but aims above the hole. It feels like many on this thread have oversimplified putting to the point that they don't even think about times when players don't aim at the hole, since every example here seems to assume the player is trying to make every putt. Its a very complicated problem. By my count, Broadie used 14 variables to model how putting would change with a larger hole in is paper. 14. Complex problems don't lend themselves to the correct answer when simplified down to a single variable.

He wasn't in the field. You don't think Jordan Spieth was one of the best, if not the best putter in the mid 10's? I linked to the Boadie paper here. He states he uses a normal distribution for directional errors in strike. If you think a normal distribution is a bad assumption, what distribution do you think is correct? Except you did try. You assumed the pro putter didn't miss the line at all from 10 ft, and had no distribution. Then you assumed the bad player only made 1 out of ever 10 putts from 10ft, and often missed by 2-3ft. Those are both distributions. Did you have data for those distribution, or a reputable source that has used a distribution like those in the past to model putting?

It may be a coincidence that the best putter from the mid 10's won that tournament on notorious bumpy greens. Or maybe it's not a coincidence.

Nearly every reply to me on here: "You are making up your own assumptions without any data to prove your point. You don't have any data or proof those assumptions and constraints are correct, so you conclusions are wrong and unfounded. Thats not science or logical. To show you why your logic is so bad, here are my assumptions and constraints that I just made up and that don't have any data to support them either, but they prove my original point so this is science and they show that I'm correct......if you can't see this you don't know what you are talking about." Good times.

Really loved this reply. I like how your argument is now a thought experiment. This is a good admission, because thought experiments are useful, but they don't prove anything, and there are lots of statements here that "bumpy greens close the gap between good putters and bad putters" as some kind of proven indisputable fact. Thought experiments don't prove things. Especially thought experiments without any data. I also like that you are allowed to use grand generalizations for your thought experiment, but if I try and state an assumption for my model, I don't have the data and I'm making it too complicated? Really? You say "you can not model the world without data, which you do not have". Yet I did use data. I used The statistics of make rates for professionals and amateurs. I used a normal distribution, which has been proven to map to many many process in the real world, and its what Mark Broadie used for his study. I used the size of the hole and reduced it for the effective size of the hole based on actual data. I used statistics to map the normal distribution to the players shot pattern so that I wasn't just imagining a player, but basing the model on data from collected thousands of actual players. What data did you use to model the world? It doesn't look like any to me, but please correct this if I missed the data you are using. What assumptions are you making with your thought experiment? On the point on the OP's original question, he askes. "would that narrow or increase the gap (or keep it the same) between the good putters and the bad putters? " Was I wrong to think that putting included the second putt and third putt? Isn't that part of what every shot counts is about? You have to think about every shot, not just the first one. Shouldn't a thought experiment be created to address the full question, and not just half of it?

This first putter who is perfect doesn't exist in human form. Nobody can make every putt perfectly in the center of the hole at dead speed on perfectly flat ground. There are statistics on the dispersion of tour pro trajectories and speed off the putter, and nobody hits it the same every time. The second putter sounds like someone who started playing golf last week. The average golfer makes over 20% of 10 footers, so you have clearly picked a very very bad putter. I also like that the bad putter here seems to get better because the bumps. As if they are more likely to hit the putt just off the edge of the hole than they are to hit it on the hole. This is a very strange assumption, and ignores how acts of skill usually get distributed in the real world. Mark Broadie didn't just use a normal distribution when he built a model of putting for kicks. Now I have been accused of setting assumptions and constraints that are unreasonable, yet you have picked a putter that does not exist in real life as your example of the good putter, and a putter who beyond terrible as your bad putter. Remind me, who is the one just making up constraints and assumptions to prove their point? But even if we stay with your strawman thought experiment, how many of those 2-3 footers that the bad putter has left do they miss because the greens are bumpy? Their make percentage goes down there too right? What about the 1-footers? They are so bad they probably miss those a decent amount, and the bumps are just going to make those putts go in less for them too. Why didn't you include the second and thirds and fourth putts as well? We are talking about golf right, where you have to hole the putt, and every shot counts? All good science tries to accurately model the world and incorporate all the important variables. It doesn't just pick the outliers and then use those to make conclusions. It also recognizes that even if the extreme conditions say one thing, the reasonable conditions closer to the middle may not.

Sounds like the same thing critics of strokes gained say as well. Funny how when that math agrees with your view, it's correct and representative of the real world, and when it doesn't, then its not representative. Which sorts of things? Models that describe how golf is played and works and understanding the variables that effect your score? My bad, that that seemed exactly like the kind of things you were interested in. I thought you would respond with what would be a better choice for those assumptions if you disagreed, but if trying to understand golf with mathematical models isn't your thing then I'm sorry I asked. I see you are interested in spelling, and good at it too. You are correct. Thank you for replying with an answer that states what you think "it is" an not just that I was wrong. Compelling to you. Plenty of other people who's first gut instinct told them that the spread would widen or stay the same now have grounds to know they had a reason to think that way. It was ok and logical to think the spread would widen or stay the same. When one gets down to it, the real answer is "it depends on how bumpy are the greens and how different are the skills the players ." You can't universally answer the question of which putter is impacted more from bumps because there are too many variables. Like all complicated problems, adjusting those variables doesn't universally lead to the same answer for any and all sets of variables. I'm glad that those people now have seen another side to this debate and that there is math and logic to support their intuition.

Mark Broadie used a normal distribution for the trajectory of the putt in this paper. He was trying to figure how how many more putts someone would make on a larger hole. (PDF) A simulation model to analyze the impact of hole size on putting in golf PDF | We develop a model of golfer putting skill and combine it with physics-based putt trajectory and holeout models to study the impact of doubling... | Find, read and cite all the research you need on ResearchGate The normal distribution shows up in a lot of places, and it does seem somewhat reasonable to use here. Plinko, interestingly enough creates a normal distribution, so at a minimum it addresses @iacas plinko quip.

The reason I'm not looking at 8 ft putts is they don't happen very often. This is Mark Broadie's stats for the distance of the first putt on the green for amateurs. Distance Occurrence of First Putt 2 1.15 3 0.46 4 0.86 5 0.86 6 0.86 7 0.89 8 0.89 9 0.89 10 1.76 15 2.62 20 1.75 25 1.25 30 1.47 40 1.28 50 0.93 Notice that 8 ft putts are not common as first putts. Whats common are longer putts. And of course there are second putts, about 15 of them, which usually occur near the hole, say around 2 to 3 feet. So why judge the answer on putts that are rare, the 8 ft putts, and not on the putts that are common? Instead shouldn't we first look at the 2-3 foot putts and then the 15-40 ft putts? That seems like the logical path doesn't it?

The disingenuous part is not necessary. Most post about putts being negatively affected 100% of the time was in response to your post 2 weeks ago, not your reply today. You said I didn't read your original post's explanation, and I was showing that your original explanation was flawed. Ok lets discuss these constraints and assumptions. Since this conversation is a bit all over the place and people are saying they can't think of a case where the better player would benefit, discussing what happens at 3 ft should make this clearer. I assumed a bell curve for putt dispersion. Is this a bad assumption? Many things in life model well with a a bell curve. Mark Broadie used this assumption in one of his prior papers, so you can ask him if he thinks that was a bad assumption. I'd like to know why you don't like this assumption if you have a problem with it. I assume Mark Broadie's values on make percentages for professionals and amateurs at 3 ft are correct. Obviously you can challenge this assumption if you like, but it seems unlikely its very wrong. I assume that the effective size of the hole is 3.4", since putt near the edge with any speed won't go in. With these three assumption, I assume I can determine the standard deviation of the players miss pattern on a typical green. If you are ok with the first three assumption this assumption should clearly be reasonable. I assumed that the balls speed when putted from 3 ft is reasonably hole-able, and will be hit at a speed to go about a foot past the hole. Most putts from 3 feet don't have a high rate of speed enough to cause major problems with the ball not going it if it hits the hole. I'm also sure you would agree that the better putter is more likely to have the correct speed, so this assumption is going in your favor, not mine. I assumed the bounce of exact half inch left or right. Now this assumption certainly depends on how bumpy the greens are. Half inch feels like a pretty good size bump though from 3 ft away. The assumption of a binary bounce makes the math simpler, but you can easily make the size of the bounce normally distributed and still see the same result directionally. I assumed every putt will be impacted by a bump. This includes putts that would normally miss the hole bumping half the time back in the hole. Why would you just assume I didn't include those? I assume the random Monte Carlo simulation will accurately how the effect of the bumps. If you don't like this assumption, then it's possible to do the math purely with cumulative distributions functions, but its often easier to understand Monte Carlo replications. If I missed an assumption you think I made, please call it out. I'm trying to be very clear and open here, and not just state "your wrong". Here is a spreadsheet that shows these calculations (binary bounce and normal bounce). You can save it to your own account and examine it further if you wish. Bumps effect on making putts - Google Sheets Sheet1 1 Hole width,4.25 Effective Cup Size,3.4 Simulation of 3 ft putt made out of 10000 with 1/2" bump. good player make % at 3 ft,99.54%,Z score,Standard Deviation,Smooth,Bumpy,Change,% Change bad player... With these assumptions from three feet, It's clearly possible, even probable, for for the bumps to impact the poor putter more than the better putter. This isn't a feeling. Its math. Please tell me where this math is wrong or where the assumptions are meaningfully flawed. Better yet, have your friend Mark tell you where this math is wrong or one of these assumptions are meaningfully flawed. Now I know that a three foot putt is just part of the picture. But we hit lots of putts (maybe most) from around this distance. And if you, or anyone else who reads this sees that bumpy greens will help the better putter from three feet relative to the poor putter, it will open their mind that this entire question of bumpy greens is not at all straight forward and obviously benefiting the bad putter.

Ask @iacas when it get bumped. I was replying his statement of a putt bumped off the center of the hole. Do you think those putts miss frequently? Do they have a negative outcome 100% of the time? Most of my putts that enter the hole 1" from the center go in.

Get the AW. It will probably blend better to the PW in terms of gaping.

Very good point.

Just bought a set of t100 which is definitely my biggest golf purchase ever. Pretty happy with them so far. Amazing how forgiving a club that looks like a blade can be these days.

A putt going at the dead center of the hole that bumps one inch left or right doesn't have a negative outcome 100% of the time. It has a negative outcome 0% of the time. It still goes in. You need to get up to 2" bumps before you need to think about negative consequences of a ball heading hitting a bump when at the dead center.