"Never up, never in"
Maybe you've heard some golf platitudes about how important it is to get the ball past the hole when you're putting. For example: "You miss 100% of the putts you leave short." Or as Arnold Palmer used to say, "Never up, never in". And so on.
I've heard all these too, and I know they're right, but they've just never seemed to sink in. Deep within me lurks an ancient terror of blowing it 10 feet past the hole and 3-putting. Even though I feel like my accuracy has improved on the greens, I still regularly leave makeable putts well short.
Then, after one too many birdie putts came up short, I realized: it should be easy to prove mathematically that if you want to lower your average total number of strokes, you should try to get your putts to finish past the hole. So that's what I did.
Setting up the model
We're going to simulate a lot of putts under different conditions to figure out the average number of strokes it takes to hole out. How should we model this? (If you don't care about these details, feel free to skip ahead to the charts.)
I'm going to assume a normal distribution. In other words, if I try to hit the ball 20 feet, the actual distances I hit the ball will be normally distributed and centered on 20 feet.
We also need to assume a standard deviation for our distribution. Let's stick with a 20-foot putt — I'll use a standard deviation of 3 feet, which means that roughly 95% of the putts will be within a range of 6 feet short to 6 feet long. That feels about right to me. I'm not a great putter, but I rarely have more than a 6-footer left after a 20-footer.
We need a couple more assumptions. First, how accurate is our putting? Assuming we actually get the ball to the hole, how likely are we to make the putt? Is it a straight putt, or is there a lot of break? For now, I'll assume 25% of the balls that make it to the hole go in. That's just a starting point, and we'll tweak it later. (In my own experience, this feels pretty generous, but maybe it's an easy putt.)
Finally, assuming we don't make the first putt, how likely are we to make the second putt? I've modeled this very simply and assumed we'll make any second putt if it's less than 5 feet, but miss if we have more than 5 feet left.
On the one hand, speaking from personal experience, I know I can definitely miss putts from inside 5 feet. But in real life, I'd make at least a few 6 and 7 footers, too. So this feels like a pretty realistic assumption. (Though note that we've also eliminated the possibility for the dreaded — dare I speak its name? — four-putt.)
Basic model results
So, with our default assumptions in place, what do we find?
For our standard 20-foot putt, here's a chart of average number of putts it takes to hole out for a given aim point — lower is better, of course. These aim points are expressed as distance from the hole, so -2 would be two feet short of the hole, and 3 is three feet past.
As you can see, the lowest part of the "dip" in this chart is roughly ~1 foot past the hole. So, to minimize our average number of putts on this standard 20-footer, we should try to hit the ball about a foot past the hole.
This simple model gives us some mathematical support for the adages listed above ("never up, never in", and so forth. But it's also a pretty flimsy model based off fly-by-night assumptions. The real power of this exercise is to see how the results change as we alter these parameters.
Tweaking the model: distance control
What happens if we change the variance in our distance control? This might correlate with being a better or worse putter, of course. But it would also change depending on how long our putt is. After all, it's easier to get within 3 feet of a target from 10 feet away than from 100 feet.
Here's the data:
The high-variance line is much flatter. It still dips a little past the hole, but there isn't much value to be had here.
In contrast, the low-variance line has a more pronounced dip. The "bottom" of the dip, the most optimal place to aim, is also further past the hole — closer to 2 feet past.
So in general, if you have a shorter putt (or better distance control), you should aim a little further past the hole than if you have a longer putt.
Tweaking the model: make percentage
What about how easy the putt is? For a dead-straight, uphill putt, maybe we have a good chance to sink it if we at least get the ball to the hole. On the other hand, if we're dealing with several feet of break, the ball isn't likely to go in even if we get the speed right. How should this affect our strategy?
We see a similar effect as with distance variability. For easier putts, it's better to be more aggressive, since we're more likely to get the benefit if we make it to the hole. By contrast, for difficult putts, the optimal aim point might be pretty close to the hole itself to limit our chance of 3-putting.
So if you have a sidewinder or a tricky double-breaker, you're better off just trying to get it to the hole at die speed. But if you've got a really makeable look, make sure you don't leave it short.
A special case: tap-ins
We can see these effects more strongly if we combine them. Consider an 18-inch tap-in. At this close range, let's say we have a 90% chance of making the putt if we get the ball to the hole. Our distance variance will also be much lower since it's such a short putt. What does our model say about this?
Obviously our average number of putts is lower in general, since this is a tap-in. But look how much more severe the "dip" is! And it's centered further out — more like 2 feet past the hole. If you think about it, this makes sense: pretty much the only way to miss a tap-in is to leave it short. And it's rare that we'd accidentally blast a shortie like this eight feet by. So for short putts, we should aim a little further past the hole to make sure we get it there, since the penalty for leaving it short is much higher.
Conclusions
This simple model backs up some of the most common putting aphorisms. If you have a short, makeable putt, make sure you don't leave it short. But if you have a long putt, a downhill putt, or a tough read, you'll minimize your score by just trying to lag it up there close enough so that you don't three-putt.
In the end, I'm not sure I really learned anything new from this exercise. But maybe seeing the math will finally help me start getting the ball to the hole more often.
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