# Why do tall golfers hit it farther?

Dave Tutelman  --  June 5, 2015

"Help me scientifically understand -- because the stats are there -- why 15 of the top 20 players on tour are 6'0"+ and are the longest drivers on tour? How does height seem to equal longer distance with these facts?"

So asked Jonathan David Muccio on June 1, 2015, in the Golf Sports Science discussion group on Facebook. A long and interesting discussion followed. Some was very traditional (levers, longer arc) and some was very scientific (torque, mathematical models). As you might expect, I come down on the scientific side. Here is my take on the question.

## First... Is it true?

Do tall golfers actually hit it farther?

As the discussion got under way, Liam Friedman pointed out, "This is only significant if less than 3 of 4 PGA tour players is over 6 feet tall." Spot on! If the 25th percentile of height on the PGA Tour is six feet or more, you would expect -- just from chance -- that 15 of the top 20 players would be six feet or taller. Same for 15 of the longest 20 drivers. That would just be an accident of the numbers.

So what is the truth here? Nobody wanted to pick up the question, so I looked around the Internet and found confirmation that this is indeed an interesting question. Interesting in two ways: it is not a statistical anomaly, and there is indeed a statistical trend that taller golfers hit it farther.

### Not a statistical anomaly

I checked by looking up PGA Tour player height statistics, and importing them into Excel. The stats available were from a much larger set than the fully-exempt PGA Tour. With the numbers there, it must have included anybody with any standing at all in the "big show", the senior tour, and the Web.com tour. But it is the data available, and I used it.

They had a huge number of players there, a page for each letter of the alphabet. I didn't want to deal with that, so I made the blatant assumption that there was no bias based on initial of last name. Then I loaded pages until I had over 400 names. Not all the data was available for all golfers; the 444 Excel rows I had contained heights for 326 of the golfers. I sorted those 326 by height and looked for the median and the quartile points. Here's what the data showed:

 Percentile Height Comment 25 5' 10" closer to 5'9" than 5'11" 50 (median) 6' 0" closer to 5'11" than 6'1" 75 6' 1"

So the 25th percentile would appear to be between 5'9" and 5'10", definitely below 6 feet. Jonathan's question of "why?" is therefore interesting, not just what we should expect statistically.

### There is indeed a trend

Continuing my Internet research, I found a page on Golf Magazine's site that had an interactive graph plotting driving distance vs height for PGA Tour players during the 2013-'14 Tour season. Here is a non-interactive picture of it. If you go to the site itself, you can identify each dot with a player.

The article says the slanted red line is the average driving distance for each height. I doubt it; I'm sure it is a best-fit straight line for driving distance vs height. But this is even better than just averages! The slope of the straight line shows how much an inch of height is worth in driving distance. Some quick measurement shows an inch of height is worth about 1.3 yards of driving distance.

But... There is a lot of scatter in the data. In statistical terms, the "R-squared" of the best-fit line is way below 1.0. In fact, you'd be hard-pressed to draw the line by eyeball; I certainly couldn't. What this says is: Height matters, but other factors matter a lot more.
Having verified that the question is interesting and meaningful, let's get back to the original question, which was...

## Why???

As I said, the speculation in the discussion group ranged from traditional buzzwords to mechanical analyses. As one of the contributors of an analysis, let me present it here and argue for it.

I used a swing model, and changed parameters one by one to reflect differences that height might mean. The model was the double pendulum of Jorgensen. (There was some discussion of that model choice. I defend it below, but for now let's just accept it.) I have a program that simulates the double pendulum model (SwingPerfect by Max Dupilka) that is quite adjustable in terms of the parameters you can set.

I set up for a "nominal" golfer 5'10" tall, then varied things one at a time to compare the 5'10" golfer with a 6'2" golfer. Some details for the nominal golfer:
• 24" arm on a 5'10", 176 pound golfer. Shoulder torque during downswing is a constant 58 foot-pounds; wrist torque is zero; we're free-wheeling.
• 45" driver of fairly normal shaft and head weights.
• Clubhead speed is 109mph. (That just worked out, given the rest of the parameters.)
We will grow this golfer from 5'10" to 6'2". That's a 5.7% increase linearly. We will grow one thing at a time, and see what happens -- what kind of difference each change makes.

### Longer arms

Let's start out by checking the most traditional of the explanations, longer arms, which is usually expressed as "increased leverage" or "bigger arc". We'll lengthen the arms 5.7%, in proportion to the taller golfer. The arms are now 25.4" long. The clubhead speed drops to 107mph, a 2mph loss. So much for that theory! We'll look more closely in the discussion below at why leverage and swing arc don't seem to help; in fact, they hurt a little bit.

Summary: longer arms reduce clubhead speed 2mph.

### Longer club

Geometry says that the taller golfer should be able to handle a proportionally longer club. (I know, I know! I do clubfitting too, and it often doesn't work that way. But, averaged over all possibilities, it does.)

We also know that a longer driver should give a higher clubhead speed. So, if we can control the longer club well enough to give good impact, we should get more distance.

I tried scaling up the club length proportionally to the height. Our 5.7% longer driver, barely legal at 47.6", gives back exactly the 2mph we lost with longer arms. That's sort of interesting. It says scaling the linear dimensions of the golfer and the club leaves clubhead speed just about the same. Sounds counterintuitive, but I'm pretty sure it has to do with the fact that we have the same torque driving a longer arm lever, so less force at the hands is applied to accelerating the club. You need to increase club length, or the lower hand-pull force will give decidedly lower clubhead speed.

Summary: a longer club gives back the 2mph we lost from longer arms.

### Golfer weight

A taller golfer in the same proportions -- with the same "build" -- as the shorter golfer will weigh more. Let's account for that.

Weight should scale with volume, if the golfers have roughly similar builds -- and volume scales as the cube of length. We increased the length 5.7%. So we must cube 1.057; we get 1.181, or a volume increase of 18.1%. The 176-pound golfer is now a 208-pound golfer. Do this and the clubhead speed drops off again. Not all the way down to 107mph, but most of the way (107.4).

Why? Changing the weight increased the golfer's MOI. It now takes more torque to rotate the torso and the extended arms. That's why the clubhead speed drops.

Summary: increasing the golfer's weight drops clubhead speed 1.5mph, almost as much as arm length increase did.

### Shoulder torque

Merely changing the golfer's weight does not convince the computer to increase the torque applied. I checked, and changing the weight had no effect on torque. But it should! More muscle mass should mean more muscular force. All other things being equal, force should go up with the cross-sectional area of the muscle, which would be proportional to the square of the height.

Not only that, but remember that the torque produced at any joint is the muscle strength times the moment arm from the joint's pivot to the muscle attachment. The larger golfer probably has proportionally larger joints. The distance from muscle attachment to pivot is longer. So the torque (force times lever arm) probably scales with the cube of the height. Why cubed? It is height squared for muscle strength, times height again for lever arm.

For the reasons above, I increased the torque by 18.1% to 68 foot-pounds. That did it! The clubhead speed went up to 115mph.

Summary: increasing the torque to reflect the golfers increased size gives a marked advantage in clubhead speed. This is the only place in the whole study where the tall golfer's clubhead speed actually increased, compared with a shorter golfer.

### Shoulder torque revisited

Mike Duffey pointed out to me, "Muscle strength scales to body weght reasonably well across an athletic population, but to the 2/3 power. I don't have access to how weight scales to height right away (especially in golfers), but that information might help adjust the predicted increase in force output. Here is a good read on the topic in athletes."

Let's incorporate this new information. We will back off from a cube scaling on height to a 2/3-power scaling on weight. In the absence of information on how weight scales with height statistically on the PGA Tour, we are left with scaling geometrically, as we did explicitly with the golfer's weight above. So scaling to the 2/3 power of weight would be the same as scaling to the square of height.

I did that. The 5.7% height scaling becomes an 11.7% strength scaling, or a shoulder torque of 65 foot-pounds. Plugging this into the computer model gives a clubhead speed of 113mph. Not as much as before, but still markedly more than we got from arm length, club length, and weight put together. We will use this number in the discussion that follows. And we will amend it as new information becomes available -- like the statistical scaling of weight to height for golfers.

I even checked to see what a 1/3 power scaling for strength would do. (That is, strength proportional to height rather than body weight.) Even that gives a gain of clubhead speed, albeit a small one, to 110mph.

### My conclusions

The reason taller golfers hit it farther is mostly because their size provides a frame for more muscle and larger lever arms at the joints. The rest of it -- larger arc, more massive body -- reduces clubhead speed, and the longer club doesn't get enough back to make up for it. But the bigger golfer is also stronger (or at least has the potential to be stronger, given proper conditioning and nutrition). If he/she realizes that potential, the result is higher clubhead speed.

Now don't go throwing Jamie Sadlowski at me... nor, for that matter, Nick Faldo. Technique matters, of course. But the question was asked in terms of trends. What does height do, all other things being equal? I'm answering in the same spirit.

## More discussion

Facebook discussions have an interesting dynamic, even if they are in technical discussion groups like this. If they draw enough interest and comments, they often break into subthreads on related topics. I'd like to comment on some of the subthreads that were still related to the original question.

### Longer arc, levers

Before we get into this topic, we need to clarify the terminology. This is especially important because the golf instruction world uses these terms as buzzwords. We need to be very specific if we are going to draw correct conclusions; vaguely-defined buzzwords will not do, not even if "everyone knows what they mean."

A "longer arc" or "bigger arc" refers to the arc traveled by the hands. It can occur two ways in the golf swing, a bigger radius (longer arms) or a bigger angle (more shoulder turn).

In the diagram, the green arc is longer than the red arc in both cases. So the term "bigger arc" is ambiguous. Let's resolve that ambiguity. When discussing a longer arc in this article, we mean a bigger radius! There's a good reason for that. A scaled-up taller golfer will have a bigger radius because of longer arms. But there is no reason to expect taller golfers to make a bigger turn. Some will, some won't, but it won't have anything to do with their height.

Now that we are explicit about what it means, here are some quotes purporting to answer the question. Some are from right at the beginning of the discussion, and others close to the end.
"Longer arms, bigger arc!"
"Bigger wheels go faster i.e wider arc"
"Let's don't try to turn this into atomic science. A smaller man, like Ben Hogan, has to have a longer swing arc and swing the club faster to match a 6'6" golfer with similar technique and abilities and a much larger swing arc. Small wheel vs large wheel. It can be done though."
"Levers, levers, levers!!!!!"
"Length of Arc, swing radius, leverage..."

... And the same thought, but with math instead of buzzwords:
"Tangential velocity v=rω. In this instance, v = club head velocity, r = radius of the lever from the point of rotation (arm and club), and ω = rotational velocity. Therefore if a taller golfer creates the same rotational velocity ω, they will hit the ball farther base as v will increase."

Simply not true! The last quote there is a good mathematical justification for the argument, but it assumes the same angular velocity for shorter and longer arms. This usual golf-instruction assumption about "leverage" is purely kinematic (just motion), and ignores kinetics (the forces that produce the motion). In fact, a longer arm loses angular velocity, unless you put in enough extra driving torque to make up the lost rotation rate. I have an article on my site proving that this scenario does not give any increased speed.

Why doesn't the speed increase proportionally to the length? The article cited above discusses the answer in detail. But here's the short form of the answer.

The leverage argument has a tacit assumption that it takes no more effort to accelerate a payload, a mass, on a long arm than on a short arm. But we know otherwise. The rotational equivalent of Newton's famous F=ma is
τ = I α
...or torque equals moment of inertia times angular acceleration. So we get the same angular acceleration from the same torque only if the moment of inertia does not increase.

But moment of inertia is proportional to the square of the radius (at least; it could be even more under many circumstances). Thus I is increasing even faster than the radius. Unless we can find more torque somewhere, "bigger wheels" or "bigger arc" will go no faster, and perhaps slower. And our computer modeling said the answer is slower for longer arms if the action is a golf swing. This means that, in order to maintain angular velocity and increase clubhead speed, you need a bigger torque τ.

So much for the longer lever theory.

### Longer club

Michael Deiters asked, "Why does a long club hit it farther than a shorter club?"

It doesn't, necessarily. Yes, the clubhead speed is greater. The ball speed might not be greater if you can't control impact on the sweet spot with the longer club. So the question might be better phrased, "Why does a long club give more clubhead speed?" I'm not just splitting hairs; I have data that this happens with some golfers -- including myself, so I'm personally sensitive to the problem.

I've done a study that goes into gory detail on the question of longer driving clubs. The short, simplified answer is this. If you can lighten the components to keep the club MOI the same, then the same swing should give the same angular velocities. And the same in-plane angular velocity means that the tangential component of clubhead speed (angular velocity times club length) will obviously go up with club length. Since angular velocity contributes about 80% of the total clubhead speed, that will boost the total clubhead speed.

Mike Duffey posted a response very similar to this.

But the longer club often has more speed even if we don't take care to maintain the moment of inertia. The reason is most of that clubhead speed is due to inertial release, not a torque driving the club from the hands. The torque driving inertial release is dependent on the length from pivot to balance point and the weight of the club. So what? Well, those are the same things that increase the MOI. As the club gets longer, both the MOI and the torque grow. So, unlike a hub-torque-driven member, the club does not lose much angular acceleration as it gets longer. True, the MOI grows faster than the torque, but that is still a lot better than the situation with the arms, which are driven by shoulder torque. So τ = I α is a more valid argument for the club than the arms.

### Why so long?

Was the increase too much? Consider... The revisited gain of 4mph of clubhead speed will give us 12 more yards for the four-inch increase in height, assuming we don't lose smash factor. That is 3 yards per inch of height. The increase in distance of the best-fit line from the Golf Magazine article was only 1.3 yards per inch of height. We now have slightly more than 2 times the gain the Tour statistics show.

Why so much? I have a few guesses, which taken together might explain the discrepancy. Most of my guesses involve the net torque increase not being the full height-cubed increase.
• My modeling did not account for the torque-velocity curve, a phenomenon well-known to biomechanics study. It says that you can't exert as much force (we are interested in torque, but there is a similar force-velocity curve) on an object that is moving away from you. The faster it is moving away, the lower the force you can exert. Consequence: A faster swing due to faster body rotation reduces the torque that the body can apply; the taller golfer is operating on a lower portion of the torque-velocity curve.
• Another biomechanical effect is that torque does not appear instantly; it builds up on an exponential basis. (Sasho MacKenzie's models use a time constant of 20msec, about 7% of the entire downswing.) My modeling does not have fine enough control of torque vs time to reflect torque ramp-up. In reality the maximum torque is not operating the whole time, but it is for the model. Why is this a height issue? Because the modeled tall golfer's faster swing takes less time -- 283msec vs 300 msec. So the ramp-up takes a greater percentage of the tall golfer's swing than the shorter golfer's.
• A taller, heavier golfer may need more effort to stay in balance. Balance often implies opposing muscles exerting force (e.g., tricep as well as bicep in the example shown), thus reducing the advantage obtained by the bigger muscles and longer lever arm.
There may also be psychological or social effects at work. (For instance, the taller golfer doesn't have to work as hard for an advantage, and so doesn't work as hard).

Or perhaps my assumptions are incorrect leading to the cube law for effective torque vs height. But a square law or even a proportional law would be enough to give the taller golfer a distance advantage.

### The double pendulum model

I took a certain amount of flak for the model I used and how I used it. Let me deal with those criticsms here. But I'm going to try to stick to points that deal with the question we are trying to answer, not just gratuitous criticisms of one model or another. Let's face it. No model is exactly the golf swing. You want to choose the model to deal with aspects of the swing that are relevant to the question being asked.

I am using the double pendulum model, a 2-dimensional model. It is old, and has been superseded in detail by 3D models. But there are a lot of questions where the 3D models just add detail to the basically correct information we learned from the double pendulum. And sometimes it adds no new information; the 2D model is accurate, as far as the particular question is concerned.

So let's look at the question we are asked: Why do taller golfers tend to driver farther than shorter golfers? If the detail added by a more complex model does not distinguish taller from shorter golfers, then it does not matter to this discussion.

For instance, Brian Manzella posted, "Wrist torque is never zero." Let me respond...

I have seen no model advocate a lot of in-plane active wrist torque. Those qualifiers -- in-plane and active -- are important.
• The 2D double pendulum model considers only in-plane torque. If the 3D models show lots of beta and gamma torque but not much alpha torque, then they are not contradicting the 2D model. They are just adding information that any 2D model would not address.
• There are three sources of in-plane wrist torque: (1) Active muscular contraction. (2) Passive hard-tissue interaction, mostly the limited range of motion for wrist cock. (3) Inertial torque applied by the club to the wrist hinge, which is unopposed by either #1 or #2. I set #1 to zero, and let #2 and #3 just happen; the model does account for them.
I haven't seen any model of a good swing that has large amounts of wrist torque that is both in-plane and due to active muscle contraction. And I will argue that, for purposes of answering the question, "Why do tall players tend to hit the ball farther?" other torques are not very interesting.
• Beta torque will happen; it is needed to keep the club on-plane, no matter how tall the golfer. Small variations in this will not affect in-plane clubhead speed.
• Gamma torque will happen; it is needed to square the clubface at impact, no matter how tall the golfer. Small variations in this will not affect in-plane clubhead speed.
So, to a first approximation, I don't see a problem with the model I used. I'll be glad to back off from that if shown a better, more detailed model that tells a different story when answering the question at hand. So far, I haven't seen that, and I am reluctant to respond further until I do.

### Extreme case

When I re-posted the article with the new section on allometric scaling, the discussion started again. Most of it was people who didn't notice the first time, but still clung to the "longer lever, bigger wheel" way of thinking. In support of that agenda, Rick Marcy asked, "Would you not agree that a 10 foot tall skinny man would have a faster swing if done correctly than DJ for example?" Very interesting question! Sometimes analyzing an extreme case can teach us a lot. So let's work this one.

We have to start by tightening the definition of "10 foot tall skinny man". Here is what we will use for the analysis:
• DJ (Dustin Johnson, a tall, big-hitting PGA Tour player and #3 golfer in the world at the time of this writing) is 6'4" tall and weighs 190 pounds. Based on scaled norms, I'm going to use 26" for the arm length in the calculations.
• Our tall comparison golfer, Skinny, is 10 feet tall. The height ratio we need for our calculations is 10'/6'4" = 1.58. Skinny is 1.58 times as tall as DJ.
• We will further assume that Skinny's limbs are all 1.58 times the length of DJ's, so he is in proportion lengthwise.
• But he is skinny! Let's make his cross-section at any section (arm, leg, torso) the same as DJ's. That has consequences for weight and strength:
• His weight is not proportional to the cube of his height, but rather directly proportional. That's because volume is proportional to height times cross-section. The height ratio is 1.58, but the cross section does not change.
• The force exerted by his muscles is the same as DJ's, because the cross-sectional area of the muscle is the same.
Let's go ahead and see the effect of one change at a time, as we plug changes into the computer model. We start with DJ, and plug in new arm length, club length, weight, and strength to get to Skinny.

 What changed? New clubhead speed Comments Start with DJ 121 mph Tweaked the torque until, at 80 ft-lbs, we got a clubhead speed over 120mph Lengthen arm to 41" 101 mph Lengthen proportional to height. We lose a lot of clubhead speed. Wingspan alone doesn't work. Lengthen club to 48" 104 mph It helps, but not much. OK, to hell with conformance to the rules. Let's lengthen the club proportionally to height. Lengthen club to 71" 120 mph Got us back almost to DJ. Weight to 300 lb 114 mph Big weight gain, but it's only proportional; he's skinny. Lost speed again. Torque gain averages somewhere between the present 80 ft-lb and 126 ft-lb Between 114 mph and 138 mph Big gain possible! See *** below for why the torque increases, and by how much.

Again, we see that almost everything we do to account for height loses clubhead speed -- until we look at the implication of increased torque driving the longer arm lever. Only then does clubhead speed show a potential increase. It is given as a range, and the bottom of the range is lower than DJ's clubhead speed. But Skinny's clubhead speed is probably higher than the bottom of the range, and the range goes well above DJ's.

So yes, Rick, 10-foot-tall Skinny will likely develop more clubhead speed than DJ. But it isn't the wingspan that does it. Without a torque boost above what DJ's muscles and joints generate, wingspan gives a speed loss, not a gain. But larger size allows Skinny to develop more torque to swing the longer lever; that and only that develops the increased speed.

 *** Here's why the torque increases, and why it's hard to compute exactly how much. (Notice that the torque increase is given as a range, not a single number.) The torque at each joint is the force the muscle can generate times the moment arm. (Reviewing, moment arm is the distance between the center-pivot of the joint and the line along which the force acts.) The force itself is the same as it was for DJ, because Skinny has the same cross-section as DJ for each limb. As for how we find the moment arm, look at the diagram: On the right, the moment arm D is parallel to the length of the forearm. The length of the limb is proportional to the height, so D will be 1.58 times what it was for DJ. Since the forces are the same, the torque will be 1.58 times what DJ generates. If all the joints were like this, then the shoulder torque for the computer model would be 80*1.58=126 foot-pounds. On the left, the moment arm d is perpendicular to the length of the forearm. Skinny's cross-section details are identical to DJ's, so Skinny's d is the same as DJ's. If all the joints were like this, then the shoulder torque would be the same 80 foot-pounds as DJ. Some joints are lengthwise, some are crosswise, and most are in between for most of the swing. Don't forget; joints move during the swing; that's their job. And as they move, we can expect changes in whether they are lengthwise or crosswise or in between. So, netting out over all the joints and the whole swing movement, the effective shoulder torque is somewhere between 80 and 126 foot-pounds. And that means the clubhead speed is somewhere between 114mph (what we got with 80 foot-pounds and Skinny's dimensions) and 138mph (the result with 126 foot-pounds). So now you might well ask: why did we have single number, not a range, with the original study? That was because we didn't assume the taller golfer was skinny; we scaled him up proportionally. So it didn't matter whether the moment arm was lengthwise or crosswise; all the dimensions were scaled up by the same amount.