What Matters for Driving Distance?
Dave Tutelman  June 5, 2012
I received an email from Reinout Schotman saying:
In golf, as you know, many dogmas (mantras) exist, which are not
properly supported by data.
I downloaded some performance data of PGAtour.com and did some
assessments.
Surprisingly I found no (statistically significant) correlation between
Launch Angle and total distance and neither for Spin and total
distance. The only thing that seems to matter is to hit it hard (swing
speed) and on the sweet spot (smash factor). And obviousely straight
and in the right direction...
The dogma "High Launch, Low Spin" does not seem to hold, at least for
PGA Tour Pro's. Is this known?
I have data and graphs in Excel to support, if you're interested.
In
support of his conclusion, Reinout presents the following scatter
plots, along with a beststraightline fit to the data. (Click on the
thumbnail for a fullsize view.)
Ball speed

Launch
Angle

Spin

I am not much surprised by Reinout's conclusion. He has correctly
identified the three launch parameters as the primary determinants of
driving distance:
 Ball speed (a combination of clubhead speed and smash factor)
 Launch angle
 Spin
Of the three, it is clear that ball speed is by
far the most important. Given a ball speed that a player can produce,
launch angle and spin tend to be just tweaks to the distance. For more
detail, see my article on driver
optimization.
To get further confirmation from the statistics, let's look at the
trend line. Reinout drew his conclusion about lack of correlation from
the slope of the trend line; the larger the slope, the greater the
correlation. But even more telling is the Rsquared value for the line;
it is a value that measures the statistical significance of any
correlation you notice. (Zero means no correlation, and one means
perfect correlation.) Here are the slopes and the Rsquared values for
our three launch parameters:

Ball
Speed

Launch
Angle

Spin

Slope

1.3
yards per mph

.06
yards per degree

.001
yards per rpm 
Rsquared

.76

.00008

.0008

We see ball speed has an advantage not only in slope but in the
statistical significance of that correlation. And not just a small
advantage either; the advantage is quite a few orders of magnitude in
both slope and Rsquared.
So the Tour stats say that ball speed affects distance way more than
does launch angle or spin. But can we explain
it  can we describe it more in mechanical terms than statistical?
Good challenge!
A Five Percent Solution
Reinout's conclusion was for total distance  though the PGAtour.com data also
includes carry distance. I have a computer simulation tool that can
answer the question for carry distance: the TrajectoWare Drive
computer program. Let's see how each of these parameters affects the
carry distance. We will also note the angle of descent, which is the
biggest single factor in how far the ball runs after landing.
We will assume a fairly typical PGA Tour golfer. He has a clubhead
speed of 115mph and uses a driver with a 10º loft. If we plug his
impact into the computer program, we get:
Launch
Parameters

Results

Ball
Speed

Launch
Angle

Spin

Carry
Distance

Descent
Angle

168.5
mph

8.9º

3195
rpm

277.2
yd

38.3º 
These will be our reference values, and we will see how they vary as we
vary the launch parameters. We will change the launch parameters up and
down by 5% of their value.

5%
down

5%
up

Total
Variation


Parameter
value

Carry
Distance

Descent
Angle

Parameter
value 
Carry
Distance 
Descent
Angle 
Carry
Distance

Descent
Angle

Ball
Speed

160.1
mph

260.0

36.0

176.9
mph

293.4

40.5

33.4

4.5

Launch
Angle

8.5º 
276.2

37.6

9.4º 
278.3

39.3

2.1

1.7

Spin

3035
rpm

277.1

37.0

3355
rpm

276.7

39.7

0.5

2.7

What do we see here? Ball
speed is fifteen times as important as anything else in producing carry
distance.
That is the simple confirmation of Reinout's observation. If I were
looking at raw data from tour players instead of a carefully controlled
computer "experiment", the playertoplayer variation would swamp out
any effect of launch angle or spin. Only ball speed would show up as a
statistically significant factor.
But that is just carry distance. Could launch angle or spin produce
enough variation in runout after landing to affect the statistics? It
doesn't look like it. True, ball speed has the biggest effect on the
angle of descent, and a high ball speed will result in a steep descent,
limiting rollout. But it is not going to come close to any significant
effect on the comparison; the carrydistance advantage of ball speed is
overwhelming.
A Standard Deviation
Reinout looked at the table above and said that I was giving too much
weight to ball speed. A 5% variation in ball speed is very difficult to
accomplish, whereas one can effect much larger changes than 5% in
launch angle and spin. He has a point! So I repeated the calculation,
using a carefully considered variation for each of the three launch
parameters.
What I did was compute the average and standard deviation for each of
the parameters. This was not hard, because Reinout had already
transferred the data from PGAtour.com
to an Excel spreadsheet that he shared with me. I just had to add a row
for standard deviation; he already had computed the averages. The
spreadsheet contained data from 186 PGA Tour players, and included the
three launch parameters, carry distance, and total distance (among
others, but angle of descent was not included). Below is a table
reflecting the bulk statistics.

Launch
Parameters

Results


Ball
Speed

Launch
Angle

Spin

Carry
Distance

Descent
Angle

Average

167
mph

10.7º

2700
rpm

273
yd

No
Data

Standard
Deviation

5.6
mph

1.3º 
222
rpm

11.5
yd

Percentage

3.4%

12.1%

8.2%

4.2%

I have also included a percentage, which is the standard deviation as a
fraction of the average. This number confirms Reinout's contention that a
blanket ±5% will give misleading values. Ball speed (the dominating
parameter in our first try) has a standard deviation less than 5%,
while launch angle and spin are both more than 5%.
So... Let's repeat the table of sensitivities we calculated above, but
this time with the "Average" values as our reference values, and vary
them up and down by an amount equal to the standard deviation. That
should give us a perfectly representative set of carry distance
variations, for the 186 players in the data.
(Note that the computer model gives a carry distance of 278.2 yards for
the launch conditions in the "Average" row. That is not the same as the
average carry distance in the data: 273 yards. We are not yet in a
position to discuss this discrepancy; we'll just accept it and do the
math.)

1
Std. Dev. down

1
Std. Dev. up

Total
Variation


Parameter
value

Carry
Distance

Descent
Angle

Parameter
value 
Carry
Distance 
Descent
Angle 
Carry
Distance

Descent
Angle

Ball
Speed

161.4
mph

266.5

36.0

172.6
mph

289.6

38.9

23.1

2.9

Launch
Angle

9.4º 
274.2

34.9

12º 
281.5

39.8

7.3

4.9

Spin

2478
rpm

277.0

35.8

2922
rpm

278.6

39.1

1.6

3.3

The ball speed still dominates, but not by as much as before. Instead
of being 15 times as important, it is only three times as important as
launch angle. But that is still a large margin. So the conclusion still
stands.
While we are looking at the data, Reinout's original conclusion was
that launch angle is not significantly correlated to distance. This
table says that, while ball speed is three times as important, there
should be a nonnegligible effect from launch angle. Why did Reinout
conclude otherwise? An answer may lie in the descent angle. Note that:
 Our table deals with carry
distance.
 Reinout's conclusion is for total
distance.
In our new table, we have amped up the relative effect of launch angle.
The first table had ball speed and launch angle at the same percentage
difference; this new table has the percentage variation of launch angle more than three
times that of ball speed. As a result, launch angle variation gives
almost twice as much difference in angle of descent as does ball speed.
And angle of descent hurts rollout after landing. Let's check this
hypothesis against the data  which has columns and graphs for both
carry distance and total distance. Here are two graphs from Reinout's
spreadsheet, showing the correlation between launch angle and distance.
Carry Distance vs Launch Angle

Total distance vs Launch Angle

This pair of scatter plots provides a very interesting result!
Specifically, look at the bestfit straight line and its slope. There
is a significant slope to the carry distance line: about an extra yard
of carry per degree of added launch angle. But this slope disappears
completely when we fit total distance
to launch angle. This says that there is a substantial negative correlation
between launch angle and rollout after landing. When we raise the
launch angle we may improve carry distance, but we do nothing for total
distance. And that observation is readily explained by our conjecture
about angle of descent.
A Fitting Observation
Let's recognize a very important fact: Tour players all have drivers that were fitted to their swings by expert clubfitters. Not one of them plays with a random, offtheshelf driver, but rather a driver optimized to that golfer.
The fitted driver is fairly important in evaluating the effect of launch
conditions on distance. Two
important points:
 It is pretty obvious that, all other things being equal,
additional ball speed turns into additional distance. A properly fitted
driver means there is nothing getting in the way of clubhead speed
turning into ball speed turning into distance. So let's not even consider the effect of ball speed; that's a given.
And it is easily confirmed by playing with the computer model, or by
noting that the statistical correlation is very strong (Rsquared is
well on its way to one; it is .76).
 For the rest, we need to understand how a proper fitting relates to the computer model. Let's look more closely at this.
Here
is a picture of the "launch space" for a clubhead speed of 86mph. It is
a threedimensional graph of distance for values of launch angle and
spin. The resulting surface is like a sheet of paper that has been bent
down at two diagonally opposite corners. (Those corners are
highlaunch/highspin and lowlaunch/lowspin. The turneddown corners
are obviously short carry distances, where nobody would ever want to
design a driver.) There is a round "ridge" of maximum distance, lying
diagonally across the other two corners, and that ridge is sloped
slightly upwards toward the highlaunch/lowspin corner.
A properly fit driver should be designed to lie along this ridge, for the golfer being fit.
I mentioned that this example represents a clubhead speed of 86mph.
Yes, I know that Tour speeds are much higher. I have played with these
graphs for
golfers from senior women to longdrive competitors. The shape is
always the same; only the numbers on the axes change. That is, a Tour
golfer's graph would have lower launch angles, lower spin, and higher
carry distance. But it would still have those two corners tucked down,
and a slanted ridge across the other two corners.
Because the shape is the same, we can learn from this graph what we
need to know about the fitting process. (I had this graph on hand from existing research,
and didn't want to bother going through the tedious work involved in
calculating and creating another publicationquality launch space
graph.)
Fitting a golfer for a driver involves finding a set of components that
work for that golfer's swing. The fitting parameters include
characteristics of both the club and the golfer, and include:
 Loft.
 Club length.
 Weight and balance.
 Shaft flex and also flex profile (how the sitffness changes over the length of the shaft).
 The golfer's clubhead speed.
 The golfer's angle of attack, and wrist bowing or cupping at impact.
 The golfer's ability to repeat the same swing; for Tour players, this tends to be very good.
The savvy clubfitter will start with a rough cut at loft, and optimize
everything else. Then he will turn back to loft, and find the ideal
loft for the golfer. This process can be looked at in launch space.
(Even though I will admit that I know of no professional clubfitter who
actually looks at driver fitting as a launch space graph.)

Here is another look at the 86mph launch space, with some clubfitting
information added. First of all, the "ridge" of maximum distance is a
red dotted line.
Then a sequence of real drivers with various lofts are plotted in
black. We can do this because, knowing the loft and the clubhead speed,
we can compute the launch angle and spin. For instance, a driver with a
fourteendegree loft swung at 86mph will propel the ball at a launch
angle of 12º with a spin of 3300rpm. So we plot a black dot on the
surface at [12º, 3300rpm] and label it "14". (And we see that the carry
distance is 188yd, the height of the launch surface of the graph at
that point.)
I computed points from 8º to 24º, and show them on the graph as black
dots connected by a black line. This is the line of feasible
performance for the driver of someone who swings the driver at 86mph at
a 0º angle of attack. We can now see fitting that golfer for driver loft as a mathematical process: find the highest point on the black curve, and note the loft that point corresponds to.
Think about this representation of fitting a golfer for driver loft.
Get comfortable with visualizing it on a 3D graph like the one above.
What follows depends on it.
Any optimization of a continuous function (like the black curve in the
graph) involves finding a place where the curve is horizontal; it
doesn't go up or down as you change where you are by small amounts. If
you look at the height of the curve (the carry distance) for lofts of
14º, 15º, and 16º, you see distances of 188, 189, and 188 yards
respectively. That is almost no change at all.
The basic mathematical approaches to optimization depend on this. They look for a "flat spot" in the function.^{[1]}
In this case, we are looking for a flat spot in the funtion of carry
distance vs loft  the black line. The middle of the flat spot is 15º,
so that is what our optimum driver should be.^{[2]}
Now take a look at the surface near the maximum carry distance. The
ridge is only sloped a little bit near the flat spot in the loft
curve. That means that distance is flat not just with loft, but also
with launch angle and spin. Let's put this back into Reinout's
observation. Since
the Tour players have wellfitted drivers, their performance in launch
space does not change much with launch angle and spin.
This is not the case everywhere in launch space. It is definitely the
case at the flat part of the curve  where a wellfitted driver lives.
But let's look at a really poorly fitted driver and see if that is
still the case. Consider our 86mph swinger trying to use a 24º driver;
at the 24º portion of the surface, small changes in spin give large
changes in distance. That is very different from what Reinout observed.
 For the maximumcarry loft of 15º, a change of 500rpm of spin gives a difference of only a yard or two.
 For a 24º driver (the "front edge" of the surface), a
change of 500rpm gives a difference of twelve yards, certainly a
significant difference.
 The
same turns out to be true for toolow a loft; for an 8º driver, a
change of 500rpm again gives a difference of twelve yards.
So Reinout's observation depends on a reasonably wellfitted driver,
certainly a reasonable assumption for a Tour player. And, if your own
driver is so illfitted that spin and launch angle make big differences
in your distance, you should be getting a new, properly fit driver post
haste.

Conclusion
Reinout Schotman has observed that, statistically from current PGA Tour
data, ball speed is a significant factor in driving distance, but
launch angle and spin are somwhere between insignificant and zero.
In this article, we have seen that:
 This is indeed an accurate statement.
 Computer modeling agrees that is the way it should be.
 This
is partly the result of Tour pros using the right driver that fits
them. If the driver were a really bad fit, then spin would be a
significant factor (and perhaps launch angle as well, though we didn't
explore that here).
Math addendum: Is this really valid?
Here's a mathematical fine point that you may or may not be interested in. If you're not into math, you don't need to understand  nor even read  this note. Skip it if it holds no interest for you.
While I was working on answering Reinout's question, I started to
wonder whether it is valid to compare a deterministic computer/physics
model with the type of statistical model Reinout gleans from the
PGAtour.com statistics. It seems to give reasonable answers, but it is
mathematically
suspect. Here's the problem; the statistical model and the computer
model are not the same, so it may be partly or perhaps even mostly luck that they
give the same answers.
 The computer model takes a set of launch conditions, and computes
a carry distance that physics says those launch conditions will produce.
 The
statistical model takes a set of data points, each of which
is an average of a season's worth of swings for one single professional
golfer. We then look at the distribution of those points. That is, the
carry distance for Bubba Watson's row on the spreadsheet is the average
of all his drives, the ball speed is the average of all his drives,
etc. All those drives are repesented as a single point in the scatter
plots  each point is the statistical summary for one player.
It may seem frivolous to question testing a deterministic
mathematical model with a statistical model. In science, theories are
tested that way all the time. When there is any randomness or outside
influences in the experimental results, scientists turn to the sort of
graphs Reinout presented  at least superficially. But this is substantially different. If we
were using statistics to test the mathematical model behind
TrajectoWare Drive, the statistical base would have one point per
measured drive  not one point representing a season's worth of
measured drives.
What does this do to the statistics we observe? At the very least, it
is probably making Rsquared much smaller than it should be. If each
data point were a single drive (rather than the average of many drives), I would expect the trend line slopes to
be pretty similar to what we see. But I would expect the points to line
up much better along that line, not scatter all over the page. And that
would result in a much better correlation of the random effects in the
experiment, an Rsquared closer to 1.0.
Why do we see so much spread in the data? Because even a single
player's driving statistics are not uniform. For instance, data will be
taken on holes where the player used a driver (certainly the intent of
Reinout's study), but also on holes where he used a 3wood, or perhaps
even an iron. Uphill and downhill. Into the wind, with the wind,
crosswind, and combinations thereof. What do we get when we average all
those drives into a single point? I honestly don't know. And that is
exactly the problem!
Think about this: When Reinout does the statistical curve fitting, he
is tacitly
assuming that the statistical fit will reflect repeated use of the
singleinstance computer model. But that assumption is mathematically
valid only if the computer model is linear. If the model is nonlinear,
then the distributions are warped by the nonlinearity, and the average
of the computed carries will not necessarily be the measured average of
carries. But we know (looking at the launch space surface) that the
function of carry distance for ball
speed, launch angle, and spin is not linear. Perhaps the restriction of
launch space implied by properly fitted drivers keeps us in a region
where the function is close enough to linear that the linearity
assumption does not do any damage.
So the fact that the computer model continue to give the same
information as the statistics might be coincidence.
More likely, it is a rough approximation to what we would get if we
gathered the statistics properly  one drive per data point. Either
way, we got lucky.
Notes:
 That is why calculus is used for
optimization; the first derivative of a function is its slope. To find a
maximum or minimum of the function, we differentiate the function. Then
we set the derivative to zero and solve for x (and/or y, z, etc). The values of [x, y, z] where the derivatives are zero is the maximum or minimum of the function.
 Actually, that is an
oversimplification  but it is a good starting point. For instance, I
usually back off about a degree  from 15º to 14º in this case  to
give up a little carry in favor of runount, because I know lower loft
gives lower angle of descent.
Last
modified  June 18,
2012
