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The last post talked about how sometimes numbers can be misleading. In this post we want to look at the best ways to approach using numbers to draw conclusions so we can avoid those issues. As we attempt to study more analytical topics in golf it is important to understand how mathematicians, physicists, engineers, and other scientists go about solving complex problems.

Mathematical Modeling

Whenever an interesting question is posed, mathematicians and physicists ask, “Is there a way I can model this using math?” A simple example of a mathematical model is calculating when you will arrive at your destination when you take a long road trip. One of the first equations taught in introductory physics is speed = (distance)/(time), or rearranging the variables, time = (distance)/(speed). So if you are driving from Minneapolis to Chicago (about 500 miles) at a speed of about 70 miles per hour, it will take you (500 miles)/(70 mph) = 7.14 hours = 7 hours, 9 minutes to arrive.

The world of physics and applied mathematics is filled with mathematical models much like our road trip model but with more complicated techniques. Models can be very useful, but also have their limitations. Here are some things to keep in mind when looking at models:

  • Models always have assumptions built in to simplify the problem — it is important to be aware of when these assumptions are valid. In our road trip model, we assumed that the speed of the car didn’t change throughout the trip; it was always going at 70 mph. This assumption is a bit unrealistic for a long trip — there will probably be bathroom breaks, traffic jams, and road construction that will slow us down. However, adding those pieces in makes the model much more complicated, if not impossible to formulate since some factors like lane closures due to accidents are impossible to predict. However, if most of the time we are going 70 mph, and there are also times we are going 80 mph to balance out the times we have to go slow, and if we don’t really care if our prediction is off by 15 minutes, then our model above still gives a good rough estimate of how long it will take us to get to Chicago.

  • No model is perfect, and there are certain points where the model will break down. Based on our model above, if I attach some illegal rocket boosters to my car so I can go 500 miles per hour, I will get from Minneapolis to Chicago in only 1 hour, right? Well, technically yes, our model does say that, but if you really did try to go 500 miles per hour along Interstate 94, chances are you wouldn’t make it to Chicago in an hour — you would probably crash your car within the first mile and then (if you survived) have to walk the rest of the way. Clearly, at 500 mph, our model does not agree with reality very well (at least for cars; it is still a good model for planes flying to Chicago at 500 mph). Thus at very high speeds the model is not very accurate. Similarly, if we drive at 10 mph, our model says it will take us 50 hours to get to Chicago. This is probably not very accurate either, since 50 hours would mean more than two days of continuous driving, and we will need to stop and get some sleep, which will add at least another 16 hours to our trip. So our model breaks down at both very low speeds and very high speeds. It is a better model for a more reasonable range of driving speeds: 30-100 mph.

  • Using models we can gain insight into why the world behaves the way it does. Why is it that it takes so much longer to drive from Minneapolis to Chicago than from Milwaukee to Chicago? Using our equation, we can see that because Milwaukee is only about 100 miles from Chicago, no matter what speed we choose (as long as it’s the same in both trips) we will get to Chicago 5 times faster if we start in Milwaukee. Why is it that traveling county roads to Chicago takes so much longer than taking the interstate? Our equation tells us that going the same distance but at a slower speed will take us more time. Thus our model tells us that faster speeds and shorter distances decrease travel time. This is a pretty obvious result, but as the models and questions get more complicated our intuition becomes less reliable, and the mathematical results become more useful.

  • Models can help us answer more complicated questions. If I go 50 mph for the first 100 miles, then 80 mph for the rest of the trip, will I get to Chicago faster than if I go 70 mph for the whole trip? We can calculate travel time = (100 miles)/(50 mph)+(400 miles)/(80 mph) = (2 hours) + (5 hours) = 7 hours. So this method is faster by about 9 minutes (though your risk of a speeding ticket is higher).

Some other examples of models

Every time I examine a new model I judge its usefulness by the factors enumerated above: What does the model do/say? What does it assume? Where does it break down? What insights or extensions can be gained from this model?

Below I apply these questions to a few sample models from a variety of fields.

Par as a model

  • What it does: It predicts what a good player will get on a given hole.
  • What it assumes: Good players will take one, two, or three shots to get on or around the green depending on the length of the hole. Once they are on or around the green, they will take two shots to hole out.
  • Where it breaks down: The definition of “good” is ambiguous. Very good players may average well under par. There is a high variance in players’ scores from hole to hole. Overall this model is a poor predictor of actual scores.
  • Insights and extensions: Comparing my score to par tells me how my game compares to this benchmark. Comparing my score on specific holes tells me where I struggle the most. While this is a pretty basic model for golf, in a future post I will show that even with just a slight extension we can learn a lot about course strategy.

Compounding interest

  • What it does: It predicts how much money you will have if you invest a certain amount on a regular basis in an investment that increases its value at a given rate.
  • What it assumes: You will continue to invest in your account regularly, your interest rate will remain constant.
  • Where it breaks down: Where any of the assumptions are not met. When emergencies require the savings to be spent. We can assume this model works best when the investment is very reliable and we are looking over a long period of time.
  • Insights and extensions: Compounding interest shows why it is important to invest early and often throughout your life.

Newtonian gravity

  • What it does: Predicts the forces between two bodies based on their masses, and can be used to predict the motion of these bodies.
  • What it assumes: Masses are not enormous (sun size ok — black hole size not ok), nor are speeds near the speed of light.
  • Where it breaks down: When either of the above assumptions are not met. When that happens we need to use Einstein’s theory of general relativity. However, for nearly all applications, Newton’s theory is very good.
  • Insights and extensions: Too many to mention. Newton’s law of gravity explains why apples fall, why the moon orbits the earth, why tides form, etc. This model was accurate enough to put men on the moon.

Nationwide polls for US presidential elections

  • What it does: Predicts the results of presidential election by polling a nationwide sample of voters
  • What it assumes: It assumes that the sample polled is representative of the actual electorate, and that the person leading in the polls will win the electoral college.
  • Where it breaks down: Since the presidency is determined by the electoral college, winning the election is only somewhat tied to the national popular vote. If the polls are close, it is more likely that someone leading in the polls could still lose the electoral college (considering the polls in each state is a better model because its assumptions are more in line with the election procedure).
  • Insights and extensions: In addition to predicting the presidency, tracking the national polls as a function of time can give insight into which events affect voters’ opinions.

Applying it to your game

As you read instruction articles or take lessons, notice all the models that are being used. Models are everywhere in golf, such as the one-plane/two-plane swing model, Phil Mickelson’s “hinge and hold” model for short game, putting on an arc vs. straight-back-straight-through, etc. Think about how these models relate to what I have discussed above. It should help you determine what questions to ask your instructor and understand when the model is useful.

As we continue to discuss a data-driven approach to golf on this site, I will return to these four questions in presenting or evaluating models in golf.