We have seen an explosion in the use of numbers and statistics in all of sports, including golf. It is not uncommon for announcers to note how a tournament leader is beating the field in total driving or strokes gained putting. This is a great step for the game in general, as we can quantify just how well a player is performing. However, with so much data available, it is hard to distinguish between what is meaningful and what is just coincidence. What is worse, data can be manipulated and selectively chosen to deceive others into believing something is true when in fact it is not. Therefore, I believe we should begin our discussion about golf statistics by discussing two common ways that numbers can be misleading.
Case 1: The hot hand gets cold1
Coming into the 2014 Masters, Bill Haas had to be confident in his game. He had won at least once every year since 2010, including the Tour Championship in 2011, and leading into Augusta Haas already had three top-10 finishes on the year. So when Haas shot 4-under 68 to lead the Masters in round one, it looked like he was taking his game to the next level of being a major competitor. All of that changed after round 2. Haas shot 78 in the second round, 10 shots higher than his first round. Clearly Haas could not handle the pressure of being near the top at a major championship. Or is that really the reason Haas fell back?
Have you ever noticed that in most golf tournaments the leader in round one is not the winner of the tournament? Most fans and commentators say this is because the pressure of winning wire-to-wire is too much to handle. This may contribute, but there are other factors in play — particularly the role of luck. To score well, a golfer has to hit good shots or get lucky breaks, and often a really good round is a combination of the two. Knowing nothing else about Bill Haas except that he shot 4 under in round 1, you would expect his good score came partly from good shots and partly from good luck. Now if you were to predict what he would do in round 2, you would expect him to hit good shots again, but you would not know whether he would get good breaks or not, so you would guess an average amount of luck. Good shots and good luck leads to better scores than good shots and average luck, so you expect Haas to do worse in round 2 than round 1 just based on statistics. In fact, if we look at the 17 players who broke par in round one, only Bubba Watson (the eventual champion), Jonas Blixt (eventual T2), Jordan Speith (eventual T2) and Fred Couples (always plays well at Augusta) broke par again in round 2. Do you think it is more likely that the other 13 players played worse in round 2 because they couldn’t handle the pressure or because they had good luck in round 1 and only average luck in round 2? I suspect the latter.
Statisticians call this phenomenon regression to the mean. It works like this: suppose you flip a coin 4 times and get 4 heads in a row. Now let’s say you flip that coin four times again. We expect you to get 2 heads and 2 tails on average, so regression to the mean means that you will probably get fewer than 4 heads this time. The same thing happens in golf. Bill Haas’s scoring average in 2014 was 70.8. After shooting 68 in round 1, he is well below his average, so we expect his score to be higher in round 2 than round 1. After shooting 78 in round two, we see his score is much higher than his average, so we expect his third round score to be lower than his second round score, and indeed he shot 74. These swings up and down the leader board likely have nothing to do with Haas getting hot, feeling the pressure and collapsing, then relaxing and playing his normal game, but they are more likely due to the random bounces inherent in the game. So next time you are on a hot streak and then make a big number, it may not have anything to do with losing your mojo; you may simply be regressing to your mean.
Case 2: False Leaders
Even on the PGA tour, getting up and down on a long bunker shot is not easy. In 2017 tour players took over 2,300 bunker shots from beyond 30 yards, and got up and down only 780 times. That is a success rate of less than 34%.
Rickie Fowler is above average even on the PGA Tour. When you look at his recent scrambling from long bunker shots though, he is phenomenal. In 2017 he got up and down from 30+ yard bunker shots an incredible 70% of the time!
Despite Rickie’s phenomenal performance, he was only 5th best on tour in 2017. The tour leader, Nick Watney, got up and down from 30+ yard bunkers 100% of the time! Based on these statistics, you might advise Nick Watney to aim for long bunkers, since he never misses!
Of course this last statement is ridiculous, but why? The numbers say that Nick Watney never misses from long bunkers, so why shouldn’t we expect him to get up and down every time? The reason we shouldn’t believe this statistic is because I left out an important piece of data: Nick Watney only attempted 4 long bunker shots in 2017. With only 4 attempts, there is a pretty good chance that he got really lucky to get up and down all 4 times.
This is an example of another statistical phenomenon: the law of large numbers. It states that the fraction of successes approaches the true probability as the number of attempts gets very very large. You can also think about the opposite direction: with only a few attempts, the success rate can fluctuate wildly.
The law of large numbers tells us that we need a lot of data for statistics to be meaningful. Nick Watney’s 4 shots don’t tell us a lot about his long bunker game. On the other hand, the 2,300 bunker shots taken by all players says that the average scrambling rate of 34% is more reliable. Scientific studies often require still more trials for their desired precision.
Assuming we are more confident in the 34% number, we can ask a different question: what is the probability that an average tour player attempting 4 long bunker shots makes all 4? The calculation is pretty simple:
\[ P(4/4) = (0.34)^4 = 0.013 = 1.3\% \]
In other words, if we gave 100 pros this challenge, it would be completely unsurprising to see one of them get up and down all 4 times. Given that 190 players attempted 30+ yard bunker shots, it is not at all surprising that someone on the PGA Tour made 4/4 bunker shots – the fact that it was Watney may just be pure luck.
What about Rickie? He made 70% of his long bunker shots, but he had 10 attempts, almost twice as many as Watney. Is this a sign that he is really good, or was he just lucky too? We can try to answer this by asking, what is the probability that an average tour player makes 7 out of 10 bunker shots? This probability is a little more complicated, but still very simple to compute:
\[ P(7/10) = {10 \choose 7}(0.34)^7(0.66)^3=0.018=1.8\% \]
So the probability of an average player making 7/10 bunker shots is nearly the same as an average player making 4/4! We would expect roughly one out of every 50 average players to do as well as Ricke did in 2017.
So based on this analysis we cannot say that Rickie Fowler or Nick Watney are any better than average. It is more likely they both were just lucky. Looking at a larger sample size supports this conclusion. In the last 5 years, Fowler has made 21 out of 58 long bunker shots (36%), while Watney has made 12 out of 41 (29%). While this is more data than before, it is still not enough to say with certainty who is the better long bunker player.
Finally, you may ask if there was anyone in 2017 whose long bunker performance can’t be explained by just chance. While we can’t definitively answer that question, we have two contenders.
First, there is Seung-Yul Noh, who got up and down 8 out of 9 times from 30+ yard bunkers. The probability of an average player doing this is 0.1%, which is ten times lower than Fowler’s or Watney’s numbers. However, there are 189 players listed in the tour stats; if we assume they are all average and compute the odds that someone among 189 gets 8 out of 9 up and down, we get a probability of a little more than 50%2. Thus although Seung-Yul Noh’s results are fairly unlikely, we would expect someone on tour to do just as well about every other year.
The second contender for a truly remarkable performance is Jim Herman, though his performance is on the opposite end of the spectrum. Herman attempted 23 long bunker shots, more than any player we have considered so far, yet he only got up and down twice. The odds of the average tour player getting up and down only twice out of 23 times is about 0.5%, which is also very low. However, we should again ask, if we gave 189 average players 23 bunker shots, how likely is it that someone would have a bad day and get up and down only twice? It turns out we would expect to see at least one out of 189 do this poorly nearly 60% of the time, just by bad luck alone.
These last two examples illustrate how hard it is to distinguish between skill and luck. It is possible that Seung-Yul Noh is especially good at long bunker shots, and Jim Herman is particularly bad. On the other hand if we assume that everyone on tour has the exact same ability on long bunkers, seeing 1 out of 189 have a really good year or a really bad year is not so unlikely. For another example, winning the lottery is very very unlikely, and we know that there is no skill involved. Yet we aren’t surprised when we hear that someone has one the lottery, because we know that there must be a winner. We wouldn’t say that because someone won, the winner must be very skilled at picking numbers. Similarly, we shouldn’t be so surprised that some equally skilled players have good years and some have bad years when dealing with small sample sizes. With just a few data points, a lot can be explained by chance. As we collect more data, the likelihood that chance plays a role decreases, and the true skill levels are revealed.
Applying it to your game
Let’s say you go out to play golf. Your front 9 is terrible; several strokes above your average. Should you try to play differently on the back 9? Probably not. You bad front 9 is probably a combination of bad shots and bad luck. Regression to the mean tells us that since there is no reason for your luck to be especially good or bad on the back 9, your back 9 will probably be better than your front 9 without you making any changes.
Suppose you are playing a pretty decent round, when suddenly out of nowhere you hook a tee shot out of bounds. What should you do next? Try to make a swing change? Hit only 3-woods for the rest of the round? Aim down the far right side of the fairway?
First of all, you should relax. Next, remember the law of large numbers. If you have only hit one bad shot, you do not have a sample large enough to conclude that something is wrong with your swing. My instructor always told me to wait until I had hit at least 3 of the same bad shot before I try to make any changes. Also remember the regression to the mean. Since this was an aberration, your next shot will probably be closer to your average.