How Streaky is Tiger Woods?

The first PGA Tour event of the year, each year, is the tournament whose ancestral name is the Tournament of Champions. (It currently sports a "Hyundai" before that phrase, having emerged from the depths of being called the "Mercedes Championships.") It's played in Hawaii, and its field consists of all and only those players who won a tournament the previous year (and who want to play in it), as you might expect from the name. This year, Dustin Johnson won the Hyundai Tournament of Champions, but he did it in slightly odd fashion. Or, well, the whole tournament happened in slightly odd fashion. For some reason, apparently, the Tour decided to shift the TOC to a Monday finish. But then Friday, Saturday, and Sunday all featured ridiculously high winds that made it impossible to play any golf. (Actually they played a few holes and then cancelled play, erasing the scores that had been put up in the little half-rounds.) Eventually they were able to play on Monday, when they got in 36 holes, and on Tuesday, when they got in another 18, making a 54-hole tournament that they could call official. And Dustin Johnson won it. My dad and I had been speculating that, if Dustin were to win this week's Hawaiian Open with a nice low score, he might set an all-time record for furthest under par in a span of seven days, since he would get in an unusually high seven rounds in that span.

Except... a few minutes ago, Dustin withdrew from the Hawaiian Open with flu-like symptoms. He had been playing rather miserably anyway, and was going to miss the cut. Ah well.

It got us thinking about how likely it is for a player to win in back-to-back weeks. You could imagine that players would feel awfully drained after a win (though not as much as if they didn't get any time off, as Dustin did here), but you could also imagine that someone who was playing well in Week X would be likely still to be playing well in Week X+1. The problem in trying to analyze this is that most players scarcely win any tournaments at all. Dustin Johnson, for instance, has 7 wins in 120 events (not counting this week, which'll push the denominator to 121). That's a winning percentage of 5.83%, or .058 in baseball format. That's low. If you assume that the two weeks are purely independent of one another, and that the 7-for-120 winning percentage represents Dustin's "true talent," he would have only an 0.34% chance of winning both of any two arbitrarily-chosen tournaments, including if those tournaments are in back-to-back weeks. In other words, if a player like Dustin Johnson played his whole career and never won in back-to-back weeks, we shouldn't be too surprised, and the difference between his having 0 back-to-back wins and having 1 or 2 of them wouldn't really be that significant.

There is one player, however, for whom that concern does not apply: Tiger Woods. He has 74 career wins in just 293 events, for a winning percentage of 25.3%, or .253, which is higher than that of the 40-120 1962 New York Mets. Each of the Mets' contests was against one other opponent, so without knowing anything else we'd expect them to have won 50% of their games. Each of Tiger's contests tends to be against over 100 other players, so knowing nothing else we'd expect him to win less than 1% of his tournaments. And yet, Tiger's got a better winning ratio than those Mets. Man they were bad, and man he's good. And, in particular, he's good enough to permit meaningful analysis of his winning tendencies when playing back-to-back weeks.

If you assume independence, Tiger should be winning both tournaments about 6.4% of the time that he plays in consecutive weeks. Also, he should fail to win a tournament about 55.9% of the time that he plays in consecutive weeks, leaving an expected 37.8% of the time that he'd win one but not the other. Over the course of his long career, he has played two weeks in a row 91 times. (Note that some of those windows overlap, i.e. if Tiger played three consecutive weeks, I count that as two separate windows, both of which include the middle week.) Our expected percentages under an assumption of independence would work out to about 5.8 two-win windows, 34.3 one-win windows, and 50.8 no-win windows. Instead, Tiger has 9 two-win windows, 28 one-win windows, and 54 no-win windows, which translate to percentages of 9.9%, 30.8%, and 59.3%, respectively.



This looks to me like data that suggests a bit of genuine streakiness: when playing in back-to-back weeks, Tiger is considerably more likely to have the same result, in terms of winning vs. not winning, in both tournaments than to have different results from one week to the next. Admittedly it's a small sample size, and not the most dramatic deviation from the "null hypothesis" numbers, but it's probably the biggest sample we'll ever get on the subject. I'm not sure this is the best way to get a confidence value, but if I just treat each window as one trial with the possible outcomes "two wins" and "not two wins," with a null hypothesis probability of "two wins" p = 6.4%, the odds of having 9 or more "two wins" outcomes in 91 trials would be about 12.6%. That's not typically considered "statistically significant," but it's enough to notice, I think. Put in more Bayesian terms, there'd be a 6.15% chance of getting exactly 9 successful trials in 91 with p = 6.4%, and a 13.88% chance of getting exactly 9 if you assume that p = 9.9%, so the evidence from Tiger's record suggests that the "independence" hypothesis is about half as likely relative to the "somewhat streaky" hypothesis than it was in our prior probability estimates. Again, definitely meaningful evidence, but not conclusive.

One final interesting thing about Tiger's history of winning in back-to-back weeks is that there's been a genuine temporal pattern. The first time Tiger pulled off this particular feat in his entire career was the autumn of 1999, when he won the National Car Rental Golf Classic/Disney, THE TOUR Championship, and the WGC-American Express Championship in three consecutive weeks in late October and early November, as part of his first run of winning in six consecutive starts. Prior to that, he was 0-for-31 in two-week windows for his career. Starting with that run of three and continuing through March of 2001, he won twice in 4 of the 13 two-week windows he played. Then he hit another 0-for-24 skid through the middle of 2006. Then, beginning with another three-in-a-row streak in the Buick Open, PGA Championship, and WGC-Bridgestone Invitational, again as part of a long streak of consecutive wins in his starts, and continuing through back-to-back wins in the 2009 Buick Open and WGC-Bridgestone Invitational, he had a span of going five for ten. Yeah, five for ten, at a stat he should be about one-for-sixteen at. Since then, however, he's had another burst of 0-for-14. So his numbers at winning in back-to-back weeks, itself a measure of streakiness, have been incredibly streaky.

Of course, this isn't utterly surprising given the overall arc of Tiger's career, which largely followed the peaks and valleys of his back-to-back-weeks performance. During his first 0-for-31 streak, i.e. his career up to but not including the 1999 Disney event, he played in 81 events and won 12 times, a 14.8% winning record, which would predict winning in back-to-back weeks just 2.2% of the time, for an expectation of 0.68 successes in those 31 trials. That's not much of an error. Then, during the 4-for-13 period from late 1999 through early 2001, Tiger played in 30 events and won 14 of them. That's a 46.6% winning rate, which would predict a 21.8% success rate at winning back-to-back, which would predict 2.83 successes in those 13 trials. That's a bit more of an error, since he won 4, but still not massive. Then came an 0-for-24 slump, spanning a period when he played in 98 events and managed 23 wins. That's a 23.5% winning rate, for a 5.5% expected back-to-back success rate, so he "should" have managed 1.32 back-to-back winnings in this period, but instead got zero. Then came the ridiculous 5-for-10 period, but this was a span in which Tiger won 20 of just 39 events. Still, that insane 51.3% winning clip would expect back-to-back wins just 26.3% of the time, for an expected—wait for it—2.63 such events in these 10 windows. He had, uh, 5 of those events. Okay, that's a pretty big miss. Finally we have his latest 0-for-14 slump, which covers a period of 45 events and just 4 wins. That he hasn't won back-to-back in that period is not surprising at all: the 8.88% winning clip projects a paltry 0.79% back-to-back winning rate, which would be just 0.11 successes in 14 trials.

So we can describe these five periods thusly: to start his career, Tiger was, though a perfectly good player, not someone who won often enough that we should ever really have expected him to win in consecutive weeks. Then came the period from late 1999 through early 2001, better known as the best run of golf anyone's ever had, when he was winning nearly half his events and accordingly had quite a few back-to-back wins, but slightly more than he should have at random. Then for the next five years Tiger was winning at a less insane clip, but still high enough that he should have had the occasional run of consecutive wins in consecutive weeks, but he didn't. Then came the second insane period when he was winning half his starts, when he had far more wins in consecutive weeks than he had any right to. Finally there has been his latest mega-slump, during which he's been such a pathetic player that we should truly have been shocked if he ever won consecutive tournaments, and indeed he hasn't. (Okay, not exactly pathetic given that he won more events than 99% even of PGA Tour players ever do; he's set himself high standards.)

I'd say that this general pattern, that he's had two of these unbelievable stretches and during them has won considerably more back-to-back events than he should have and other than that hasn't done it at all, despite being a sufficiently winning player that he easily could have, gives even more evidence for the streakiness hypothesis. Of course, part of that is that Tiger's whole career has been really streaky, which you can tell just by looking at the year 2000 or, alternately, the years 2010 and 2011. But the consecutive-weeks thing seems to have been additionally streaky on top of the general ebbs and flows of his career: when he's been good, which is to say, good by his own ridiculous standards, he's done more back-to-back winning even than you'd expect given how ridiculously good he was at the time, and conversely, when he hasn't been insanely good, he hasn't won in consecutive weeks at all, even though he's a good enough golfer in those "slumps" that he easily could have. It looks like not only is there streakiness, there's streakiness in the streakiness.