Myth vs. Math: Most Important Set

This is the final installment in a five-part series called Myth vs. Math. In this series, I am taking a look at five widely-accepted statements that tennis writers, analysts, fans, and commentators frequently make. I'm checking to see if these statements hold up against the numbers. The first four statements in this series were "big servers have a notable advantage in tiebreakers," "Novak Djokovic has the best defensive return in the game right now," "Rafael Nadal tends to get tougher draws than Djokovic since making his return from injury," and "The top players play the big points better than their opponents." The final statement being put to the test is "The first set is the most important set in a best-of-5 match."
The Myth

It's inevitable. In almost every single televised tennis match, the commentators will mention some stat about first sets near or after the conclusion of the first set. They say something along the lines of "In 42 meetings between Nadal and Djokovic, the winner of the first set has gone on to win 35 times," or "Roger
Federer wins 93% of his matches when he wins the first set." Then they go on to talk about how important winning the first set is, claiming that it is somehow more important than any other set in a tennis match.

Commentators and their viewers aren't the only ones guilty of thinking this way. Howard Bryant, one of ESPN's best tennis writers, wrote an article called 'Don't lose the first set.' Even the ATP World Tour website fell into the trap, including in each players' match record tab their record both after winning and losing the first set.

The line of thinking makes sense at first glance, and the numbers seem to back up the claim. Winning the first set gives a player the lead and a mental edge through the rest of the match. At worst, the player is going into the third set tied at one set apiece.

Not everyone believes the first set is the most important. Some would argue that the fifth is the most important set, because it is the deciding set. Others argue in favor of the third set, because barring retirement, a third set is guaranteed to be played and it is closer to the end of the match than the first or second set. Before doing the research, I believed that the second set would be the most important, because the player is either ahead 2-0 or tied at 1-1 with the momentum.

On paper, all five sets are of equal value. It takes three sets to win a best-of-5 match and all five sets are an opportunity to win one of those three.

The Math

In my research I looked at all 508 men's singles matches played at each of the last four majors. Then I looked at the winning percentage of players who won the first set, players who won the second set, and so on. Results varied depending on the round that the match was played in and which of the majors the match was played at.

For example, the winner of the first set went on to win the match at the US Open almost six percent more than at the Australian Open and over seven percent more at Wimbledon. Meanwhile, at Roland Garros, the second and third sets were more important than at any other tournament.

Here are the total results for the four majors:

The winner of the first set went 402-100 (80.08%)
The winner of the second set went 406-91 (81.70%)
The winner of the third set went 389-103 (79.07%)
The winner of the fourth set went 198-36 (84.62%)
The winner of the fifth set went 87-0 (100%)

Even though the winner of the fourth set and fifth set had higher winning percentages, only 86 of the 508 matches even went to a fifth set and 234 lasted at least four sets. In over half of the matches, the fourth set had no impact on the outcome and that's the case more so in fifth sets.

The gap in importance for the first three sets isn't that big in the first set of data. That's because so many matches are won in straight sets, and in all of those matches, the winning percentage for each of the first three sets is 100%.

So to see how big the difference in importance of the first three sets is, I'm going to look only at matches that went the distance. Those are the closest matches where every set matters, but also, even the loser of the match won two sets.

Here are the records from the four majors in five-set matches

*The winner of a five-set match wins 50% of the first two sets, so anything above 50% implies greater importance

The winner of the first set went 47-40(54.02%)
The winner of the second set went 48-39(55.17%)
The winner of the third set went 35-52(40.23%)
The winner of the fourth set went 51-36 (58.62%)
The winner of the fifth set went 87-0 (100%)

Obviously, in a five-set match, the fifth set is going to be the most important because it is the deciding set. After that, on paper, the first four sets should all be of equal importance. However, the fourth set is clearly more important, followed in order by the second set and the first set. Then, the third set is by far less important than any other set.

Conclusion

Based on the math, I would say that the order of importance of sets goes 2nd, 1st, 4th, 3rd, 5th. Of course, another reasonable person could look at the same numbers and come up with a different order.

There will be disagreement on where in the order the fourth and fifth sets go, because not all best-of-5 matches go all five sets. However, we are able to judge the first three sets on equal ground. The math clearly shows that the second set is slightly more important than the first set and the third set is much less important than the previous two.

So the claim that the first set is the most important set in best-of-5 matches is not true. Obviously, every set is important, even in best-of-5. So when stats suggest that the first set is important, it doesn't mean it is any more important than the rest.

Fun Facts

In my research I found some other interesting facts, so I thought I would share them:

-While slightly more than half of all matches at majors are played in the first round, 76% of the retirements in the last four majors came in the first round with none coming in the second week.

-The Australian Open, which was absurdly hot this year, produced twice as many retirements and withdrawals as any of the other three slams

-Roland Garros had the most two-set comebacks at six, while Wimbledon had the least at two.

-In matches where the same player won the first two sets, 33 matches went five sets. The player who won the first two sets went 17-16 in the deciding set.

-The least common way to win a five-set match was by dropping the second and fourth sets, while winning the first, third, and fifth, occurring half as often as a two-set comeback.

-Roland Garros had the most straight-set matches at 76, while the US Open had the least at 56.

-At the US Open, the winner of the first set went on to win 83.46% of the time, while the winner of the second set only won the match 75.4% of the time (impact of the crowd?). In every other tournament the second set winner won more often than the first set winner.

-At Roland Garros, each set is more important, with the average win percentage of the winners of each of the first four sets at 84.42%. US Open was the lowest at 79.48%, while the other two were slightly above 80%. This is likely because less matches at Roland Garros go five sets than the other three tournaments.

-The round of 16 produced the highest percentage of straight set victories (59.38%) other than the semifinals. Mostly because of the dominance of the top tier of the game.

-Each of the first three round have the exact same percentage of five-set matches, and the quarterfinals is the only round with a higher percentage.

-Nobody has come back from two sets down in the second week of a major in the last 12 months. Tsonga did it to Federer in Wimbledon and Djokovic also did it to Federer at the US Open in 2011. Has anyone done it since?

-At three of the last four major finals, the winner of the first set did not win the match.