Author: jsackmann

Podcast Episode 16: The Second Half of our Second Week Chat

Episode 16 of the Tennis Abstract Podcast, with Carl Bialik, didn’t quite work out as planned — my microphone malfunctioned for much of the first half of the recording — but the second half of our conversation could be salvaged. Thus, this episode is missing the big news of the second week, the all-American women’s semifinals, but we still touched on a variety of burning US Open topics, like the youngsters making news in New York and the inevitable hypotheticals of the players who weren’t able to participate in Flushing this year.

This is a shorter-than usual episode, clocking in at 34 minutes; we hope to return on Friday with another mid-Slam update. Thanks for listening!

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Quantifying Cakewalks, or The Time Rafa Finally Got Lucky

Italian translation at settesei.it

During this year’s US Open, much has been made of some rather patchy sections of the draw. Many great players are sitting out the tournament with injury, and plenty of others crashed out early. Pablo Carreno Busta reached the quarterfinals by defeating four straight qualifiers, and Rafael Nadal could conceivably win the title without beating a single top-20 player.

None of this is a reflection on the players themselves: They can play only the draw they’re dealt, and we’ll never know how they would’ve handled a more challenging array of opponents. The weakness of the draw, however, could affect how we remember this tournament.  If we are going to let the quality of the field color our memories, we should at least try to put this year’s players in context to see how they compare with majors in the past.

How to measure draw paths

There are lots of ways to quantify draw quality. (There’s an entire category on this blog devoted to it.) Since we’re interested in the specific sets of opponents faced by our remaining contenders, we need a metric that focuses on those. It doesn’t really matter that, say, Nick Kyrgios was in the draw, since none of the semifinalists had to play him.

Instead of draw difficulty, what we’re after is what I’ll call path ease. It’s a straightforward enough concept: How hard is it to beat the specific set of guys that Rafa (for instance) had to play?

To get a number, we’ll need a few things: The surface-weighted Elo ratings of each one of a player’s opponents, along with a sort of “reference Elo” for an average major semifinalist. (Or finalist, or title winner.) To determine the ease of Nadal’s path so far, we don’t want to use Nadal’s Elo. If we did that, the exact same path would look easier or harder depending on the quality of the player who faced it.

(The exact value of the “reference Elo” isn’t that important, but for those of you interested in the numbers: I found the average Elo rating of every slam semifinalist, finalist, and winner back to 1988 on each of the three major surfaces. On hard courts, those numbers are 2145, 2198, and 2233, respectively. When measuring the difficulty of a path to the semifinal round, I used the first of those numbers; for the difficulty of a path to the title, I used the last.)

To measure path ease, then, we answer the question: What are the odds that an average slam semifinalist (for instance) would beat this particular set of players? In Rafa’s case, he has yet to face a player with a weighted-hard-court Elo rating above 1900, and the typical 2145-rated semifinalist would beat those five players 71.5% of the time. That’s a bit easier than Kevin Anderson‘s path the semis, but a bit harder than Carreno Busta’s. Juan Martin del Potro, on the other hand, is in a different world altogether. Here are the path ease numbers for all four semifinalists, showing the likelihood that average contenders in each round would advance, giving the difficulty of the draws each player has faced:

Semifinalist   Semi Path  Final Path  Title Path  
Nadal              71.5%       49.7%       51.4%  
del Potro           9.1%        7.5%       10.0%  
Anderson           69.1%       68.9%       47.1%  
Carreno Busta      74.3%       71.2%       48.4%

(We don’t yet know each player’s path to the title, so I averaged the Elos of possible opponents. Anderson and Carreno Busta are very close, so for Rafa and Delpo, their potential final opponent doesn’t make much difference.)

There’s one quirk with this metric that you might have noticed: For Nadal and del Potro, their difficulty of reaching the final is greater than that of winning the title altogether! Obviously that doesn’t make logical sense–the numbers work out that way because of the “reference Elos” I’m using. The average slam winner is better than the average slam finalist, so the table is really saying that it’s easier for the average slam winner to beat Rafa’s seven opponents than it would be for the average slam finalist to get past his first six opponents. This metric works best when comparing title paths to title paths, or semifinal paths to semifinal paths, which is what we’ll do for the rest of this post.

Caveats and quirks aside, it’s striking just how easy three of the semifinal paths have been compared to del Potro’s much more arduous route. Even if we discount the difficulty of beating Roger Federer–Elo thinks he’s the best active player on hard courts but doesn’t know about his health issues–Delpo’s path is wildly different from those of his semifinal and possible final opponents.

Cakewalks in context

Semifinalist path eases of 69% or higher–that is, easier–are extremely rare. In fact, the paths of Anderson, Carreno Busta, and Nadal are all among the ten easiest in the last thirty years! Here are the previous top ten:

Year  Slam             Semifinalist               Path Ease  
1989  Australian Open  Thomas Muster                  84.1%  
1989  Australian Open  Miloslav Mecir                 74.2%  
1990  Australian Open  Ivan Lendl                     73.8%  
2006  Roland Garros    Ivan Ljubicic                  73.7%  
1988  Australian Open  Ivan Lendl                     72.2%  
1988  Australian Open  Pat Cash                       70.1%  
2004  Australian Open  Juan Carlos Ferrero            69.2%  
1996  US Open          Michael Chang                  68.8%  
1990  Roland Garros    Andres Gomez                   68.4%  
1996  Australian Open  Michael Chang                  66.2%

In the last decade, the easiest path to the semifinal was Stan Wawrinka‘s route to the 2016 French Open final four, which rated 59.8%. As we’ll see further on, Wawrinka’s draw got a lot more difficult after that.

Del Potro’s draw so far isn’t quite as extreme, but it is quite difficult in the historical context. Of the nearly 500 major semifinalists since 1988, all but 15 are easier than his 9.1% path difficulty. Here are the top ten, all of whom faced draws that would have given the average slam semifinalist less than an 8% chance of getting that far:

Year  Slam             Semifinalist              Path Ease  
2009  Roland Garros    Robin Soderling                1.6%  
1988  Roland Garros    Jonas Svensson                 1.9%  
2017  Wimbledon        Tomas Berdych                  3.7%  
1996  Wimbledon        Richard Krajicek               6.4%  
2011  Wimbledon        Jo Wilfried Tsonga             6.6%  
2012  US Open          Tomas Berdych                  6.8%  
2017  Roland Garros    Dominic Thiem                  6.9%  
2014  Australian Open  Stan Wawrinka                  7.0%  
1989  Roland Garros    Michael Chang                  7.1%  
2017  Wimbledon        Sam Querrey                    7.5%

Previewing the history books

In the long term, we’ll care a lot more about how the 2017 US Open champion won the title than how he made it through the first five rounds. As we saw above, three of the four semifinalists have a path ease of around 50% to win the title–again, meaning that a typical slam winner would have a roughly 50/50 chance of getting past this particular set of seven opponents.

No major winner in recent memory has had it so easy. Nadal’s path would rate first in the last thirty years, while Carreno Busta’s or Anderson’s would rate in the top five. (If it comes to that, their exact numbers will depend on who they face in the final.) Here is the list that those three men have the chance to disrupt:

Year  Slam             Winner                  Path Ease  
2002  Australian Open  Thomas Johansson            48.1%  
2001  Australian Open  Andre Agassi                47.6%  
1999  Roland Garros    Andre Agassi                45.6%  
2000  Wimbledon        Pete Sampras                45.3%  
2006  Australian Open  Roger Federer               44.5%  
1997  Australian Open  Pete Sampras                44.4%  
2003  Australian Open  Andre Agassi                43.9%  
1999  US Open          Andre Agassi                41.5%  
2002  Wimbledon        Lleyton Hewitt              39.9%  
1998  Wimbledon        Pete Sampras                39.1%

At the 2006 Australian Open, Federer lucked into a path that was nearly as easy as Rafa’s this year. His 2003 Wimbledon title just missed the top ten as well. By comparison, Novak Djokovic has never won a major with a path ease greater than 18.7%–harder than that faced by more than half of major winners.

Nadal has hardly had it easy as he has racked up his 15 grand slams, either. Here are the top ten most difficult title paths:

Year  Slam             Winner                Path Ease  
2014  Australian Open  Stan Wawrinka              2.2%  
2015  Roland Garros    Stan Wawrinka              3.1%  
2016  Us Open          Stan Wawrinka              3.2%  
2013  Roland Garros    Rafael Nadal               4.4%  
2014  Roland Garros    Rafael Nadal               4.7%  
1989  Roland Garros    Michael Chang              5.0%  
2012  Roland Garros    Rafael Nadal               5.2%  
2016  Australian Open  Novak Djokovic             5.4%  
2009  US Open          J.M. Del Potro             5.9%  
1990  Wimbledon        Stefan Edberg              6.2%

As I hinted in the title of this post, while Nadal got lucky in New York this year, it hasn’t always been that way. He appears three times on this list, facing greater challenges than any major winner other than Wawrinka the giant-killer.

On average, Rafa’s grand slam title paths haven’t been quite as harrowing as Djokovic’s, but compared to most other greats of the last few decades, he has worked hard for his titles. Here are the average path eases of players with at least three majors since 1988:

Player           Majors        Avg Path Ease  
Stan Wawrinka         3                 2.8%  
Novak Djokovic       12                11.3%  
Rafael Nadal         15                13.6%  
Stefan Edberg         4                14.6%  
Andy Murray           3                18.8%  
Boris Becker          4                18.8%  
Mats Wilander         3                19.8%  
Gustavo Kuerten       3                22.0%  
Roger Federer        19                23.5%  
Jim Courier           4                26.4%  
Pete Sampras         14                28.9%  
Andre Agassi          8                32.3%

If Rafa adds to his grand slam haul this weekend, his average path ease will take a bit of a hit. Still, he’ll only move one place down the list, behind Stefan Edberg. After more than a decade of battling all-time greats in the late rounds of majors, it’s fair to say that Nadal deserved this cakewalk.

Update: This post reads a bit differently than when I first wrote it: I’ve changed the references to “path difficulty” to “path ease” to make it clearer what the metric is showing.

Nadal and Anderson advanced to the final, so we can now determine the exact path ease number for whichever one of them wins the title. Rafa’s exact number remains 51.4%, and should he win, his career average across 16 slams will increase to about 15%. Anderson’s path ease to the title is “only” 41.3%, which would be good for ninth on the list shown above, and just barely second easiest of the last 30 US Opens.

Measuring the Impact of the Serve in Men’s Tennis

By just about any measure, the serve is the most important shot in tennis. In men’s professional tennis, with its powerful deliveries and short points, the serve is all the more crucial. It is the one shot guaranteed to occur in every rally, and in many points, it is the only shot.

Yet we don’t have a good way of measuring exactly how important it is. It’s easy to determine which players have the best serves–they tend to show up at the top of the leaderboards for aces and service points won–but the available statistics are very limited if we want a more precise picture. The ace stat counts only a subset of those points decided by the serve, and the tally of service points won (or 1st serve points won, or 2nd serve points won) combines the effect of the serve with all of the other shots in a player’s arsenal.

Aces are not the only points in which the serve is decisive, and some service points won are decided long after the serve ceases to have any relevance to the point. What we need is a method to estimate how much impact the serve has on points of various lengths.

It seems like a fair assumption that if a server hits a winner on his second shot, the serve itself deserves some of the credit, even if the returner got it back in play. In any particular instance, the serve might be really important–imagine Roger Federer swatting away a weak return from the service line–or downright counterproductive–think of Rafael Nadal lunging to defend against a good return and hitting a miraculous down-the-line winner. With the wide variety of paths a tennis point can follow, though, all we can do is generalize. And in the aggregate, the serve probably has a lot to do with a 3-shot rally. At the other extreme, a 25-shot rally may start with a great serve or a mediocre one, but by the time by the point is decided, the effect of the serve has been canceled out.

With data from the Match Charting Project, we can quantify the effect. Using about 1,200 tour-level men’s matches from 2000 to the present, I looked at each of the server’s shots grouped by the stage of the rally–that is, his second shot, his third shot, and so on–and calculated how frequently it ended the point. A player’s underlying skills shouldn’t change during a point–his forehand is as good at the end as it is at the beginning, unless fatigue strikes–so if the serve had no effect on the success of subsequent shots, players would end the point equally often with every shot.

Of course, the serve does have an effect, so points won by the server end much more frequently on the few shots just after the serve than they do later on. This graph illustrates how the “point ending rate” changes:

On first serve points (the blue line), if the server has a “makeable” second shot (the third shot of the rally, “3” on the horizontal axis, where “makeable” is defined as a shot that results in an unforced error or is put back in play), there is a 28.1% chance it ends the point in the server’s favor, either with a winner or by inducing an error on the next shot. On the following shot, the rate falls to 25.6%, then 21.8%, and then down into what we’ll call the “base rate” range between 18% and 20%.

The base rate tells us how often players are able to end points in their favor after the serve ceases to provide an advantage. Since the point ending rate stabilizes beginning with the fifth shot (after first serves), we can pinpoint that stage of the rally as the moment–for the average player, anyway–when the serve is no longer an advantage.

As the graph shows, second serve points (shown with a red line) are a very different story. It appears that the serve has no impact once the returner gets the ball back in play. Even that slight blip with the server’s third shot (“5” on the horizontal axis, for the rally’s fifth shot) is no higher than the point ending rate on the 15th shot of first-serve rallies. This tallies with the conclusions of some other research I did six years ago, and it has the added benefit of agreeing with common sense, since ATP servers win only about half of their second serve points.

Of course, some players get plenty of positive after-effects from their second serves: When John Isner hits a second shot on a second-serve point, he finishes the point in his favor 30% of the time, a number that falls to 22% by his fourth shot. His second serve has effects that mirror those of an average player’s first serve.

Removing unforced errors

I wanted to build this metric without resorting to the vagaries of differentiating forced and unforced errors, but it wasn’t to be. The “point-ending” rates shown above include points that ended when the server’s opponent made an unforced error. We can argue about whether, or how much, such errors should be credited to the server, but for our purposes today, the important thing is that unforced errors aren’t affected that much by the stage of the rally.

If we want to isolate the effect of the serve, then, we should remove unforced errors. When we do so, we discover an even sharper effect. The rate at which the server hits winners (or induces forced errors) depends heavily on the stage of the rally. Here’s the same graph as above, only with opponent unforced errors removed:

The two graphs look very similar. Again, the first serve loses its effect around the 9th shot in the rally, and the second serve confers no advantage on later shots in the point. The important difference to notice is the ratio between the peak winner rate and the base rate, which is now just above 10%. When we counted unforced errors, the ratio between peak and base rate was about 3:2. With unforced errors removed, the ratio is close to 2:1, suggesting that when the server hits a winner on his second shot, the serve and the winner contributed roughly equally to the outcome of the point. It seems more appropriate to skip opponent unforced errors when measuring the effect of the serve, and the resulting 2:1 ratio jibes better with my intuition.

Making a metric

Now for the fun part. To narrow our focus, let’s zero in on one particular question: What percentage of service points won can be attributed to the serve? To answer that question, I want to consider only the server’s own efforts. For unreturned serves and unforced errors, we might be tempted to give negative credit to the other player. But for today’s purposes, I want to divvy up the credit among the server’s assets–his serve and his other shots–like separating the contributions of a baseball team’s pitching from its defense.

For unreturned serves, that’s easy. 100% of the credit belongs to the serve.

For second serve points in which the return was put in play, 0% of the credit goes to the serve. As we’ve seen, for the average player, once the return comes back, the server no longer has an advantage.

For first-serve points in which the return was put in play and the server won by his fourth shot, the serve gets some credit, but not all, and the amount of credit depends on how quickly the point ended. The following table shows the exact rates at which players hit winners on each shot, in the “Winner %” column:

Server's…  Winner %  W%/Base  Shot credit  Serve credit  
2nd shot      21.2%     1.96        51.0%         49.0%  
3rd shot      18.1%     1.68        59.6%         40.4%  
4th shot      13.3%     1.23        81.0%         19.0%  
5th+          10.8%     1.00       100.0%          0.0%

Compared to a base rate of 10.8% winners per shot opportunity, we can calculate the approximate value of the serve in points that end on the server’s 2nd, 3rd, and 4th shots. The resulting numbers come out close to round figures, so because these are hardly laws of nature (and the sample of charted matches has its biases), we’ll go with round numbers. We’ll give the serve 50% of the credit when the server needed only two shots, 40% when he needed three shots, and 20% when he needed four shots. After that, the advantage conferred by the serve is usually canceled out, so in longer rallies, the serve gets 0% of the credit.

Tour averages

Finally, we can begin the answer the question, What percentage of service points won can be attributed to the serve? This, I believe, is a good proxy for the slipperier query I started with, How important is the serve?

To do that, we take the same subset of 1,200 or so charted matches, tally the number of unreturned serves and first-serve points that ended with various numbers of shots, and assign credit to the serve based on the multipliers above. Adding up all the credit due to the serve gives us a raw number of “points” that the player won thanks to his serve. When we divide that number by the actual number of service points won, we find out how much of his service success was due to the serve itself. Let’s call the resulting number Serve Impact, or SvI.

Here are the aggregates for the entire tour, as well as for each major surface:

         1st SvI  2nd SvI  Total SvI  
Overall    63.4%    31.0%      53.6%  
Hard       64.6%    31.5%      54.4%  
Clay       56.9%    27.0%      47.8%  
Grass      70.8%    37.3%      61.5%

Bottom line, it appears that just over half of service points won are attributable to the serve itself. As expected, that number is lower on clay and higher on grass.

Since about two-thirds of the points that men win come on their own serves, we can go even one step further: roughly one-third of the points won by a men’s tennis player are due to his serve.

Player by player

These are averages, and the most interesting players rarely hew to the mean. Using the 50/40/20 multipliers, Isner’s SvI is a whopping 70.8% and Diego Schwartzman‘s is a mere 37.7%. As far from the middle as those are, they understate the uniqueness of these players. I hinted above that the same multipliers are not appropriate for everyone; the average player reaps no positive after-effects of his second serve, but Isner certainly does. The standard formula we’ve used so far credits Isner with an outrageous SvI, even without giving him credit for the “second serve plus one” points he racks up.

In other words, to get player-specific results, we need player-specific multipliers. To do that, we start by finding a player-specific base rate, for which we’ll use the winner (and induced forced error) rate for all shots starting with the server’s fifth shot on first-serve points and shots starting with the server’s fourth on second-serve points. Then we check the winner rate on the server’s 2nd, 3rd, and 4th shots on first-serve points and his 2nd and 3rd shots on second-serve points, and if the rate is at least 20% higher than the base rate, we give the player’s serve the corresponding amount of credit.

Here are the resulting multipliers for a quartet of players you might find interesting, with plenty of surprises already:

                   1st serve              2nd serve       
                    2nd shot  3rd  4th     2nd shot  3rd  
Roger Federer            55%  50%  30%           0%   0%  
Rafael Nadal             31%   0%   0%           0%   0%  
John Isner               46%  41%   0%          34%   0%  
Diego Schwartzman        20%  35%   0%           0%  25%  
Average                  50%  30%  20%           0%   0%

Roger Federer gets more positive after-effects from his first serve than average, more even than Isner does. The big American is a tricky case, both because so few of his serves come back and because he is so aggressive at all times, meaning that his base winner rate is very high. At the other extreme, Schwartzman and Rafael Nadal get very little follow-on benefit from their serves. Schwartzman’s multipliers are particularly intriguing, since on both first and second serves, his winner rate on his third shot is higher than on his second shot. Serve plus two, anyone?

Using player-specific multipliers makes Isner’s and Schwartzman’s SvI numbers more extreme. Isner’s ticks up a bit to 72.4% (just behind Ivo Karlovic), while Schwartzman’s drops to 35.0%, the lowest of anyone I’ve looked at. I’ve calculated multipliers and SvI for all 33 players with at least 1,000 tour-level service points in the Match Charting Project database:

Player                 1st SvI  2nd SvI  Total SvI  
Ivo Karlovic             79.2%    56.1%      73.3%  
John Isner               78.3%    54.3%      72.4%  
Andy Roddick             77.8%    51.0%      71.1%  
Feliciano Lopez          83.3%    37.1%      68.9%  
Kevin Anderson           77.7%    42.5%      68.4%  
Milos Raonic             77.4%    36.0%      66.0%  
Marin Cilic              77.1%    34.1%      63.3%  
Nick Kyrgios             70.6%    41.0%      62.5%  
Alexandr Dolgopolov      74.0%    37.8%      61.3%  
Gael Monfils             69.8%    37.7%      60.8%  
Roger Federer            70.6%    32.0%      58.8%  
                                                    
Player                 1st SvI  2nd SvI  Total SvI  
Bernard Tomic            67.6%    28.7%      58.5%  
Tomas Berdych            71.6%    27.0%      57.2%  
Alexander Zverev         65.4%    30.2%      54.9%  
Fernando Verdasco        61.6%    32.9%      54.3%  
Stan Wawrinka            65.4%    33.7%      54.2%  
Lleyton Hewitt           66.7%    32.1%      53.4%  
Juan Martin Del Potro    63.1%    28.2%      53.4%  
Grigor Dimitrov          62.9%    28.6%      53.3%  
Jo Wilfried Tsonga       65.3%    25.9%      52.7%  
Marat Safin              68.4%    22.7%      52.3%  
Andy Murray              63.4%    27.5%      52.0%  
                                                    
Player                 1st SvI  2nd SvI  Total SvI  
Dominic Thiem            60.6%    28.9%      50.8%  
Roberto Bautista Agut    55.9%    32.5%      49.5%  
Pablo Cuevas             57.9%    28.9%      47.8%  
Richard Gasquet          56.0%    29.0%      47.5%  
Novak Djokovic           56.0%    26.8%      47.3%  
Andre Agassi             54.3%    31.4%      47.1%  
Gilles Simon             55.7%    28.4%      46.7%  
Kei Nishikori            52.2%    30.8%      45.2%  
David Ferrer             46.9%    28.2%      41.0%  
Rafael Nadal             42.8%    27.1%      38.8%  
Diego Schwartzman        39.5%    25.8%      35.0%

At the risk of belaboring the point, this table shows just how massive the difference is between the biggest servers and their opposites. Karlovic’s serve accounts for nearly three-quarters of his success on service points, while Schwartzman’s can be credited with barely one-third. Even those numbers don’t tell the whole story: Because Ivo’s game relies so much more on service games than Diego’s does, it means that 54% of Karlovic’s total points won–serve and return–are due to his serve, while only 20% of Schwartzman’s are.

We didn’t need a lengthy analysis to show us that the serve is important in men’s tennis, or that it represents a much bigger chunk of some players’ success than others. But now, instead of asserting a vague truism–the serve is a big deal–we can begin to understand just how much it influences results, and how much weak-serving players need to compensate just to stay even with their more powerful peers.

Podcast Episode 15: Return of the Podcast

The long-awaited Episode 15 of the Tennis Abstract Podcast is Carl Bialik’s and my lightning-round update on the summer of tennis and a midway-point checkup of the US Open. I boldly select Diego Schwartzman and Anastasija Sevastova as my picks to win it all, while Carl says much more sensible things.

This is a shorter-than usual episode, clocking in at 34 minutes; we hope to return on Friday with another mid-Slam update. Thanks for listening!

Click to listen, subscribe on iTunes, or use our feed to get updates on your favorite podcast software.

How Much Does Height Matter in Men’s Tennis?

Italian translation at settesei.it

Clearly, height matters. On average, tall players can serve faster and more effectively than can shorter players. And usually, short players who succeed on tour do so by returning and moving better than their taller colleagues. The conventional wisdom is that height is an advantage, but only up to a point. An inch or two above six feet (a range between 185 and 190 cm) is good, but much more than that is too much. No player above 6’4″ (193 cm, Marat Safin) has ever reached No. 1 in the ATP rankings.

While 5’7″ (170 cm) Diego Schwartzman‘s surprise run to the US Open quarterfinals has brought this issue to the forefront, pundits and fans talk about it all the time. This is a topic crying out for some basic data analysis, yet as is too often the case in tennis, some really simple work is missing from the conversation. Let’s try to fix that.

When I say “basic,” I really mean it. We all know that tall men hit more aces than short men. But how many? How strong is the relationship between height and, say, first serve points won? In this post, I’ll show the relationship between height and each of nine different stats, from overall records to serve- and return-specific numbers.

For my dataset, I took age-25 seasons from 1998 to 2017 in which the player completed at least 30 tour-level matches. (I used only one season per player so that the best players with the longest careers wouldn’t be weighted too heavily.) That gives us 156 player-seasons, from Hicham Arazi and Greg Rusedski in 1998 up to Schwartzman and Jack Sock in 2017. There aren’t very many players at the extremes, so I lumped together everyone 5’8″ (173 cm) and below and did the same with everyone 6’5″ (196 cm) and above. I also grouped players standing 5’10” with those at 5’9″, because there were only four 5’10” guys in the dataset.

That gives us nine “height levels”: one per inch from 5’8″ to 6’5″ with the exception of 5’10”. (The ATP website displays heights in meters, but its database must record and/or store them in inches, because every height translates to something close to an integer height in inches. For example, no player is listed at 174 cm, or 5’8.5″.) Some individual heights are certainly exaggerated, as male athletes and their organizations tend to do, but we have to make do with the information available, and we may assume that the exaggerations are fairly consistent.

Let’s start with the most basic building block of tennis, the match win. There is a reasonably strong relationship here, although the group of players at 6’1″ is nearly as good as the tallest subset. In each of these graphs, height is given on the horizontal axis in centimeters, from 173 (the 5’8″ and below group) up to 196 (the 6’5″ and higher group).

There is a similar, albeit slightly weaker, relationship when we look at the level of single points. Since a small difference in points results in a larger difference in matches won (at the extreme, winning 55% of points translates to nearly a 100% chance of winning the match) this isn’t a surprise. At the match level, r^2 = 0.38, and at the point level, below, r^2 = 0.27:

(If you’re wondering how all of the averages are above 50%, it’s because the sample is limited to player-seasons with at least 30 matches. A fair number of those matches are against players who aren’t tour regulars, and the regulars–the guys in this sample–win a hefty proportion of those matches.)

Serve stats

Now we get to confirm our main assumptions. Taller players are better servers, and the gap is enormous, ranging from 60% of service points won for the shortest players up to nearly 70% for the tallest:

As strong as that relationship is (r^2 = 0.81), the relationship between height and ace rate is stronger still, at r^2 = 0.83:

Aces don’t tell the whole story–the stat with the strongest correlation to height is first serve points won (r^2 = 0.92) as you can see here:

But this is where things start to get interesting. Nearly every inch makes a player more effective on the first serve, but opponents are able to negotiate tall players’ second serves much more successfully. There remains a modest relationship with height (r^2 = 0.18), but it is the weakest of all the stats presented here:

It’s nice to be tall, as anyone who has seen John Isner casually spin a second-serve ace out of the reach of an unlucky opponent. But except in the tallest category, height doesn’t confer much of a second-serve advantage. Players standing 6’4″ (193 cm) win about as many second-serve points as do players at 5’9″ (175 cm). That doesn’t mean that the second serves of the shorter players are just as good–they probably aren’t–but that shorter players tend to possess other skills that they can leverage in second-serve points, which usually last longer. For the purposes of today’s overview, it doesn’t really matter why short players are able to negate the advantage of height on second serve points, just that they are clearly able to do so.

Return stats

We wouldn’t be having this conversation–and David Ferrer wouldn’t be headed to a likely place in the Hall of Fame–if the inverse relationship between height and return effectiveness weren’t nearly as strong as the positive one between height and serving prowess. “Nearly” is the key word here. The relationship between height and overall return points won is almost as strong (r^2 = 0.74) as that of height and overall service points won, but not quite:

Schwartzman is doing more than his part to hold up the left side of that trendline: He is both the shortest player in the top 50 and the best returner. On first serve points, however, there’s only so much the returner can do, so while shorter players still have an advantage, it is less substantial. The relationship here is a bit weaker, at r^2 = 0.63:

It follows, then, that the relationship between height and second-serve return points won must be stronger, at r^2 = 0.77:

The overall and first-serve return point graphs make clear just how much worse the tallest players are than the rest of the pack. The graphs exaggerate it a bit, because I’ve grouped players from 6’5″ all the way up to 6’11”, and the Isners of the sport are considerably less effective than players such as Marin Cilic. Still, we find plenty of confirmation for the conventional wisdom that a height of 6’2″ or 6’3″ (188 cm to 190 cm) allows for players to remain effective on both sides of the ball, while a small increase from there can be a disadvantage.

A note on selection bias

It’s easy to lapse into shorthand and say something like, “shorter players are better returners.” More precisely, what we mean is, “of the players who have become tour regulars, shorter players are better returners.” They have to be, because it is nearly impossible for them to be top-tier servers. If they’ve cracked the top 50, they must have developed a world-class return game. The shorter the player, the more likely this is true.

The same logic is considerably weaker if we descend a couple rungs lower on the ladder of tennis skill. In collegiate tennis, it’s still an advantage to be tall–as Isner can attest–but a player such as 5’10” Benjamin Becker can serve as well as nearly all the competition he will face at that level.

One more note on selection bias

My choice to use each player’s age-25 season might understate the ability of either short or tall players. It is possible that certain playing styles result in earlier or later peaks, meaning that while tall players could be better at age 25, shorter players may be superior at age 28. There are anecdotes that support the argument in both directions, so I don’t think it’s a major issue, but it is one worthy of additional study.

Further reading

A guest post on this blog earlier this year posed the question, Are Taller Players the Future of Tennis?

I didn’t mention serve speed in the above, but here’s a quick study of the fastest serves and their correlation with height.

Podcast Episode 14: Wimbledon Preview

Episode 14 of the Tennis Abstract Podcast is Carl Bialik’s and my Wimbledon preview! We highlight the favorites, the overrated, and the underrated, along with a look at some of the most intriguing matchups. Along the way, we talk about the difficult of making grass court forecasts, and speculate about how players’ consistency changes with age.  Enjoy!

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The Men Are Old, and The Best Men Are Even Older

Italian translation at settesei.it

It’s been one of the main talking points in men’s tennis for years now: The sport is getting older. Every year, a bigger slice of Grand Slam draws are taken up by thirty-somethings, and now, the entire big four has entered their fourth decades.

I don’t want to belabor the point. But my interest was piqued by an observation from commentator Chris Fowler this week:

When we talk about the sport getting older, this is what we really mean — the best guys are getting up in years.

When we calculate the average age of a draw, or the number of 30-somethings, we weight every player equally. Democratic as it is, it gives most of the weight to guys who are looking for flights home before middle Sunday. As substantial as the overall age shift has been over the last decade, the shift at the top of the game has been even more dramatic.

To quantify the shift, I calculated what I’ll call the “projected winner age” (PWA) of every Wimbledon men’s field from 1991 to 2017. This captures in one number the notion that Fowler is hinting at. We take a weighted average of all 128 men in the main draw, weighted by their chances of winning the tournament, as determined by grass-court Elos at the start of the event.

For example, last year’s Wimbledon men’s draw had an average age of 28.5 years, but a projected winner age of 30.0. We don’t yet know the exact average age of this year’s draw (it looks to be about the same, maybe a tiny bit younger), but we can already say that the PWA is 31.4.

An observer a decade ago would’ve thought such a number was insane. Here are the average ages and PWAs for the last 27 Wimbledons men’s events:

As recently as 2011, there wasn’t much difference between average age and PWA. Until 2015, the difference had never been greater than two years. Now, the difference is almost three years, and the point of comparison–average age–is nearly its own all-time high.

A lot of this, of course, is thanks to the big four. Even as the aging curve has shifted, allowing for late bloomers such as Stan Wawrinka, the biggest stars of the late ’00s–Roger Federer and Rafael Nadal–have declined even less than the revised aging curve would imply. In a sport hungry for new winners, we might have to settle for winners who are newly in their 30s.

Podcast Episode 13: Kvitova vs Barty, Serena vs McEnroe, and Federer vs The Field

In the Episode 13 of the Tennis Abstract Podcast, Carl Bialik and I start with Petra Kvitova’s first title back from injury, and her chances as a floater in the Wimbledon draw, along with other returning grass-court threats Victoria Azarenka and Sabine Lisicki.

We move on to what we hope is a sensible, fact-based discussion of John McEnroe’s comments about how Serena Williams would fare on the men’s tour. It runs from about 19:00 to 50:00 — if you want to skip it, we understand. Finally, we talk Federer’s title in Halle, and how much his Wimbledon chances are aided by Wimbledon’s seeding formula, which moves him into the top four. (For more on that, see my post from earlier today.)

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Measuring the Impact of Wimbledon’s Seeding Formula

Italian translation at settesei.it

Unlike every other tournament on the tennis calendar, Wimbledon uses its own formula to determine seedings. The grass court Grand Slam grants seeds to the top 32 players in each tour’s rankings, and then re-orders them based on its own algorithm, which rewards players for their performance on grass over the last two seasons.

This year, the Wimbledon seeding formula has more impact on the men’s draw than usual. Seven-time champion Roger Federer is one of the best grass court players of all time, and though he dominated hard courts in the first half of 2017, he still sits outside the top four in the ATP rankings after missing the second half of 2016. Thanks to Wimbledon’s re-ordering of the seeds, Federer will switch places with ATP No. 3 Stan Wawrinka and take his place in the draw as the third seed.

Even with Wawrinka’s futility on grass and the shakiness of Andy Murray and Novak Djokovic, getting inside the top four has its benefits. If everyone lives up to their seed in the first four rounds (they won’t, but bear with me), the No. 5 seed will face a path to the title that requires beating three top-four players. Whichever top-four guy has No. 5 in his quarter would confront the same challenge, but the other three would have an easier time of it. Before players are placed in the draw, top-four seeds have a 75% chance of that easier path.

Let’s attach some numbers to these speculations. I’m interested in the draw implications of three different seeding methods: ATP rankings (as every other tournament uses), the Wimbledon method, and weighted grass-court Elo. As I described last week, weighted surface-specific Elo–averaging surface-specific Elo with overall Elo–is more predictive than ATP rankings, pure surface Elo, or overall Elo. What’s more, weighted grass-court Elo–let’s call it gElo–is about as predictive as its peers for hard and clay courts, even though we have less grass-court data to go on. In a tennis world populated only by analysts, seedings would be determined by something a lot more like gElo and a lot less like the ATP computer.

Since gElo ratings provide the best forecasts, we’ll use them to determine the effects of the different seeding formulas. Here is the current gElo top sixteen, through Halle and Queen’s Club:

1   Novak Djokovic         2296.5  
2   Andy Murray            2247.6  
3   Roger Federer          2246.8  
4   Rafael Nadal           2101.4  
5   Juan Martin Del Potro  2037.5  
6   Kei Nishikori          2035.9  
7   Milos Raonic           2029.4  
8   Jo Wilfried Tsonga     2020.2  
9   Alexander Zverev       2010.2  
10  Marin Cilic            1997.7  
11  Nick Kyrgios           1967.7  
12  Tomas Berdych          1967.0  
13  Gilles Muller          1958.2  
14  Richard Gasquet        1953.4  
15  Stanislas Wawrinka     1952.8  
16  Feliciano Lopez        1945.3

We might quibble with some these positions–the algorithm knows nothing about whatever is plaguing Djokovic, for one thing–but in general, gElo does a better job of reflecting surface-specific ability level than other systems.

The forecasts

Next, we build a hypothetical 128-player draw and run a whole bunch of simulations. I’ve used the top 128 in the ATP rankings, except for known withdrawals such as David Goffin and Pablo Carreno Busta, which doesn’t differ much from the list of guys who will ultimately make up the field. Then, for each seeding method, we randomly generate a hundred thousand draws, simulate those brackets, and tally up the winners.

Here are the ATP top ten, along with their chances of winning Wimbledon using the three different seeding methods:

Player              ATP     W%  Wimb     W%  gElo     W%  
Andy Murray           1  23.6%     1  24.3%     2  24.1%  
Rafael Nadal          2   6.1%     4   5.7%     4   5.5%  
Stanislas Wawrinka    3   0.8%     5   0.5%    15   0.4%  
Novak Djokovic        4  34.1%     2  35.4%     1  34.8%  
Roger Federer         5  21.1%     3  22.4%     3  22.4%  
Marin Cilic           6   1.3%     7   1.0%    10   1.0%  
Milos Raonic          7   2.0%     6   1.6%     7   1.7%  
Dominic Thiem         8   0.4%     8   0.3%    17   0.2%  
Kei Nishikori         9   1.9%     9   1.7%     6   1.9%  
Jo Wilfried Tsonga   10   1.6%    12   1.4%     8   1.5%

Again, gElo is probably too optimistic on Djokovic–at least the betting market thinks so–but the point here is the differences between systems. Federer gets a slight bump for entering the top four, and Wawrinka–who gElo really doesn’t like–loses a big chunk of his modest title hopes by falling out of the top four.

The seeding effect is a lot more dramatic if we look at semifinal odds instead of championship odds:

Player              ATP    SF%  Wimb    SF%  gElo    SF%  
Andy Murray           1  58.6%     1  64.1%     2  63.0%  
Rafael Nadal          2  34.4%     4  39.2%     4  38.1%  
Stanislas Wawrinka    3  13.2%     5   7.7%    15   6.1%  
Novak Djokovic        4  66.1%     2  71.1%     1  70.0%  
Roger Federer         5  49.6%     3  64.0%     3  63.2%  
Marin Cilic           6  13.6%     7  11.1%    10  10.3%  
Milos Raonic          7  17.3%     6  14.0%     7  15.2%  
Dominic Thiem         8   7.1%     8   5.4%    17   3.8%  
Kei Nishikori         9  15.5%     9  14.5%     6  15.7%  
Jo Wilfried Tsonga   10  14.0%    12  13.1%     8  14.0%

There’s a lot more movement here for the top players among the different seeding methods. Not only do Federer’s semifinal chances leap from 50% to 64% when he moves inside the top four, even Djokovic and Murray see a benefit because Federer is no longer a possible quarterfinal opponent. Once again, we see the biggest negative effect to Wawrinka: A top-four seed would’ve protected a player who just isn’t likely to get that far on grass.

Surprisingly, the traditional big four are almost the only players out of all 32 seeds to benefit from the Wimbledon algorithm. By removing the chance that Federer would be in, say, Murray’s quarter, the Wimbledon seedings make it a lot less likely that there will be a surprise semifinalist. Tomas Berdych’s semifinal chances improve modestly, from 8.0% to 8.4%, with his Wimbledon seed of No. 11 instead of his ATP ranking of No. 13, but the other 27 seeds have lower chances of reaching the semis than they would have if Wimbledon stopped meddling and used the official rankings.

That’s the unexpected side effect of getting rankings and seedings right: It reduces the chances of deep runs from unexpected sources. It’s similar to the impact of Grand Slams using 32 seeds instead of 16: By protecting the best (and next best, in the case of seeds 17 through 32) from each other, tournaments require that unseeded players work that much harder. Wimbledon’s algorithm took away some serious upset potential when it removed Wawrinka from the top four, but it made it more likely that we’ll see some blockbuster semifinals between the world’s best grass court players.

Unpredictable Bounces, Predictable Results

Italian translation at settesei.it

These days, the grass court season is the awkward stepchild of the tennis calendar. It takes place almost entirely within a single country’s borders, lasts barely a month, and often suffers from the absence of top players who prefer to rest after the French Open.

The small number of grass court events makes the surface problematic for analysts, as well. The surface behaves differently than hard or clay courts and rewards certain playing styles, so it’s reasonable to assume that many players will be particularly good or bad on grass. But with 90% of tour-level matches contested on other surfaces, many players don’t have much of a track record with which we can assess their grass-court prowess.

I was surprised, then, to find that grass court results are rather predictable. Elo-based forecasts of ATP grass court matches are almost as accurate as hard court predictions and considerably more effective than clay court forecasts. Even when we use “pure” surface forecasts–that is, predicting matches using ratings which draw only on results from that surface–grass court forecasts are a bit better than clay court predictions.

I started with a dataset of the roughly 50,000 ATP matches from 2000 through last week, excluding retirements and withdrawals. As a benchmark, I used official ATP rankings to make predictions for each of those matches. 66.6% of them were right, and the Brier score for ATP rankings over that span is .210. (Brier score measures the accuracy of a set of forecasts by averaging the squared error of each individual forecast, so a lower number is better. To put tennis-specific Brier scores in context, in 2016, ATP rankings had a .208 Brier score, and aggregate betting odds had a .189 Brier score.)

Let’s break that down by surface and compare the performance of ATP rankings, Elo, and surface-specific Elo. “F%” is the percentage of matches won by the favorite–as determined by that system, and “Br” is Brier score:

Surface  ATP F%  ATP Br  Elo F%  Elo Br  sElo F%  sElo Br  
Hard      67.3%   0.207   68.0%   0.205    68.5%    0.202  
Clay      66.1%   0.211   67.1%   0.211    67.0%    0.213  
Grass     66.0%   0.215   67.6%   0.207    68.5%    0.207

All three rating systems do best on hard courts, and for good reason: official rankings and overall Elo are more heavily weighted toward hard court results than they are clay or grass. Surface-specific Elo does best on hard courts for a similar reason: more data.

Already, though, we can see the unexpected divergence of clay and grass courts, especially with surface-specific Elo. It’s possible to explain overall Elo’s better performance on grass courts due to the presumed similarly between hard and grass–if a player excels on one, he’s probably good on the other, even if he’s horrible on clay.  But that doesn’t explain sElo doing better on grass than on clay. There are 3.3 times as many tour-level matches on clay than on grass, so even allowing for the fact that players choose schedules to suit their surface preferences, almost everyone is going to have more results on dirt than on turf. More data should give us better results, but not here.

We can improve our forecasts even more by blending surface-specific ratings with overall ratings. After testing a wide range of possible mixes, it turns out that equally weighting Elo and sElo provides close to the best results. (The differences between, say, 60/40 and 50/50 are extremely small on all surfaces, so even where 60/40 is a bit better, I prefer to keep it simple with a half-and-half mix.) Here are the results for weighted surface Elos for all three surfaces:

Surface  ATP F%  ATP Br  
Hard      68.6%   0.202  
Clay      68.0%   0.207  
Grass     69.8%   0.196

Now grass courts are the most predictable of the major surfaces! Even when we use a weighted average of Elo and sElo, grass court forecasts rely on less data than those of the other surfaces–the surface-specific half of the grass court forecasts uses less than one-third the match results of clay court predictions and less than one-fifth the results of hard court forecasts. In fact, we can do at least as well–and perhaps a tiny bit better–with even less data: A 50/50 weighting of grass-specific Elo and hard-specific Elo is just as accurate as the half-and-half mix of grass-specific and overall Elo.

Regardless of the exact formula, it remains striking that we can predict ATP grass court results so accurately from such limited data. Even if one-third of ATP events were played on grass, I still wouldn’t have been surprised if grass court results turned out to be the least predictable. The more a surface favors the server–and it’s hardest to break on grass–the tighter the scoreline will tend to be, introducing more randomness into the end result. Despite that structural tendency, we’re able to pick winners as successfully on grass as on the more common surfaces.

Here’s my theory: Even though there aren’t many grass court events, the conditions at those few tournaments are quite consistent. Altitude is roughly sea level, groundskeepers follow the lead of the staff at Wimbledon, and rain clouds are almost always in sight. Compare that homogeneity to the variety of hard courts or clay courts. The high-altitude hard courts in Bogota are nothing like the slow ones in Indian Wells. The “clay” in Houston is only nominally equal to the crushed brick of Roland Garros. While grass courts are almost identical to each other, clay courts are nearly as different from each other as they are from other surfaces.

It makes sense that ratings based on a uniform surface would be more accurate than ratings based on a wide range of surfaces, and it’s reassuring to find that the limited available data doesn’t cancel out the advantage. This research also suggests a further path to better forecasts: grouping hard and clay matches by a more precise measure of surface speed. If 10% of tour matches are sufficient to make accurate grass court predictions, the same may be true of the slowest one-third of clay courts. More data is almost always better, but sometimes, precisely targeted data is best of all.