How can we identify the best shots in tennis? At first glance, it seems like a simple problem. Thanks to the shot-by-shot data collected for over 3,500 matches by the Match Charting Project, we can look at every instance of the shot in question and see what happened. If a player hits a lot of winners, or wins most of the ensuing points, he or she is probably pretty good at that shot. Lots of unforced errors would lead us to conclude the opposite.

A friend recently posed a more specific question: Who has the best smash in the men’s game? Compared to other shots such as, say, slice backhands, smashes should be pretty easy to evaluate. A large percentage of them end the point–in the contemporary men’s game (I discuss the women’s game later on), 69% are winners or induce forced errors–which reduces the problem to a straightforward one.

The simplest algorithm to answer my friend’s question is to determine how often each player ends the point in his favor when hitting a smash–that is, with a winner or by inducing a forced error. Call the resulting ratio “W/SM.” The Match Charting Project (MCP) dataset has at least 10 tour-level matches for 80 different men, and the W/SM ratio for those players ranges from 84% (Jeremy Chardy) all the way down to 30% (Paolo Lorenzi). Both of those extremes are represented by players with relatively small samples; if we limit our scope to men with at least 90 recorded smashes, the range isn’t quite as wide. The best of the bunch is Jo-Wilfried Tsonga, at 79%, and the “worst” is Ivan Lendl, at 57%. That isn’t quite fair to Lendl, since smash success rates have improved quite a bit over the years, and Lendl’s rate is only a couple percentage points below the average for the 1980s. Among active players with at least 90 smashes in the books, Stan Wawrinka brings up the rear, with a W/SM of 65%.

We can look at the longer-term effects of a player’s smashes without adding much complexity. It’s ideal to end the point with a smash, but most players would settle for winning the point. When hitting a smash, ATPers these days end up winning the point 81% of the time, ranging from 97% (Chardy again) down to 45% (Lorenzi again). Once again, Tsonga leads the pack of the bigger-sample-size players, winning the point 90% of the time after hitting a smash, and among active players, Wawrinka is still at the bottom of that subset, at 77%.

Here is a list of all players with at least 90 smashes in the MCP dataset, with their winners (and induced forced errors) per smash (W/SM), errors per smash (E/SM), and points won per smash (PTS/SM):

PLAYER W/SM E/SM PTS/SM Jo-Wilfried Tsonga 78% 6% 90% Tomas Berdych 76% 6% 88% Pete Sampras 75% 7% 86% Roger Federer 73% 7% 86% Rafael Nadal 69% 7% 84% Milos Raonic 73% 9% 82% Andy Murray 67% 6% 82% Kei Nishikori 68% 11% 81% David Ferrer 71% 9% 81% Andre Agassi 67% 8% 80% Novak Djokovic 66% 9% 80% Stefan Edberg 62% 12% 78% Stan Wawrinka 65% 10% 77% Ivan Lendl 57% 13% 71%

These numbers give us a pretty good idea of who you should back if the ATP ever hosts the smash-hitting equivalent of baseball’s Home Run Derby. Best of all, it doesn’t commit any egregious offenses against common sense: We’d expect to see Tsonga and Roger Federer near the top, and we’d know something was wrong if Novak Djokovic were too far from the bottom.

**Smash opportunities**

Still, we need to do better. Almost every shot made in a tennis match represents a decision made by the player hitting it: topspin or slice? backhand or run-around forehand? approach or stay back? Many smashes are obvious choices, but a large number are not. Different players make different choices, and to evaluate any particular shot, we need to subtly reframe the question. Instead of vaguely asking for “the best,” we’d be better served looking for the player who gets the most value out of his smash. While the two questions are similar, they are not the same.

Let’s expand our view to what we might call “smash opportunities.” Once again, smashes make our task relatively straightforward: We can define a smash opportunity simply as a lob hit by the opponent.* In the contemporary ATP, roughly 72% of lobs result in smashes–the rest either go for winners or are handled with a different shot. Different players have very different strategies: Federer, Pete Sampras, and Milos Raonic all hit smashes in more than 84% of opportunities, while a few other men come in under 50%. Nick Kyrgios, for instance, tried a smash in only 20 of 49 (41%) of recorded opportunities. Of those players with more available data, Juan Martin Del Potro elected to go for the overhead in 61 of 114 (54%) of chances, and Andy Murray in 271 of 433 (62.6%).

** Using an imperfect dataset, it’s a bit more complicated; sometimes the shots that precede smashes are coded as topspin or slice groundstrokes. I’ve counted those as smash opportunities as well.*

Not all lobs are created equal, of course. With a large number of points, we would expect them to even out, but even then, a player’s overall style may effect the smash opportunities he sees. That’s a more difficult issue for another day; for now, it’s easiest to assume that each player’s mix of smash opportunities are roughly equal, though we’ll keep in mind the likelihood that we’ve swept some complexity under the rug.

With such a wide range of smashes per smash opportunities (SM/SMO), it’s clear that some players’ average smashes are more difficult than others. Federer hits about half again as many smashes per opportunity as del Potro does, suggesting that Fed’s attempts are more difficult than Delpo’s; on those more difficult attempts, Delpo is choosing a different shot. The Argentine is very effective when he opts for the smash, winning 84% of those points, but it seems likely that his rate would not be so high if he hit smashes as frequently as Federer does.

This leads us to a slightly different question: Which players are most effective when dealing with smash opportunities? The smash itself doesn’t necessarily matter–if a player is equally effective with, say, swinging volleys, the lack of a smash would be irrelevant. The smash is simply an effective tool that most players employ to deal with these situations.

Smash opportunities don’t offer the same level of guarantee that smashes themselves do: In the ATP these days, players win 72% of points after being handed a smash opportunity, and 56% of the shots they hit result in winners or induced forced errors. Looking at these situations takes us a bit off-track, but it also allows us to study a broader question with more impact on the game as a whole, because smash opportunities represent a larger number of shots than smashes themselves do.

Here is a list of all the players with at least 99 smash opportunities in the MCP dataset, along with the rate at which they hit smashes (SM/SMO), the rate at which they hit winners or induced forced errors in response to smash opportunites (W/SMO), hit errors in those situations (E/SMO), and won the points when given lobs (PTW/SMO). Like the list above, players are ranked by the rightmost column, points won.

PLAYER SM/SMO W/SMO E/SMO PTW/SMO Jo-Wilfried Tsonga 80% 68% 13% 80% Roger Federer 84% 66% 13% 78% Pete Sampras 86% 68% 15% 78% Tomas Berdych 75% 66% 16% 76% Milos Raonic 85% 67% 14% 76% Novak Djokovic 81% 60% 13% 75% Kevin Anderson 66% 57% 12% 74% Rafael Nadal 74% 57% 16% 73% Andre Agassi 77% 62% 17% 73% Boris Becker 85% 59% 18% 72% Stan Wawrinka 79% 58% 15% 72% Kei Nishikori 72% 57% 17% 70% Andy Murray 63% 52% 15% 70% Dominic Thiem 66% 52% 11% 70% David Ferrer 71% 57% 17% 69% Pablo Cuevas 73% 54% 14% 67% Stefan Edberg 81% 52% 23% 65% Bjorn Borg 81% 41% 20% 63% JM del Potro 54% 48% 19% 60% Ivan Lendl 74% 45% 28% 59% John McEnroe 74% 43% 24% 56%

The order of this list has much in common with the previous one, with names like Federer, Sampras, and Tsonga at the top. Yet there are key differences: Djokovic and Wawrinka are particularly effective when they respond to a lob with something other than an overhead, while del Potro is the opposite, landing near the bottom of this ranking despite being quite effective with the smash itself.

The rate at which a player converts opportunities to smashes has some impact on his overall success rate on smash opportunities, but the relationship isn’t that strong (r^2 = 0.18). Other options, such as swinging volleys or mid-court forehands, also give players a good chance of winning the point.

**Smash value**

Let’s get back to my revised question: Who gets the most value out of his smash? A good answer needs to combine how *well* he hits it with how *often* he hits it. Once we can quantify that, we’ll be able to see just how much a good or bad smash can impact a player’s bottom line, measured in overall points won, and how much a great smash differs from an abysmal one.

As noted above, the average current-day ATPer wins the point 81% of the time that he hits a smash. Let’s reframe that in terms of the probability of winning a point: When a lob is flying through the air and a player readies his racket to hit an overhead, his chance of winning the point is 81%–most of the hard work is already done, having generated such a favorable situation. If our player ends up winning the point, the smash improved his odds by 0.19 points (from 0.81 to 1.0), and if he ends up losing the point, the smash hurt his odds by 0.81 (from 0.81 to 0.0). A player who hits five successful smashes in a row has a smash worth about one total point: 5 multiplied by 0.19 equals 0.95.

We can use this simple formula to estimate how much each player’s smash is worth, denominated in points. We’ll call that Point Probability Added (PPA). Finally, we need to take into account how often the player hits his smash. To do so, we’ll simply divide PPA by total number of points played, then multiply by 100 to make the results more readable. The metric, then, is PPA per 100 points, reflecting the impact of the smash in a typical short match. Most players have similar numbers of smash *opportunities*, but as we’ve seen, some choose to hit far more overheads than others. When we divide by points, we give more credit to players who hit their smashes more often.

The overall impact of the smash turns out to be quite small. Here are the 1990s-and-later players with at least 99 smash opportunities in the dataset along with their smash PPA per 100 points:

PLAYER SM PPA/100 Jo-Wilfried Tsonga 0.17 Pete Sampras 0.11 Tomas Berdych 0.11 Roger Federer 0.10 Rafael Nadal 0.05 Milos Raonic 0.04 Juan Martin del Potro 0.02 Andy Murray 0.01 Kevin Anderson 0.01 Kei Nishikori 0.00 David Ferrer 0.00 Andre Agassi 0.00 Novak Djokovic -0.02 Stan Wawrinka -0.07 Dominic Thiem -0.07 Pablo Cuevas -0.10

Tsonga reigns supreme, from the most basic measurement to the most complex. His 0.17 smash PPA per 100 points means that the quality of his overhead earns him about one extra point (compared to an average ATPer) every 600 points. That doesn’t sound like much, and rightfully so: He hits fewer than one smash per 50 points, and as good as Tsonga is, the average player has a very serviceable smash as well.

The list gives us an idea of the overall range of smash-hitting ability, as well. Among active players, the laggard in this group is Pablo Cuevas, at -0.1 points per 100, meaning that his subpar smash costs him one point out of every thousand he plays. It’s possible to be worse–in Lorenzi’s small sample, his rate is -0.65–but if we limit our scope to these well-studied players, the difference between the high and low extremes is barely 0.25 points per 100, or one point out of every 400.

I’ve excluded several players from earlier generations from this list; as mentioned earlier, the average smash success rate in those days was lower, so measuring legends like McEnroe and Borg using a 2010s-based point probability formula is flat-out wrong. That said, we’re on safe ground with Sampras and Agassi; the rate at which players convert smashes into points won has remained fairly steady since the early 1990s.

**Lob-responding value**

We’ve seen the potential impact of smash skill; let’s widen our scope again and look at the potential impact of smash *opportunity* skill. When a player is faced with a lob, but before he decides what shot to hit, his chance of winning the point is about 72%. Thus, hitting a shot that results in winning the point is worth 0.28 points of point probability added, while a choice that ends up losing the point translates to -0.72.

There are more smash opportunities than smashes, and more room to improve on the average (72% instead of 81%), so we would expect to see a bigger range of PPA per 100 points. Put another way, we would expect that lob-responding skill, which includes smashes, is more important than smash-specific skill.

It’s a modest difference, but it does look like lob-responding skill has a bigger range than smash skill. Here is the same group of players, still showing their PPA/100 for smashes (SM PPA/100), now also including their PPA/100 for smash opportunities (SMO PPA/100):

PLAYER SM PPA/100 SMO PPA/100 Jo-Wilfried Tsonga 0.17 0.18 Roger Federer 0.10 0.16 Pete Sampras 0.11 0.16 Milos Raonic 0.04 0.12 Tomas Berdych 0.11 0.09 Kevin Anderson 0.01 0.08 Novak Djokovic -0.02 0.07 Rafael Nadal 0.05 0.03 Andre Agassi 0.00 0.01 Stan Wawrinka -0.07 0.00 Kei Nishikori 0.00 -0.03 Andy Murray 0.01 -0.03 Dominic Thiem -0.07 -0.05 David Ferrer 0.00 -0.06 Pablo Cuevas -0.10 -0.12 Juan Martin del Potro 0.02 -0.19

Djokovic and Delpo draw our attention again as the players whose smash skills do not accurately represent their smash *opportunity* skills. Djokovic is slightly below average with smashes, but a few notches above the norm on opportunities; Delpo is a tick above average when he hits smashes, but dreadful when dealing with lobs in general.

As it turns out, we can measure the best smashes in tennis, both to compare players and to get a general sense of the shot’s importance. What we’ve also seen is that smashes don’t tell the entire story–we learn more about a player’s overall ability when we widen our view to smash opportunities.

**Smashes in the women’s game**

Contemporary women hit far fewer smashes than men do, and they win points less often when they hit them. Despite the differences, the reasoning outlined above applies just as well to the WTA. Let’s take a look.

In the WTA of this decade, smashes result in winners (or induced forced errors) 63% of the time, and smashes result in points won about 75% of the time. Both numbers are lower than the equivalent ATP figures (69% and 81%, respectively), but not dramatically so. Here are the rates of winners, errors, and points won per smash for the 14 women with at least 80 smashes in the MCP dataset:

PLAYER W/SM E/SM PTS/SM Jelena Jankovic 73% 9% 83% Serena Williams 72% 13% 81% Steffi Graf 61% 9% 81% Svetlana Kuznetsova 70% 10% 79% Simona Halep 66% 11% 76% Caroline Wozniacki 61% 16% 74% Karolina Pliskova 62% 18% 74% Agnieszka Radwanska 54% 13% 74% Angelique Kerber 57% 15% 72% Martina Navratilova 54% 13% 71% Monica Niculescu 50% 15% 70% Garbine Muguruza 63% 19% 70% Petra Kvitova 59% 22% 68% Roberta Vinci 58% 14% 68%

Historical shot-by-shot data is less representative for women than for men, so it’s probably safest to assume that trends in smash success rates are similar for men than for women. If that’s true, Steffi Graf’s era is similar to the present, while Martina Navratilova’s prime saw far fewer smashes going for winners or points won.

Where the women’s game really differs from the men’s is the difference between smash opportunities (lobs) and smashes. As we saw above, 72% of ATP smash opportunities result in smashes. In the current WTA, the corresponding figure is less than half that: 35%. Some of the single-player numbers are almost too extreme to be believed: In 12 matches, Catherine Bellis faced 41 lobs and hit 3 smashes; in 29 charted matches, Jelena Ostapenko saw 103 smash opportunities and tried only 10 smashes. A generation ago, the gender difference was tiny: Graf, Martina Hingis, Arantxa Sanchez Vicario, and Monica Seles all hit smashes in at least three-quarters of their opportunities. But among active players, only Barbora Strycova comes in above 70%.

Here are the smash opportunity numbers for the 17 women with at least 150 smash opportunities in the MCP dataset. SM/SMO is smashes per chance, W/SMO is winners (and induced forced errors) per smash opportunity, E/SMO is errors per opportunity, and PTS/SMO is points won per smash opportunity:

PLAYER SM/SMO W/SMO E/SMO PTW/SMO Maria Sharapova 12% 57% 11% 76% Serena Williams 55% 58% 18% 72% Steffi Graf 82% 52% 17% 71% Karolina Pliskova 47% 52% 16% 70% Simona Halep 14% 41% 11% 69% Carla Suarez Navarro 25% 33% 9% 69% Eugenie Bouchard 29% 50% 18% 68% Victoria Azarenka 35% 52% 17% 67% Angelique Kerber 39% 42% 14% 66% Garbine Muguruza 43% 51% 18% 66% Monica Niculescu 57% 41% 19% 65% Petra Kvitova 48% 50% 19% 65% Agnieszka Radwanska 44% 42% 18% 65% Johanna Konta 30% 47% 21% 64% Caroline Wozniacki 36% 44% 18% 64% Elina Svitolina 14% 38% 14% 63% Martina Navratilova 67% 42% 26% 58%

It’s clear from the top of this list that women’s tennis is a different ballgame. Maria Sharapova almost never opts for an overhead, but when faced with a lob, she is the best of them all. Next up is Serena Williams, who hits almost as many smashes as any active player on this list, and is nearly as successful. Recall that in the men’s game, there is a modest positive correlation between smashes per opportunity and points won per smash opportunity; here, the relationship is weaker, and slightly *negative*.

Because most women hit so few smashes, there isn’t quite as much to be gained by using point probability added (PPA) to measure WTA smash skill. Graf was exceptionally good, comparable to Tsonga in the value she extracted from her smash, but among active players, only Serena and Victoria Azarenka can claim a smash that is worth close to one point per thousand. At the other extreme, Monica Niculescu is nearly as bad as Graf was good, suggesting she ought to figure out a way to respond to more smash opportunities with her signature forehand slice.

Here is the same group of women (minus Navratilova, whose era makes PPA comparisons misleading), with their PPA per 100 points for smashes (SM PPA/100) and smash opportunities (SMO PPA/100):

PLAYER SM PPA/100 SMO PPA/100 Maria Sharapova 0.03 0.21 Serena Williams 0.09 0.15 Steffi Graf 0.15 0.14 Karolina Pliskova -0.01 0.09 Carla Suarez Navarro 0.04 0.08 Simona Halep 0.00 0.07 Eugenie Bouchard -0.02 0.03 Victoria Azarenka 0.08 0.00 Angelique Kerber -0.03 -0.02 Garbine Muguruza -0.07 -0.03 Petra Kvitova -0.07 -0.04 Monica Niculescu -0.13 -0.06 Caroline Wozniacki -0.01 -0.07 Agnieszka Radwanska -0.02 -0.07 Johanna Konta -0.12 -0.08 Elina Svitolina 0.01 -0.09

The table is sorted by smash opportunity PPA, which tells us about a much more relevant skill in the women’s game. Sharapova’s lob-responding ability is well ahead of the pack, worth better than one point above average per 500, with Serena and Graf not far behind. The overall range among these well-studied players, from Sharapova’s 0.21 to Elina Svitolina’s -0.09, is somewhat smaller than the equivalent range in the ATP, but with such outliers as Sharapova here and Delpo on the men’s side, it’s tough to draw firm conclusions from small subsets of players, however elite they are.

**Final thought**

The approach I’ve outlined here to measure the impact of smash and smash-opportunity skills is one that could be applied to other shots. Smashes are a good place to start because they are so simple: Many of them end points, and even when they don’t, they often virtually guarantee that one player will win the point. While smashes are a bit more complex than they first appear, the complications involved in applying a similar algorithm to, say, backhands and backhand opportunities, are considerably greater. That said, I believe this algorithm represents a promising entry point to these more daunting problems.