We all know that double faults aren’t good, but it’s less clear just how bad they are. Over the course of an entire match, a single point here or there doesn’t seem to matter too much, especially when a double fault creeps in at a harmless moment, like 40-love. Yet many missed second serves are far more costly. Let’s try to quantify the impact of tennis’s most enervating outcome.

To do this, we need to think in terms of *win probability.* In each match, a player wins a certain percentage of service points and a certain percentage of return points. If those rates are sufficiently dominating–say, Mihaela Buzarnescu’s 65% of service points won and 59% of return points won in last week’s San Jose final–the player’s chance of winning the match is 100%. No matter how unlucky or unclutch she was, those percentages result in a win. But in a close contest, in which both players win about 50% of points (often referred to as “lottery matches”), the result is heavily influenced by clutch play and luck. In Buzarnescu’s tour de force, flipping the result of a single point would be meaningless. But in a tight match, like the Wimbledon semifinal between John Isner and Kevin Anderson, a single point could mean the difference between a spot in the championship match and an early flight home.

My aim, then, is to measure the average win probability impact of a double fault. To take another example, consider last week’s Washington quarter-final between Andrea Petkovic and Belinda Bencic. Bencic won nearly 51% of total points–59% of her service points and 42% on return–but lost in a third-set tiebreak. Those serve and return components were enough to give her a 56.3% chance of winning the match: claiming more than half of total points usually results in victory, but so close to 50%, there’s plenty of room for things to go the other way.

I refer to this match because double faults played a huge role. Bencic tallied 12 double faults in 105 service points, a rate of 11.4%, more than double the WTA tour average of 5.1%. Had she avoided those 12 double faults and won those points at the same rate as her other 93 service points, she would have ended up with a much more impressive service-points-won rate of 67%. Combined with her 42% rate of return points won, that implies an 87% chance of winning the match–more than 30 percentage points higher than her actual figure! Roughly speaking, each of her 12 double faults cost her a 2.5% chance (30% divided by 12) of winning the match.

A double fault rate above 10% is unusual, but a cost of 2.5% per offense is not. When we run this algorithm across the breadth of the ATP and WTA tours, we find that the cost of double faults adds up fast.

**Tour averages**

Using the method I’ve described above–replacing double faults with average non-double-fault service points–and taking the average of all tour-level matches in 2017 and 2018 through last week’s tournaments, we find that the average WTA double fault costs a player 1.83% of a win. Put another way, every 55 additional double faults subtracts one match from the win column and adds one to the loss column.

In the men’s game, the equivalent number is 1.99% of a win. The slightly bigger figure is due to the fact that men, on average, win more service points, so the difference between a double fault and a successful service offering is greater.

There is, however, an alternative way we could approach this. By comparing double faults to all other service points, we’re trading a lot of the double faults for *first *serve outcomes. We might be more interested in knowing how a player would fare if his or her *second* serve were bulletproof–still eliminating double faults, but replacing them specifically with second serves instead of a generic mix of service points.

In that case, the algorithm remains very similar. Instead of replacing double faults with non-double-fault serve points, we replace them with non-double-fault *second* serve points. Then the cost of a double fault is a little bit less, because second serve points result in fewer points won than service points overall. The second-serve numbers are 1.61% per double fault in the women’s game and 1.70% per double fault in the men’s game. For the remainder of this post, I’ll stick with the generic service points, but one approach is not necessarily better than the other; they simply measure different things.

**Building a player-specific stat**

Odious as double faults are, they are not completely avoidable. Very few players are able to sustain a double fault rate below 2%, and tour averages are around twice that. Since the beginning of 2017, the ATP average has been about 3.9%, and the WTA average roughly 5.1%, as we saw above.

We can measure players by considering their match-by-match double fault rates *compared to tour average. *In Bencic’s unfortunate case, her 12 double faults were 6.7 more than a typical player would’ve committed in the same number of service points. In contrast, in the same match, Petkovic recorded only 3 double faults in 102 service points, 2.2 double faults *fewer* than an average player would have.

We know that each WTA double fault affects a player’s chances of winning the match by 1.83%, so compared to an average service performance, Bencic’s excessive service errors cost her about a 17% chance of winning (6.7 times 1.83%), while Petkovic’s stinginess increased her own odds by about 6.6% (2.2 times 1.83%).

Repeat the process for every one of a player’s matches, and you can assemble a longer-term statistic. Let’s start with the WTA players who, since the start of last season, have cost themselves the most matches (“DF Cost”–negative numbers are bad), along with those who have most improved their lot by avoiding double faults:

Player DF% DF Cost Kristina Mladenovic 7.7% -3.84 Daria Gavrilova 7.9% -3.77 Jelena Ostapenko 7.7% -3.58 Petra Kvitova 8.1% -3.01 Camila Giorgi 8.3% -2.63 Oceane Dodin 10.2% -2.51 Donna Vekic 7.0% -1.91 Venus Williams 6.7% -1.71 Coco Vandeweghe 6.4% -1.60 Aliaksandra Sasnovich 6.7% -1.55 … Agnieszka Radwanska 2.3% 1.27 Sloane Stephens 2.1% 1.43 Caroline Wozniacki 3.2% 1.43 Barbora Strycova 3.5% 1.47 Elina Svitolina 3.9% 1.48 Simona Halep 3.5% 1.53 Qiang Wang 2.6% 1.54 Anastasija Sevastova 3.1% 1.57 Carla Suarez Navarro 2.1% 1.67 Caroline Garcia 3.6% 1.82

And the same for the men:

Player DF% DF Cost Benoit Paire 6.2% -4.51 Ivo Karlovic 5.8% -3.63 Fabio Fognini 5.0% -2.38 Denis Shapovalov 6.3% -2.26 Grigor Dimitrov 5.1% -2.25 Gael Monfils 5.0% -2.22 David Ferrer 5.2% -2.06 Jeremy Chardy 5.3% -2.00 Fernando Verdasco 4.8% -1.94 Jack Sock 4.8% -1.73 … Roger Federer 2.1% 0.88 Tomas Berdych 2.9% 0.89 Juan Martin del Potro 2.8% 0.93 Albert Ramos 3.1% 0.97 Pablo Carreno Busta 2.2% 1.07 Richard Gasquet 2.6% 1.12 John Isner 2.6% 1.23 Dusan Lajovic 1.9% 1.23 Denis Istomin 1.9% 1.23 Philipp Kohlschreiber 2.5% 1.24

**Situational double faults**

These aggregate numbers have the potential to hide a lot of information. They consider only two things about each match: how many double faults a player committed, and how close the match was. This statistic would treat Bencic the same whether she hit nine of her double faults at 40-love, or nine of her double faults in the third-set tiebreak. Yet the latter would have a colossally greater impact.

While this is an important limitation to keep in mind, it appears that double faults are distributed relatively randomly. That is, most players do not hit a majority of their double faults in particularly high- or low-leverage situations. The player lists displayed above show both the most basic stat–double fault percentage–along with my more complex approach. For players with at least 20 matches since the beginning of last season, double fault rate is very highly correlated with the match-denominated cost of double faults. (For men, r^2 = 0.752, and for women, r^2 = 0.789.) In other words, most of the variance in double fault cost can be explained by the *number *of double faults, leaving little room for other factors, such as the importance of the situation when double faults are committed.

That said, there’s plenty of room for additional analysis into those specific sitations. Instead of taking a match-level look at win probability, as I have here, one could identify the point score of every single one of a player’s double faults, and see how each event affected the win probability of that match. I suspect that, for most players, that would amount to a whole lot of extra complexity for not a lot of added insight, but perhaps there are some players who are uniquely able to land their second serve when it matters most, or particularly prone to double faults at key moments. This match-level look has made it clear how costly double faults can be, and it’s possible that for some players, missed serves are even more damaging than that.

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