The more gut-wrenching the moment, the more likely it is to stick in memory. We easily recall our favorite player double-faulting away an important game; we quickly forget the double fault at 30-0 in the middle of the previous set. Which one is more common? The mega-choke or the irrelevancy?

There are three main factors that contribute to double faults:

- Aggressiveness on second serve. Go for too much, you’ll hit more double faults. Go for too little, your opponent will hit better returns.
- Weakness under pressure. If you miss this one, you lose the point. The bigger the point, the more pressure to deliver.
- Chance. No server is perfect, and every once in a while, a second serve will go wrong for no good reason. (Also, wind shifts, distractions, broken strings, and so on.)

In this post, I’ll introduce a method to help us measure how much each of those factors influences double faults on the ATP tour. We’ll soon have some answers.

**In-game volatility**

At 30-40, there’s more at stake than at 0-0 or 30-0. If you believe double faults are largely a function of server weakness under pressure, you would expect more double faults at 30-40 than at lower-pressure moments. To properly address the question, we need to attach some numbers to the concepts of “high pressure” and “low pressure.”

That’s where *volatility* comes in. It quantifies how much a point matters by considering several win probabilities. An average server on the ATP tour starts a game with an 81.2% chance of holding serve. If he wins the first point, his chances of winning the game increase to 89.4%. If he loses, the odds fall to 66.7%. The volatility of that first point is defined as the difference between those two outcomes: 89.4% – 66.7% = 22.7%.

(Of course, any number of things can tweak the odds. A big server, a fast surface, or a crappy returner will increase the hold percentages. These are all averages.)

The least volatile point is 40-0, when the volatility is 3.1%. If the server wins, he wins the game (after which, his probability of winning the game is, well, 100%). If he loses, he falls to 40-15, where the heavy server bias of the men’s game means he still has a 96.9% chance of holding serve.

The most volatile point is 30-40 (or ad-out, which is logically equivalent), when the volatility is 76.0%. If the server wins, he gets back to deuce, which is strongly in his favor. If he loses, he’s been broken.

**Mixing in double faults**

Using point-by-point data from 2012 Grand Slam tournaments, we can group double faults by game score. At 40-0, the server double faulted 3.0% of points; at 30-0, 4.2%; at ad-out, 2.8%.

At any of the nine *least* volatile scores, servers double faulted 3.0% of points. At the nine *most* volatile scores, the rate was only 2.7%.

(At the end of this post, you can find more complete results.)

To be a little more sophisticated about it, we can measure the correlation between double-fault rate and volatility. The relationship is obviously negative, with an r-squared of .367. Given the relative rarity of double faults and the possibility that a player will simply lose concentration for a moment at any time, that’s a reasonably meaningful relationship.

And in fact, we can do better. Scores like 30-0 and 40-0 are dominated by better servers, while weaker servers are more likely to end up at 30-40. To control for the slightly different populations, we can use “adjusted double faults” by estimating how many DFs we’d expect from these different populations. For instance, we find that at 30-0, servers double fault 26.7% more than their season average, while at 30-40, they double fault 28.6% less than average.

Running the numbers with adjusted double fault rate instead of actual double faults, we get an r-squared of .444. To a moderate extent, servers limit their double faults as the pressure builds against them.

**More pressure on pressure**

At any pivotal moment, one where a single point could decide the game, set, or match, servers double fault less than their seasonal average. On break point, 19.1% less than average. With set point on their racket, 22.2% less. Facing set point, a whopping 45.2% less.

The numbers are equally dramatic on match point, though the limited sample means we can only read so much into them. On match point, servers double faulted only 4 times in 296 opportunities (1.4%), while facing match point, they double faulted only 4 times in 191 chances (2.2%).

**Better concentration or just backing off?**

By now, it’s clear that double faults are less frequent on important points. Idle psychologizing might lead us to conclude that players lose concentration on unimportant points, leading to double faults at 40-0. Or that they buckle down and focus on the big points.

While there is surely some truth in the psychologizing–after all, Ernests Gulbis is in our sample–it is more likely that players manage their double fault rates by changing their second-serve approach. With a better than 9-in-10 chance of winning a game, why carefully spin it in when you can hit a flashy topspin gem into the corner? At break point, there’s no thought of gems, just fighting on to play another point.

And here, the numbers back us up, at least a little bit. If players are avoiding double faults by hitting more conservative second serves on important points, we would expect them to lose a few more second serve points when the serve lands in play.

It’s a weak relationship, but at least the data suggests that it points in the expected direction. The correlation between in-game volatility and percentage of second serve points won is negative (r = -0.282, r-squared = 0.08). Complicating the results may be the returner’s conservative approach on such points, when his initial goal is simply to keep the ball in play, as well.

Clearly, chance plays a substantial role in double faults, as we expected from the beginning. It’s also clear that there’s more to it. Some players do succumb to the pressure and double fault some of the time, but those moments represent the minority. Servers demonstrate the ability to limit double faults, and do so as the importance of the point increases.

*Full results, from 2012 Grand Slam main draw matches. ‘Adj DF%’ is the ratio of actual double faults to expected double faults, considering the season average of each player in the sample. ‘Vol’ is volatility, as described in the post above. In the bottom rows of the table, ‘MP’ is match point, ‘SP’ is set point, and ‘Fcg’ means ‘facing,’ so ‘Fcg MP’ tallies points where the returner held match point.*

SCORE DF% Adj DF% VOL 0-0 3.4% 1.01 22.7% 0-15 3.5% 1.03 33.3% 0-30 3.1% 0.88 39.9% 0-40 2.7% 0.76 31.1% 15-0 3.5% 1.07 16.7% 15-15 2.9% 0.85 29.5% 15-30 3.1% 0.89 44.8% 15-40 3.4% 0.99 48.6% 30-0 4.2% 1.27 9.5% 30-15 3.4% 1.00 20.9% 30-30 3.2% 0.94 42.7% 30-40 2.5% 0.71 76.0% 40-0 3.0% 0.93 3.1% 40-15 3.3% 0.99 8.7% 40-30 3.5% 1.04 24.0% DEUCE 3.1% 0.91 42.7% AD IN 3.2% 0.95 24.0% AD OUT 2.8% 0.79 76.0% BREAK PT 2.8% 0.81 Fcg MP 2.2% 0.61 Fcg SP 1.9% 0.55 GAME PT 3.3% 0.98 MP 1.4% 0.45 SP 2.4% 0.78