It has been fairly well established that modern tennis is a game of short rallies. Rallies shorter than 4 shots – i.e. serve, return, one additional shot in the court by the server, and one additional shot in the court by the returner – account for about 70% of all rallies in a match. About 20% of the points then tend to fall into the 5-8 range, and only about 10% of the points will be the extended rallies we see in the highlight reels.
With that being said, I wanted to tackle the following two questions:
- Are the rally length distributions similar regardless of whether the point started with a first serve or a second serve? In other words, are points that start with a second serve longer? And if so, longer by how many strokes?
- Knowing that the first serve winning percentages can climb upwards of 80%, while the second serve winning percentages hover around 50%, how exactly do returners make up that 30%? Specifically, do the returners tend to win their points off of second serve returns in short rallies, quickly following the return? Or are they winning extended baseline battles?
To answer those questions, I looked at the 2020 US Open men’s singles point-by-point statistics, collected by Jeff Sackmann. The dataset contains 27,395 individual points tagged with features such as rally length, serve direction, or distance covered by both players, and can be found here. (If you want to see the code for the analysis below, click here).
Below are the rally lengths broken up by whether the point began with a first or a second serve. Note that I’m ignoring rallies of length 0 – double faults.
Rally Length Following a First Serve (2020 US Open Men’s Singles)
Rally Length Following a Second Serve (2020 US Open Men’s Singles)
Three things immediately stand out:
- Both graphs are significantly right-skewed, confirming the prevalence of the 0-4 rally length on both first and second serve points. Overall, there just weren’t that many rallies longer than 5 strokes, regardless of what serve was put in play.
- However, looking within the 0-4 range itself, there is a difference. In the first serve graph, we see two spikes at rally lengths 1 and 3; intuitively that makes sense – both 1 and 3 stroke rallies are points for the server. In particular, a rally length of 1 is either and ace or an unreturned serve, while rally length of 3 is a “Serve + 1” pattern by the server, where the “+1” results in either a winner, or an error by the returner.
- Looking at the 0-4 range in the second serve graph, we see a smaller contribution of the 1 stroke rally – i.e. second serves get returned more often than first serves – but the key here is the spike in the rally of length 2. A rally of length 2 is a point for the returner – serve in play, followed by either a return winner, or an error by the server on shot number 3. Also, we have further confirmation of the importance of the “Serve +1” patterns for the server, with the rally of length 3 being the most common following a second serve in play.
Here is a table outlining some of the same data:
| Serve in Play | Rally Length (Mode) | Rally Length 25th Percentile | Rally Length Median | Rally Length 75th Percentile | 0-4 Rallies | 5-8 Rallies | 9+ Rallies |
| 1st | 1 | 1.0 | 2.0 | 3.0 | 88.25% | 7.83% | 3.92% |
| 2nd | 3 | 2.0 | 3.0 | 4.0 | 81.66% | 11.08% | 7.26% |
Were second serve rallies longer at the 2020 US Open? Yes, but with a caveat. The mode, 25th, 50th, and 75th percentile values were all higher for the second serve points. We also see a bigger proportion of the rallies going past 5 strokes on the second serve. With that being said, over 80% of second serve points were still contested in that 0-4 range. Interestingly enough, the decrease of about 6.5% in the 0-4 point length between the first and second serve points is about evenly distributed among the 5-8 and 9+ rally lengths; both increased in the neighborhood of about 3.2%.
Now that we know the answer to the question of “are second serve rallies longer” is “yes, but..,” let’s take a look at where points won on the return come from. To do that, we need to look at “even” rally lengths: remember that rallies of length 2, 4, 6 etc. all indicate a point won by the returner. Comparing the frequencies with which rallies of a given length occur on first or second serves will give us insight as to where the returners do most of their damage.
| Serve / Rally Length | 2 | 4 | 6 | 8 |
| 1st | 16.67% | 6.86% | 1.84% | 0.97% |
| 2nd | 26.68% | 9.62% | 2.91% | 1.69% |
| Difference | +10.01% | +2.76% | +1.07% | +0.72% |
Looking at the table above, it is clear that the rally length of 2 – i.e. serve in play, followed by a return winner, or an error by the server on stroke 3 – was the biggest difference maker for the returners. About 26.7% of all points that started with a second serve were of length 2, as opposed to only 16.7% of points that started with a first serve. This difference of 10% drops off dramatically once the rally gets to 4 strokes and longer, meaning the effect of the serve and return diminishes pretty quickly after the third and fourth stroke of the rally.
I always wondered whether the players with high second serve return win percentages were good returners, or simply good baseliners, who put the return in play and grind out long rallies. This data suggests the former: good second serve return win percentage will probably be driven by strong performance in the 2 and 4 length rallies, highlighting the effect of the return itself.
A simple takeaway from this exercise is as follows: as you sit down to watch the Australian Open in the next few weeks, you have two options. If you want to see tennis for what it is and watch the whole matches, it will be a lot of serving and returning and not much else most of the time. If, on the other hand, you are a rally aficionado, I would just wait for the highlights.
Michal's Musings 
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