# Statistical Analysis of Roger Federer’s Second Serve

Hey guys, this is a post from Gaurav who I am sure you remember posted here a couple of months ago with a statistical breakdown of the Federer vs. Nadal rivarly. This time he has put together an analysis of Roger's second serve which is very interesting.

We all know Roger has an extremely reliable second serve just from what we see on our screens, but how does it stack up statistically and could he be doing more with it? Take it away Gaurav…

It was a few months back and I was watching a match- now I donβt remember which one, but what I do remember, is that one of the two was getting pretty badly creamed on his second serve. The match went something like this- Fault, Second serve, BOOM, return winner, server watches winner fly by. Repeat.

The funny thing was that it wasnβt like he had a bad first serve. In fact when it did land (if it ever did- he was so traumatized by this stage that first serves were a distant dream) he won a majority of the points fairly easily. To the extent that the commentator in the box, who, to until this point, was quite happy doling out inane observations to any one who hadnβt already had the foresight to press the mute button, said something actually insightful. It went something like this.

βYou know, Robby, it might make more sense for him to go for more on his second serve. Itβs not like it could get any worse than it already is (which was pretty true). At the very worst, heβs going to double fault, and lose the point. If it goes inβ¦ well then itβs a different story altogether.β

This remark, as unremarkable as it may be, came back to me when I was watching Federer play Murray in Shanghai last year. Murray was dispatching Federerβs second serve, sending it back with compound interest, to the point where Federer was forced to go for something extra on his serve- in one game resulting in three double faults in a row. (Needless to say he got broken in that game. Murray could have sat in his chair and heβd still have the break)

Jokes aside, it was a result of the pressure Murray was applying on Federerβs second serve. Now Federerβs second serve is widely believed to be the best in the business. Federerβs no pushover on his second serve, but here was Murray, walloping Federerβs serve as though it was a football travelling in slo- mo.

Novak has led the way in the return department. If there was one facet to the game that still had scope for improvement, it was the second serve return. Watching him pulverize Nadal on serve has to be one of the highlights of any tournament.

Back to Roger and the second serve, the remark got me thinking about whether it made sense for a player (not Roger in particular) to go for more on the second serve, risking a double fault, so as to (perhaps) ensure a higher probability of a winning a point.

So I got thinking and decided to have a look at the numbers. I managed to pull out Federerβs serve statistics.

Federer Serve | 1st serve | 2nd serve |
---|---|---|

% Serve landed | 64 | 100 |

Avg speed (mph) | 125 | 95 |

Relative speed | 100 | 76 |

% Points Won | 78 | 59 |

The probability of a player winning a point (on serve) is mathematically equal to the chance of winning the point on the first serve OR doing a fault AND THEN winning the point on his second serve.

For those of you who were fond of math and probability in school, to put this in an equation, the Probability of winning a point equals:

**Prob(Landing First serve) * Prob(Winning point once first serve has landed)**

**+**

**Prob(fault) * Prob(Landing second serve) * Prob(Winning point once second serve has landed)**

In Federerβs case, inputting these values we see that on average:

=((0.64)*(0.78)) + ((0.36)*(1.00)*(0.59))

**=0.71124**

Telling us that Federer wins **71.12%** of his points on serve.

Now the point of this article is to see, whether Federer could, in anyway, improve the percentage of points won on serve. Assuming that his first serve is the limit of his serving prowess, there is nothing that can be done to improve it any further. Therefore we go on to the second serve (the second half of the equation)

Does it make more sense for Roger to try and go for more on his second serve? Will this enhance his chance of winning a point beyond the current 71% that he is at or is he already serving at the optimum level his abilities allow? Letβs have a look at the numbers.

By plotting the figures of Federerβs serve present in the table into the graph, and assuming linearity, we can derive the table below (which is actually far less daunting than it looks) which basically gives us a rough estimate of what percentage of points we could expect Federer to win with regards to the speed of the serve.

Here I should admit one of the limitations of the analysis- itβs purely based on the speed and fails to take into account factors such as spin. What I do feel however is that speed is typically compensated for by additional spin.

Also, two absolutely identical serves with respect to speed and spin might have different results depending on the ballβs proximity to the lines. A first serve would imbibe that additional risk, a second generally not.

What I am trying to get at is that there are limitations to the analysis, but for all practical purposes, it should work within the stated assumptions and shouldnβt really have an impact on the end result of the study.

a) Actual Speed | b) Relative Speed* | c) % Landed | d) % Points Won | e) % of fault and winning second serve | f) % of first serves landed AND points won | g) You win Point (=e+f) |
---|---|---|---|---|---|---|

95 | 76 | 100 | 58.9 | 21.204 | 49.92 | 71.124 |

96.25 | 77 | 98.5 | 59.675 | 21.160755 | 49.92 | 71.080755 |

97.5 | 78 | 97 | 60.45 | 21.10914 | 49.92 | 71.02914 |

98.75 | 79 | 95.5 | 61.225 | 21.049155 | 49.92 | 70.969155 |

100 | 80 | 94 | 62 | 20.9808 | 49.92 | 70.9008 |

101.25 | 81 | 92.5 | 62.775 | 20.904075 | 49.92 | 70.824075 |

102.5 | 82 | 91 | 63.55 | 20.81898 | 49.92 | 70.73898 |

103.75 | 83 | 89.5 | 64.325 | 20.725515 | 49.92 | 70.645515 |

105 | 84 | 88 | 65.1 | 20.62368 | 49.92 | 70.54368 |

106.25 | 85 | 86.5 | 65.875 | 20.513475 | 49.92 | 70.433475 |

107.5 | 86 | 85 | 66.65 | 20.3949 | 49.92 | 70.3149 |

108.75 | 87 | 83.5 | 67.425 | 20.267955 | 49.92 | 70.187955 |

110 | 88 | 82 | 68.2 | 20.13264 | 49.92 | 70.05264 |

111.25 | 89 | 80.5 | 68.975 | 19.988955 | 49.92 | 69.908955 |

112.5 | 90 | 79 | 69.75 | 19.8369 | 49.92 | 69.7569 |

113.75 | 91 | 77.5 | 70.525 | 19.676475 | 49.92 | 69.596475 |

115 | 92 | 76 | 71.3 | 19.50768 | 49.92 | 69.42768 |

116.25 | 93 | 74.5 | 72.075 | 19.330515 | 49.92 | 69.250515 |

117.5 | 94 | 73 | 72.85 | 19.14498 | 49.92 | 69.06498 |

118.75 | 95 | 71.5 | 73.625 | 18.951075 | 49.92 | 68.871075 |

120 | 96 | 70 | 74.4 | 18.7488 | 49.92 | 68.6688 |

121.25 | 97 | 68.5 | 75.175 | 18.538155 | 49.92 | 68.458155 |

122.5 | 98 | 67 | 75.95 | 18.31914 | 49.92 | 68.23914 |

123.75 | 99 | 65.5 | 76.725 | 18.091755 | 49.92 | 68.011755 |

125 |
100 |
64 |
77.5 |
17.856 |
49.92 |
67.776 |

What we see is that Roger is actually serving at the most optimum level his body and prowess allow him to (yellow highlighted row). As we go down the table, the (second) serve speeds are increasing (column a).

Corresponding to that, as the speed increases, the percentage of serves landed falls (column c). These two factors combine to give us the probability of winning the second serve (column e) and in turn, the point (column g).

It shouldnβt come as any surprise really, given how technically gifted Federer is, that he is maximizing his potential. In fact, if he tried going for his average first serve in place of his second, it would be an absolute disaster, right there at the bottom of the table, with most recent evidence coming from the outcome of the game in Shanghai against Murray.

This isnβt to say that all players on tour might be maximizing their chance of success. There might be a case for a certain player to have a look at his figures and see whether it makes more sense for him to try and go for a little more on his second delivery.

Also, a crucial parameter that Iβve mentioned a few times that Iβd like to bring up now, is a players physical and technical prowess. At an early age in oneβs career, those are the bars that are progressively raised over time, as one is moulded into the player that he eventually becomes.

The older one gets, the harder it gets to raise this level, not just due to the physical limitations placed by age, but also by the limits of your natural ability.

How could Federer improve his? Perhaps by going for serves over the lowest part of the net, taking a break from competitive tennis and hitting the gym for a couple of months or just praying that heβs a late bloomer and gains a few inches in height!

Itβs a trade off he probably knows better than any of us and we should leave it to him. Because more often than not, heβs shown the entire world how it should be done. Thereβs a reason why heβs the greatest of all timeβ¦

## Appendix:

**Relative speed**

The speed of Federerβs serve divided by his average first serve speed. Multiplied by 100 (Used just for ease of calculation)

For example- His average first serve speed equals 125 mph. So the relative speed of his average first serve would equal 125 divided by 125 and multiplied by 100, giving us 100.

Similarly his relative second serve speed would equal 95/ 125*100 = 76.

**The relationship between speed and percentage points won**

This is an interesting one and is where the limitations stated above come into play. Iβd like to call it factor Ex (you know from FedEx :P). When looking at Rogerβs figures I realized that there was a close relationship between the speed of his serve and the percentage of points that he won.

This value could be calculated by dividing the Percentage of points won by the average speed of the serve. In fact for both, his first and his second serve, this value came to 62, meaning that Federerβs percentage of winning a point is 0.62 times the speed of the serve in miles per hour.

i.e: Factor Ex= Percentage of Points Won / Speed of Serve (mph) * 100

For his first serve, taking the values from table number 1: 78/125*100= 62.4

Similarly, for the second: 59/95*100= 62.1

Incredibly interesting if you ask me and if you have any questions then let me know in the comments below.

Note from Jonathan: Here's some additional analysis I was kindly emailed by Dr Harry Harper who read Gaurav's article and put a little something together himself:

Guaravβs article about second serves was something I have been interested in for quite some time, and itβs prompted me to revisit this area, to see if Fed really has found the optimum second serve.

Taking his equations, we see that:

Probability of winning a point equals:

**Prob(Landing First serve) * Prob(Winning point once first serve has landed)**

**+**

**Prob(fault) * Prob(Landing second serve) * Prob(Winning point once second serve has landed)**

And taking numbers from the best tennis stats website ever, http://www.tennisabstract.com/.

We can make this table:

1stIn | 1st% | 2ndIn | 2%-InP | SPW |
---|---|---|---|---|

63.10% | 76.70% | 94.60% | 60.40% | 69.48% |

Where 1stIn is Prob(Landing First serve),

1st% is Prob(Winning point once first serve has landed)

2ndIn is Prob(Landing second serve)

2%-InP is Prob(Winning point once second serve has landed, i.e. is βIn Playβ)

SPW% is Probability of winning a point on serve

Note that:

Prob(fault) =100%-1stIn

And that 2ndIn =/= 100%

Ok, so thereβs nothing new so far, same formulae, different numbers.

This is where things start getting interesting. Suppose nothing changes on Fedβs first serve, but that he makes changes to his second. This will affect two quantities: 2ndIn , and 2%-InP, which in turn will change SPW. So if we make these two the variables, we get:

SPW = 48.4% + 36.9%*2ndIn*2%-InP

Where the x, and y axis are the 2ndIn and 2%-InP axes respectively, and height is SPW.

But this graph doesnβt really tell us much, except that SPW is best when both 2ndIn and 2%-InP are at their greatest, which is fairly unexciting. It just says that a server is at their best if they can get all their second serves in and win all resulting points.

This is because weβve ignored so far Guaravβs best insight, the Ex Factor. In other words, that 2ndIn and 2%-InP are closely related, because they are both measurements of what we call a βlatent variableβ, something which we could study instead, because it is the main reason why the other two variables change. The latent variable? Speed.

What Iβm saying here, is that variations to speed give us variations in 2ndIn and 2%-InP, and that if we take the relationships between speed and each of these individually, we can replace the two variables with just the one, and find out what the optimum speed Fed should serve at is.

So, first up Speed vs 2ndIn. That is, how speed affects the chance of getting the ball in.

Assuming, as Guarav did, linearity, and taking his numbers of 95 mph for 2nd serve, and 125mph for 1st serve, we can construct this graph:

Note how the graph flattens on either end, as we assume 100% success on a serve of 85mph, and 0% at 190mph, since otherwise weβd have a negative probability, or one greater than 100%, which is ridiculous.

This graph gives the relationship 2ndIn = 1.9435-0.0105*Speed, for 89Speed186, and 0 or 1 otherwise.

Now Speed vs 2%-InP. That is, how speed affects the chance of winning the point, should the ball land in.

Assuming again, as Guarav did, linearity, and taking his numbers of 95 mph for 2nd serve, and 125mph for 1st serve, we can construct this graph:

This graph gives the relationship 2%-InP = 0.005433*Speed + 0.087833, for Speed less than 168, and 1 otherwise.

Right then, now we can substitute back into the original, to find that:

SPW = 48.4% + 36.9%*2ndIn*2%-InP

= 48.4% + 36.9%*(1.9435-0.0105*Speed)*(0.005433*Speed + 0.087833)

For 89 Speed 168. The equation is slightly different outside that interval.

Time for the moment of truth: the graph of Speed vs SPW:

So, the maximum point?

At speed = 90mph, where SPW=69.65%

Currently, as we saw above, Rogerβs at 95mph, which gives him 69.48%.

The potential increase of taking off 5mph is 0.17% more points won on serve. Itβd only make a difference once in every 588 service points.

First of all, thatβs pretty insignificant.

Secondly, the linearity assumption is a pretty big assumption to make. It suggests that at 90mph, Roger will make 99.85% of 2nd serves in, which is quite ridiculous. No-one, I mean no-one makes more than about 95% of 2nd serves in, which is very close to what Roger makes currently anyway.

Things like being human, and dealing with pressure mean that the speed/2ndIn graph should max out at about 95%, because no-one is going to make 99.85% of 2nd serves. Thatβs ridiculous. Look at the current 2ndIn top ten:

Player | 2ndIn |
---|---|

Victor Hanescu [ROU] | 0.95 |

Denis Istomin [UZB] | 0.948 |

Roger Federer [SUI] | 0.946 |

Philipp Kohlschreiber [GER] | 0.943 |

Florian Mayer [GER] | 0.942 |

Novak Djokovic [SRB] | 0.942 |

Tomas Berdych [CZE] | 0.941 |

Jo Wilfried Tsonga [FRA] | 0.939 |

Jarkko Nieminen [FIN] | 0.937 |

Janko Tipsarevic [SRB] | 0.935 |

Guess whoβs right up there? Roger #3!

The fact that Fed is so close to that perfect number of 69.65%, suggests that he has indeed found the ideal serve speed.

In fact, if we make the maximum 2nd serve landed percentage 95%, rather than 100%, then the graph looks like this:

And the maximum is right where Fed serves, at 95mph.

## Conclusion

There is no profit to be made from speeding up the second serve.

Mathematically there is profit to be made from slowing it down, but it is almost insignificant, and it is unlikely that the theory will translate in the real world because Roger is human.

Therefore Rogerβs second serve speed is currently **peRFect**.

Naturally.

first

Maiden title!

Wow Gaurav.

What happens if he DECREASES the speed of the second serve? There’s a point where this would be suicidal, of course, but is there a range where it might help?

Suicidal would be correct. Unless he decides to do a Hingis and pull out the infamous underarm serve. That might be quite interesting actually.

He should do that vs. Dull on clay. He stands so far back he would win the point. Also would psych him out.

Now, to add to that, the slower out wide serves for example on the deuce court give amazing results against opponents who stand far back. If you draw a straight line, you could say that as the speed goes down, the percentage points won also go down. Wrong! In this scenario, a lower speed against a certain opponent will let you win a higher percentage of points. So, the straight line will not remain that straight after all.

The same serve may actually give you worse results against a opponent who stands closer to the baseline. Maybe a fast T serve will give you an advantage.

Really interesting stuff. Good work Gaurav.

It’s so painful and frustrating watching Murray annihilate rogers second serve.

Fiasco had some success putting more juice into his second serve against Murray in the wimbledon QF. He was really going for the lines under pressure. Sure he had lots of DF but I reckon in terms of that match it paid off and he could easily have won.

Hi Gaurav… A very interesting article on Feds second serve. I liked how you have incorporated the mathematics with equations and graphs and used 100 in your calculation to keep it fairly simple for me at least as i only got a grade C in GCSE. Again well done and maybe you should send this post to Roger and his team direct as they may find it productive and put it to good use?!!? π

Ok, I see what you did there. It’s an extrapolation of the second serve, all the way limiting to the speed of the first serve. It’s very linear, as you’ve admitted and you’ve not accounted for spin.

So, it’s easy for you to say, given the linearity of the numbers, that Federer’s second serve is at its optimum. That’s not necessarily true. You can potentially increase speed by changing the line and at the same time increasing the length of the line you take. That can also be achieved by changing where Federer stands for his second delivery. Point I’m trying to make is, increasing the second serve speed doesn’t necessarily mean that the % second serve landed number will reduce.

Again, going for serves over the lower part of the net is not a solution because it would put Federer much further away (closer to the alley) than he would want to. That will work fine on grass and actually I think he does stand a bit wider on grass because the skid factor reduces the opponents options. But, on clay or hard courts, he will be open for a down the line or a sharper cross court. Plus, the T serve will miss more often than not if the delivery tends to be shorter. If he chooses to stay where he is and continues going down the lower part, then his serves become a lot more predictable.

I don’t know if I made sense with all that. A big part of Federer’s woes are the racquets his opponents use. They are also much fitter and conditioned. Courts speeds have also been a big factor. I doubt there is a sane way to improve the second serve. If he does that, it’s reasonable to assume his opponents can find a way to counter it. Actually, they will find a way to counter it.

Federer’s only sane option is to improve the points he wins on his opponents second serve. Racquet, fitness, and age contribute in extraordinary way to achieve that.

Good points, Sid. And thanks for the other day.

Gaurav, two questions. re Roger’s serve statistics:

1) Shouldn’t the percentage of second serves landed be something less than 100% to reflect double faults? Or are double faults included in the percentage of points won (59%)?

2) What period do these stats cover? His entire professional career?

Hey Arif,

You’re right about double faults, but the proportion of double faults is next to negligible, so there’s no point including it in the analysis.

For eg: You could say that Janowicz throws in a significant amount of double faults. There I would say that he is employing precisely what this analysis is looking to solve for and serving at a faster pace on his second, risking a double fault. For him, the optimum probably lies at perhaps a 5% faster first serve, allowing for the probability of the serve landing only say, 95%, but winning him more than sufficient points to cover that risk.

And yes, these cover his entire professional career. I had to scour the net for these, but they seem reliable enough.

*…faster ‘second’ serve, allowing…

Does this mean Federer’s time is over and his fans should give up hope and let him go? You know, just grow up and stop being cheerleaders? π

No way man. Cheerleader for life!!!!!

Same here, Gaurav. I was taking a dig at our friend Arif here, trying to elicit a response from him. Think he will take the bait? π

Also, I’ll reply to my big comment so as to explain why I don’t comply with the table you’ve listed. More of that later!

Hi Dr. Harper!!

Loved your additional insights. Will go over some of the numbers again in order to ensure that I’ve understood everything correctly, but from what I can see, you’re spot on. And the assumptions seem fair, and the limitations mentioned upfront.

I wasn’t able to pull out his second serve stats. It doesn’t change anything, but just makes it all the more accurate. Thanks for that. I’m actually quite surprised to see that people actually land around 95% of second serves. I was under the impression that it was close to around 99ish, which is why I just assumed it to be 100 for simplicity. But s great table to be aware of.

Anyhoo, back to the article. Would love discussing more technical stuff with you, if you have some more ideas.

Sure Gaurav. There are heaps more topics we can look into too. We just need time and the itch to investigate.

And about double faults, whilst they themselves are rare, and almost insignificant, once that first we’ve is out, they become considerably more likely, naturally.

Wow Gaurav, great job!

Though must admit I had to read the post for at least 5 times to just get a grip on the subject. Just me as a woman or poor old brain no longer takes these numbers easily…and to post intelligent comment here, haha, reading even 100 times wouldn’t be enough, sorry. For boys here it seems like a daily discussion…hah? I’m impressed!

Well done anyway π

OMG more numbers…hat off to you guys.

OlΓ‘ Jon!

OMG!

As an engineer, the only thing that I want to say is….

I LOVE YOUR ANALYSIS, SOOO PROFESSIONAL!

Great job, Jon and Gaurav! Great job π

Nothing to do with me this one, I just put the post together. All Gaurav and Harry π

Say you’re playing a game of poker with eight other friends of yours in your basement, and you decide to play all the way to infinity, and also make sure that there will be no bets, so nobody folds, which means everyone gets to see the river card and show their hands. Who do you think will win the most?

Now, let’s come to this analysis. Help me understand, somebody, anybody, wouldn’t the “optimum” speed of the second serve also depend on the opponents playing style, surface, and the equipment limitation that Roger has etc.? Yes, I understand the graphs above have been plotted based on Roger’s career stats. Call it intuition, but I still believe Roger’s second serve is not at it’s optimum and there is room for improvement.

Why do we assume the results to be linear on a second serve? Let me explain. Don’t you think that perhaps against a certain opponent, on a certain surface, on a certain side of the court, there are advantages to be gained by increasing the speed of the serve that could be significant? I am not even talking about location here folks, I’m just talking a about a specific scenario.

I am not a mathematician. But my conclusion is, there really is no way to plot a graph and say that someones second serve is at its optimum. It’s not something as simple as using linear calculations for specific second serve speeds. You can’t just draw a straight line with a specific slope is what I’m saying.

I don’t intend to diminish Gaurav’s work here. But, I’m the most qualified to refute it because. Why? Because I have without a doubt, the worst second serve in the history of the game π So, I know for a fact that variations of speeds give different results against different opponents. The height of the opponent plays a big factor to for example π

Conclution: Tennis is more than just plotting a graph and we cannot assume that Roger’s second serve is at its optimum.

Sid,

I think you’re missing the point of the analysis. Compare it to an article Jon wrote about which racket has served Federer the best. Now you can argue to the end of time that it depended on who his opponent was, or what his playing style was and how far back they stood from the net, or whether they drank two sips of a mango smoothie on the day or what not- but for sure as hell not on his racket. Nobody’s denying all those factors. Sure they play their part. But take the analysis for what it is, and not for what it isn’t.

Every analysis has a scope, and limitations, both of which are mentioned here. If stats could figure out a sport, everyone would have mathematically had their games figured out. But we all know that that isn’t the case. Stats and analyses like this can assist a player in getting his game as closed to ‘figured out’ as possible.

And plus, it’s just for fun- I would think of all people on this blog, you know it better than any of us. Just because we don’t have access to all the data you mentioned, or no way to incorporate some elements mathematically into an analysis doesn’t mean we shouldn’t try whatsoever.

Gaurav, yes I do have problems with the straight line, but, the numbers you pulled out are not wrong. Your extrapolation of those numbers is oversimplified though. I have a problem only with the conclusion that Roger’s second serve is at its optimum. No, it’s not. It’s a big reason he is suffering against a lot of the top players. There’s plenty of room for improvement.

I found out that in three matches at the end of the AO and the first two at Wimbledon, his average second serve speed was 97 or more. In one match, it reached a 99 average. There must be a reason he believed 95 wasn’t the optimum speed to go with. Agreed that you’re talking about a 95 average over his career though.

I’m not diminishing your effort by any means, but I’ll stand vociferously against your final assessment that Roger has found the optimum second serve and it cannot be improved upon π

It’s all for fun, I know. And I also agree that we can only do so much with the data available to us. Federer’s second serve is far from PeRFect π

Cheers!

P.S. Whoever drinks mango smoothie before facing Roger needs to know that it ain’t gonna work!

I think Silver Moonlight hit the nail on the head. Given Roger’s current ability, his second serve is optimal.

But, the reason I started out this analysis in the first place was because I, like you, and a lot of other people started feeling that given the improved returners of today (Murray, Novak), Roger needed to improve his second serve.

I think the key word here is ‘given his current state.’ If he could figure out how to improve the probability of landing faster (read better) second serves, that would of course be a different story altogether. Now whether it’s changing his serve position, using a different tennis racket, hitting the gym- that’s something he knows and needs to work on, not just on his second serve, but in all aspects of his game, if he wants to make another run at the top of men’s tennis.

Ok, so an important thing to clear up, I think, is that what these graphs and plots suggest is not that the serve cannot be improved.

It’s more that the current serve speed is optimal for the current ability level.

So if all he did was serve faster- not work on a better, faster, serve – just because, say, he was under pressure in a particular match, it would not be helpful. But he could improve his raw serving ability, which would change the graph completely, and that would of course be helpful.

I think that’s right, correct me if I’m wrong though.

Also, if you’re interested , there’s some very interesting stuff on a related topic here:

http://heavytopspin.com/category/serve-speed/

Got it. So you’re saying that we should use the word “optimum” to convince ourselves. A few months from now, we may see an increase in Roger’s second serve speed to say 98 with a better % points won in comparison to the table. We could then simply reverse engineer and say, “Oh well, that’s the optimum speed for Roger’s second serve. Not much improvement can be expected”.

Spot on!!

Except…Roger would improve his second serve win percentage by increasing his average second serve speed to say, 98 mph, proving that 95 mph wasn’t in fact the optimum number, belying our original claim that it indeed was?

In which case we will have to travel back in time, modify this table to reflect 98 mph as the optimum second serve speed, which would then be proven to be a fact in the future, thus creating some sort of predestination paradox? Or, maybe one of the other paradoxes?

I’m losing my mind, I swear π

Perfect.

Unfortunately, if Roger cannot do anything more on this second serve in terms of percentage points won, then it doesn’t mean it’s optimized, it means he has let it deteriorate, and has not been proactive enough for a myriad of reasons.

I’m not buying into this π

Great work, Gaurav: must have been long work hours on this one!

However, isn’t just saying that the speed impact is linear is just WAY too simplistic? It’d be more like exponential to me:

Take 4 serves: one a 80 mph, 85 mph, 120 mph, 125 mph. There are 5mph differences between the serves, but the difference between 120 and 125 makes the serve A LOT harder to return. and that’s just talking about the speed: as you said yourself, it also depends on spin, but also the opponent, the court surface, the weather conditions, on it goes on and on and on. You just can’t play tennis with math! π

Way to go on the math though!

So, come on you guys – what are the improvements that you feel Roger should be making on his second serve, and why?

(Sid, I’ll take “and he needs to increase his racquet head size in order to make this work” as a given!!! Just wondering what all the fine minds here think “this” is.)

He needs to get younger, and build some hunger, if you know what I mean π

Hey Gaurav, just want to say nice article. It looks like it took you a lot of time to figure it out and calculate all and make all these formulas and charts. Nice and well done !!!

Now, that’s all I have to say. Simply because….. I don’t understand most of it (all of it). Way above my head !!!

Don’t worry, it’s a girl thing !!!

Maybe Roger reads Jonathan’s site too and looks at this and comes up with a solution??

Now see, that’s not even fair, Katyani. You make us read your semi infinitely long comments and now you want to just chicken out on this one? Do I hear clucking?

http://www.youtube.com/watch?v=FJ_Mt7Zm5hM

We demand you read the entire post and write a 25 line comment. Anything less will not be acceptable π

Hahah

Wow Sid,

I did not visit this site for a couple of days and your reaction is what I see first???

Hmmm….

Sid, you want me to make a long comment about this article?? Well, I have one answer for you.

A very wise person (THAT WOULD BE ME) once said you should only talk and write about things you know (passion and love for Roger) and leave the other things (strings, racketchange, first and second serve) for people who know about that (that would be all of you people) !!!

I am already embarrassing myself with my love for Roger, so I will not make a fool out of myself for writing about things I really don’t know much about !!!

But….. don’t worry, those semi infinitely long comments will be back !!!

And when they are back, please read them and reply to them once in a while !!!

Ps: Sid…. you are SO lucky I don’t know how to link video’s !!! Yet !!!

And Jonathan, what is this? Already three new articles? What, is there a holiday in England??

Wow, must be good to be English and living in England right now.

Andy winning Wimby, looks like Froome will win the Tour the France and the Royal baby beeing due at any moment !!! Lucky people….

Katyani, if you ever talk about the royal baby, or even so much as mention it, I will not read any of your comments, guaranteed π

Seriously, all humans are equal, there is no “royal” for crying out loud. This is one of the things that’s really stupid with Britain. The royal baby is nothing but just another unit that will be pumped into this planet. Why should the royal family thrive on the hard earned money of the so called “common” people?

Sid…… Roger that !!!

Just a short example to show that we don’t get ‘optimum’ by definition from this method.

I had a look at a player who we know isn’t exactly the best out there, Jeremy Chardy, (in fact, he’s got the worst 2nd serve accuracy of the current top 50), in his four set defeat of Ryan Harrison in Wimbledon Round 1 this year.

Applying the method shows that he would have been better off in the match doing two first serves. He served at 102mph on his first, and 85mph on his 2nd. If he had gone for the full 102mph on his 2nd doing two first serves, sure he would have double faulted more, but he would have gone from 71.8% service points won, to 75.2% service points won, an increase of 3.4%, which is quite significant.

It goes to show how ineffective his second serve is, and how the method is not entirely pointless, because it does say that Chardy, unlike Federer, really needs to do a lot of work on his second serve.

Once again, a complete misunderstanding. Nobody is denying that the table in question has been built with the help of data that was available, with a lot of assumptions of course. The example you gave is an exception rather than the rule because Chardy’s second delivery as you’ve mentioned was giving extremely poor results. If you build that table for the Top 4, you could come to the conclusion that their second delivery is optimized. Well, was there any doubt? It’s like reverse engineering and stating the obvious.

Each of these four have their weaknesses. It’s incorrect to assume that the second serves of each of these players is tuned for optimum results.

Is there any way you can tell me that a first serve, sent at 95 mph, against a certain opponent, on a certain surface, will give you a % points won exactly as your table suggests? No, you can’t. Similarly, can you prove that with the same conditions, it will give exactly the same result against a completely different opponent? No, you can’t.

Conversely, can you prove that a second serve sent at 101 mph, against a certain opponent, on a certain surface, give you the exact % serve in, and % points won value as the corresponding 95 mph first serve? No, you can’t. For starters, the 101 mph second serve is a completely different ball than a 101 mph first serve.

In my opinion, we cannot conclude that Federer has found the optimum second serve speed. At the same time, I’m not denying the point you’re trying to get across with your analysis.

This is non-Federer stuff, but I simply had to share it…

http://www.youtube.com/watch?v=ZzDB70d9AUU