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Discussion Starter #1
A little while ago I made a 2D scatter plot showing the stats of players based on their return and serve win percentage. It generated a bit of discussion and some good points were made by various MTF contributors. However, much more statistics about players are available from the ATP pages, and ideally one would like to visualize the basic four parameters (first serve and return win percentage together with the second serve and return win percentage) in a single plot. However, the best that can be done is 3D-visualization. Essentially no more than 3 parameters are possible in a single figure, at least if we avoid the more fancy options like color-coding. Below the current top eight players are shown together with John Isner (JI) and Diego Schwartzman (DS), Lorenzo Sonego (LS) and Yoshihito Nishioka, where these additional four are included mainly to show how much spread there can be in the data. The player initials are used for the identification of each dot, and I think you know the initials for rest (the current top eight). The horizontal axis ranging from 65-80 % gives the first serve win percentage, and not unexpectedly John Isner is the leader here, as seen by the dot in the lower right corner. The axis labeled "first serve return win %" ranging from 23-35 % is indeed the win percentage against first serve, where you can see that Rafael Nadal, Novak Djokovic, and Diego Schwartzman are doing really well.

The vertical axis ranging from 48-56 % is the average of the wins on the second serve and the win percentage against the second serve, as indicated by the label. There are essentially two reasons I have bunched these stats together. Firstly, we only have three dimensions, and thus we have to do something along these lines to display everything in a single 3D-plot. Secondly, to some extent return win percentage against second serve and win percentage on your own second serve depend on the same skill set, the ability to win the point from a more or less neutral position. Players like Isner, that often get a major advantage after the second serve, is the exception, after all.

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So what can we learn, if anything, from the above plot?

* The thing that stands out the most is the superiority of big three. The ideal point is the upper corner (furthest from the viewer) which corresponds to a more or less invincible player, leading all the stat categories. As can be seen, the big three are much closer to this ideal point than anybody else.

* It might be tempting to use data like the above to get new input to questions like "who is really the best". However, I don't think the plot is very good for that purpose, and we already have the ranking that most likely gives a better measure for the performance during 2019 (and the stat here is for 2019 only). For one thing, higher-ranked players have earned their win percentage against tougher opposition, since they typically have won more matches, and therefore statistics like this imply slightly different things for different players. Also, the player performance depends on how many first serves you put in play, and this is not at all measured by the stat displayed here.

* An example of interesting data is when a player with a good serve have a fairly modest win percentage on the first serve. This applies to Zverev in particular, who has a good first serve but has better performance in other categories (note that Zverev does better against the first serve than many others).

So, do you see anything else of interest in the plot? Are there additional players that you would like to see here?
 

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As a research scientist, I've generally avoided 3D scatterplots like the plague, instead much preferring correlation matrices, or even PCA plots.

You could separate the second serve return and second serve win pts as separate variables and just do two 3D scatterplots side by side. Your base assumption that 2nd serve return and serve measures the same thing does not quite hold for me. For one, the pressure is greater on the server in that circumstance (in general), and the returner may have advantage if they are playing aggressively on return.

I think one thing you can correlate is that the better their first serve return is, the better their second serve return stats would also be relative to others (this you can run a separate plot for).
 

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Discussion Starter #3
As a research scientist, I've generally avoided 3D scatterplots like the plague, instead much preferring correlation matrices, or even PCA plots.
Different plots are good for different purposes, I guess, and I think the 3D-scatter plot has some visual appeal, even if I seldom (or more like never) use them myself professionally. However, I agree that the second serve win percentage and the second serve return win percentage probably are not so well correlated as I would hope. Making a principal component analysis to clarify things could possibly help, but I think it would be a bit of overkill in this case. Your other suggestion, to use two sets of data in parallel (or in the same plot) is probably a more straightforward solution to keep all the data. Maybe I will do an update of the figure without throwing away information.
 

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Discussion Starter #4
So, as per MWW:s suggestion I included all the data, not to misrepresent anything. Maybe the figure becomes somewhat less appealing visually, but now all the information is kept. Basically I just added a vector (arrow) to each player, where the length indicates the difference between the win percentage on second serve and the win percentage against second serve. Not surprisingly, Isner has a long arrow pointing upwards (that is, the win percentage on the second serve is greater). Of course, Diego Schwartzman has a long arrow pointing downwards (wins on second serve return greater). Most players have slightly larger win percentage on the second serve than against, and hence we see a bit more of the upward pointing arrows. The only player we see without a visible arrow is Daniel Medvedev. The reason is that he has the same win percentage against the second serve as on the second serve (a zero length arrow which cannot be seen).

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