An Average Tennis Essay
This post is about Rahul and his tired right arm. One night, not long ago, he couldn’t sleep. At around 5:00 a.m., he got up and decided to head down to the tennis courts to hit some serves. He got all his electronic equipment together (speed gun, camera, etc.) and fired up his 1959 Ford Edsel. He hit 88 serves before his arm started to hurt.
So, that is our setup. We have 88 data points to work with. All we have to do is find the average speed of the serves he hit, and then we are done. I can tell you that the average was 95.8 mph. Thank you for dropping by my blog. Stop back in a couple days for another exciting and informative post!
Wait…you know that I wouldn’t be writing about the average speed of tennis serves unless I had something interesting to say. Take a look at this Stem and Leaf Plot:
| 8 44455 |
| 8 666677 |
| 8 8888999999 |
| 9 0000000000111111 |
| 9 2222223 |
| 9 4 |
| 9 |
| 9 88999 |
| 10 0000111111 |
| 10 2222223333333 |
| 10 44445555 |
| 10 6677 |
| 10 888 |
I imagine most people have seen these before. It is my understanding that most kids in elementary school get exposed to this handy tool. Between you and me, I didn’t see these until I took a Ph.D. course in statistics. This type of display wasn’t invented until the 1970s, and it took time for their use to become widely adopted.
For those of you new to this type of figure, they are pretty simple to explain. If you look at the top line of the plot, you see “8 44455.” The 8 is the stem, and 44455 forms the leaf. This means that the serve speeds were 84,84,84,85,85. Do you see how that works? That is about all there is to it. The bottom line reads “10 888.” This means that Rahul hit three serves at 108 mph.
A Stem and Leaf Plot is a very nice way to visualize a data set. Making a picture or some type of image is always helpful when dealing with large batches of numbers. The visual representation of data is a hallmark of Exploratory Data Analysis, an approach to statistics that I heavily endorse. After all, it is much easier to study a picture than it is a large table or string of numbers.
Whenever I am tasked with analyzing data, I first turn to the Stem and Leaf Plot. Look at the plot carefully, and you will see why. I said earlier that the average speed of Rahul’s serves was 95.8 mph. Do you see the two peaks on the plot? Are there any serves that were hit 95 or 96 mph? The answer is no; he didn’t hit any serves that speed, so how can an average of 95.8 be representative of the data? The short answer is it isn’t, the longer answer is coming up.
Any data set with two peaks, like this one, must be broken into two separate batches. Instead of averaging the speed of Rahul’s serves, we need to look more closely at the data to see what might be going on. It is apparent to me that Rahul was practicing both his first and second serve. The first serves were the speedy ones. The slower serves were his second serves; in those instances, he was more concerned with spin than speed.
Here is the proper Stem and Leaf Plot for the second serves. The data has only one peak, and the average speed is now 89.13 mph. This makes more sense, doesn’t it? Just by looking at the plot, I would guess that the average should be around 90 mph.
| 8 44455 |
| 8 666677 |
| 8 8888999999 |
| 9 0000000000111111 |
| 9 2222223 |
| 9 4 |
As for the first serves, this is what we end up with. The average speed works out to 102.7 mph. That seems about right when looking at the plot.
| 9 88999 |
| 10 0000111111 |
| 10 2222223333333 |
| 10 44445555 |
| 10 6677 |
| 10 888 |
Once again, calculating averages is not as simple and straightforward as it seems it should be. If you ever need to find the average of a set of numbers, I suggest you first make a Stem and Leaf Plot. That way, you will know if it is proper to treat the numbers as a set or if they need to be broken apart.
One last thought, I know Rahul, and there is no way he hits any of his serves that hard. My guess is the entire data set is in kph, not mph, but that is a story for another post.
