Before continuing, let's make it clear what is meant by an "Event." For our purposes, an Event begins when a dragonfly hits the water in a splash-dunk. If the dragonfly rises after the splash-dunk and performs a spin-dry, then that was a one-splash-dunk Event. If the dragonfly does two splash-dunks, then rises for a spin-dry, it's a two-splash-dunk Event, and so on. Each time we see an Event beginning, we keep count of the number of splash-dunks before the spin-dry.
Secondly, let's give a reminder why the dragonflies are splash-dunking in the first place: They are bathing. When dragonflies perch they hold their wings straight out, and they cannot clean them. The wings collect various types of debris, and the way to clean them is to plunge into the water one or more times.
Here are the results for splash-dunk events in 2017:
|Splash-Dunk Events for 2017. Total number of Events is 86; average number of splash-dunks per Event is 2.41.|
This is a typical distribution, with the number of splash-dunks per event ranging from 1 to 7. Compare this with the cumulative results for 2011-2017:
|Splash-Dunk Events for 2011-2017. Total number of Events is 688; average number of splash-dunks per Event is 2.32.|
These are the results from observations of 688 splash-dunk Events. The average number of splash-dunks per Event is 2.32. Notice that more Events have just a single splash-dunk than any other number. In addition, the record number of splash-dunks in an Event is still 8—the number of splash-dunks in the famous event associated with the constipated darner. You can read details about the constipated darner at the following link: http://thedragonflywhisperer.blogspot.com/search/label/constipated.
Each year we see that there seems to be a bit of an excess in the number of Events around 3 and 4 splash-dunks. Let's look at this a bit more carefully. We begin with an exponential fit to the data:
The red dots are the data points, and the blue line is an exponential fit of the form a Exp[–bx], with the parameters a and b taking on the values a = 425 Events and b = 0.488 Events/splash-dunk. This shows clearly that the data is generally exponential in its fall-off, but with an excess of Events at 3 and 4. This has been a significant feature of the data each year.
The exponential fall off implies that each splash-dunk is an independent occurrence; that is, after each splash-dunk there is a certain probability that the dragonfly will do another splash-dunk independent of how many splash-dunks it has already performed. For the most part, this seems to be a valid description of the splash-dunk behavior. For some reason, however, there is a greater probability that a dragonfly will perform 3 or 4 splash-dunks. Perhaps fewer than 3 splash-dunks is usually not enough to clean the dragonfly, whereas more than 4 splash-dunks starts to get tiring, making the 3 to 4 range sort of a "sweet spot" for the dragonflies.