It is a study we would like to do. The plan is to select a random sample of domains for each length and query a source, e.g., Alexa, show a confidence interval by length for domain traffic.
The stats hypothesis would be to test whether length can be used as a predictor of traffic.
I was taken aback by the 13 character length, I wonder how many of those are parked, what the average length is for the top 100,1000, 10000 sites according to Alexa/Google/Etc. I suspect there there is a nice split between the parked longer names and the shorter actually used as a website names.
The stats hypothesis would be to test whether length can be used as a predictor of traffic.