How many applications are usually received for a given PhD position?
There was over 80 for the position I applied for a few years ago, at a Dutch neuroscience institute with a good reputation. I imagine there would be more at world famous universities in the US. I hear that most of the time, most applicants aren't that good, even the top 10% who make it to the interview. In my case the best candidate got the position, and the one ranked second (me) got an offer for a position later on... so in the end it was one out of 40.