This is my guess from 5 years of training at several schools and skimming my Facebook friends that train/used to train.
White belt: Around half will make it through the first month. Of those, about a third make it to blue.
Blue belt (16.5% of those that start): About a third will make it to purple. Most of those that drop will do so in their first year as blue. Those that make it longer then that tend to stick it out.
Purple belt (5.5% of those that start): About half make it to brown. This seems to be for three reasons:
1) life -- purple is where people need to be very dedicated and invest a lot more time. Up until this point you can pretty much just show up to class and drill whatever your instructor shows. After purple though it seems that you need to invest a lot of your time researching, planning, and drilling your own style (at least, this is how I see it as, as a purple myself). Also, a lot of life changes occurred between white and purple. People start grad school, get married, move, have kids, etc.
2) Lost interest -- I know a number of people who left for other martial arts where they felt they were progressing much faster. Progress at purple is much slower, especially if you are "top dog" at your school.
3) Injury -- I know a few people who dropped out due to injuries sustained right after getting their purple. Taking a few months off right getting purple is hard -- blue belts gun for you and there is the perception that a purple belt is an unbeatable jedi master, irregardless of injury, age, or difference in size.
Brown (2.75% of those that start): Almost everyone makes the jump from brown to black. I actually can't think of anyone who did not... Those that made it to this stage are incredibly dedicated and have just spend three to five years at purple alone, so that last year at brown is almost nothing. I'd say at least 80% make it to black.
Black (2.2% of those that start)
Note: All this is based on training at serious, established BJJ schools, with dedicated black belt instructors. There are a couple schools in my area, such as a university club and a couple other hobbyist schools mostly run by colored belts, where almost no one makes it to blue, and going past that requires training elsewhere. Including those schools, the numbers above would drop substantially from white to purple. Include those, maybe ~2% in my area make it to purple?