There are several variables you need to consider.
One criteria is athletic ability. Let's face it, some people are more athletic than others. Not only can they perform techniques better with little training, but, at least early on, they get more out of their training. Part of the success in many martial arts is the ability to do movements at high speed, with the appropriate balance of relaxation and flexing. Compared to less athletic students, many athletic people can just move faster, and more quickly learn to control muscle relaxation and flexing at high speed.
A second criteria is determination. Some people are really motivated to work hard and some are less so. Some get more out of a practice than others.
A third criteria is age. In general, young people are more adept at learning physical movements than older people.
A fourth criteria is amount of time training. And this is tough to measure, because the real issue is not time in the dojo, but total time training. Some people spend 10 or more hours a week in the dojo. Many only spend 3 or 4. In addition, many people might train a half hour or more every day in addition to what they do in the dojo. We all know that these people who train intensively are going to progress more quickly.
All these factors influence how long it will take a student to become competent at some required set of skills. If the test for black belt comes after achieving some level of competence in a specific set of skills, then it would be difficult to draw out an average amount of time to black belt. Because if age, athletic ability, total hours training and intensity of training were all taken into account, then some people are going to accomplish in three years what it takes others 6 or more years for others to do.
It is my experience that the many systems ignore many of these issues and don't test once some given level of ability is achieved. Rather, rank promotion comes in a more orderly, easy to measure progression, one based on time in the dojo. It's just like going to school. By and large students move from one grade to the next based on time, not ability. Each year a pool of people, with all different skill levels, move up a notch together.
Therefore, promotion becomes based on attendance. I would argue that if you look at the average (mean) time spent in the dojo training, of all students across all Martial Arts, it is probably not much more than 4 hours per week, and could be less. Let's suppose a school requires a student to be in the dojo training an average of 4 hours per week over 4 years (allowing 2 weeks illness and vacation.) That would translate into 800 hours until black belt.
My guess is that this number of hours in the dojo might actually be a bit high for many karate systems. I know of a school that uses time in the dojo as the base criteria to be qualified to test for black belt, and it is much closer to 300 hours, but the students are expected to train extensively outside the dojo.
