What data about grades should I tell the class? Make public? I have had a few professors who announced to the class the mean and median scores for each exam, but I haven't seen anyone go further than this (e.g., announce this information for homework too, give a histogram of all the scores, etc.). Also, this information has always been kept semi-private, in the sense that it is announced verbally in class, without an explicit instruction not to share it, but still making it unlikely that the information would spread beyond the class.

I'm wondering what to take into consideration when deciding what sort of information to provide, and how. Would it be acceptable to put the mean and median scores for each exam on my (publicly available) website, for example? Or is that violating the privacy of my students in some way? Might they be ashamed, as a group, if they've all done extraordinarily badly? But as their teacher, I would have to also be ashamed if that were the case - so perhaps deciding to publicize how my students are doing ought to be considered as a sign of confidence that my teaching will make them do well. 

I'm sure there are other issues I'm not thinking of regarding this. Your experiences (either as a student or as a teacher) with classes that gave this sort of information, and any other thoughts, would be welcome.

Edit: I'd be teaching an undergraduate math class. But regardless of my personal situation, I'd like to hear whether you think the field of study, or the level of the course, affect the answer to the question.
Revealing information about grades is normally directed by some combination of university policy and government regulations (if one is teaching at a public institution of higher learning). 

Where I teach, for instance, all grades for written exams are officially published, but semi-anonymized by replacing the students' names with their student ID numbers. In addition, the aggregate statistics are also published, along with the rubric for converting "raw" scores into final grades. 

My general belief, however, is that grades are an "internal" matter, that should not generally be publicized beyond the extent required by regulations. There is no need for everyone in the world to see the grade distribution.