As a teacher, I want to give students the privacy to respond to their score individually. For a while I did exactly that, returning tests without discussing scores with the class. However, I realized that students developed skewed views of how the class was doing. Most often students with high scores assumed the rest of the class had scored as high as they had, and students with lower scores assumed the rest of the class was doing much better than them. I wanted to give my students some information to inform their opinions.
One way to do this while still respecting the privacy of individual grades is to share the median test score with students. The idea is that students will know how they scored relative to their peers as factual information - not as judgement. Hopefully this understanding will lead to a realistic idea of what is possible for the student.
Outside of school, people still have emotional reactions based on their scores — though "score" is most often interpreted as "income." I wonder how people's feelings about their income would change by learning statistics about the median household income in their neighborhood. I suspect that most people do not have an accurate perception of how much money they make relative to people around them. Could this be a way to increase empathy by giving people a more accurate picture of the incomes of people around them? Could the calculations involved be part of a math lesson? Hmmm...