Well, take decibels for example. They are a log scale physical intensity, so averaging them make no sense whatsoever (e.g. absolute silence is negative infinity dB). I would argue these scores are more like labels than actual numbers (i.e. a 3 plus a 5 doesn't really equal an 8 in any real sense). You can of course take the mean of any collection of numbers, but I've heard many a statistician lament such careless practices. The median is at least more easily interpreted for cases like this.