Mathematically speaking Jeff is correct - you can't take an average of averages and expect it to represent the true overall average (although it might!) - it all depends on what you are averaging. Also, your comment that your method does not skew the overall grade by placing too much weight on the Art department could also be flawed in that it places more weight on the grade of a single Business Studies tutor than of three Art or 2 Maths tutors. Also, are the grades given by each tutor equivalent - do they all represent the outcome of a similar number of tests? You can only truly take a mathematical average of comparable values - don't try comparing apples with oranges.
Consider this scenario: the student takes three tests for each of the three Art tutors, three test each for the Maths tutors, and a single test for the Business Studies tutor. If the student has a bad day in an Art or Maths test, it only effects 1 out of many tests for that subject and will have little effect on his overall grade as you are calculating it, but if he flunks the single Business Studies test, it would grossly skew your overall mark, an effect not noticed if you took a true mathematical average of all grades received.
The moral? Be careful that what you are calculating is what your users really expect!