Is there any data whose mean is impossible to calculate?

Answer 1

Yes, plenty.

Any non-numerical data, for example a survey of favourite colours - it is nonsense to talk about the "mean average" colour (though it is perfectly reasonable to talk about the "mode average" colour, which would be the colour that most people surveyed said was their favourite) - besides how could you calculate it: you can't add Red to Green to Blue to Yellow and divided by 4 (for example).

Also, beware of datasets which look numeric but aren't. For example: shoe sizes - they use numbers as labels*; it would be nonsense for a shoe shop to try to find the mean average shoe size when considering what shoes have been bought.

* The UK shoe sizes are based on increments of length of a barleycorn (13 inch)

Answer 2

If the set is finite, and the individual numbers are well-defined, then you can always calculate the arithmetic mean. In the case of certain infinite sets, I believe there are circumstances where the average is not defined.