Yes, it can be a problem and I agree more context like median or a full histogram should be more common, but I have never seen anyone claim that the average was 2 in your example.
While the average value was 3.4, I don't think it would be unfair to suggest the average person would have scored a 2 (or whatever the relevant scenario is).
I was always bemused that the average family size was 2.4 children as no family fits that average ever. With mode and medium that would not happen.
Of note I have no idea what the average family size is children wise, just going on longstanding data from UK that in itself may be out of date, though does highlight the point.
Yet if we set up a game with a penalty proportional to the distance between your guess and the true number of children for a family, and you guess the median or mode, you'll be right more often than I (because I will never be right), but you will also continually be losing money to my guess of the mean.
What matters in real life is generally not how often you are correct, but rather by how much you are wrong. What you need to minimise is not your error rate, but the consequences when you are wrong. You are free to make an infinite amount of mistakes, as long as you are sure to make them in such a way that they are comparatively insignificant.
> Yet if we set up a game with a penalty proportional to the distance between your guess and the true number of children for a family, and you guess the median or mode, you'll be right more often than I (because I will never be right), but you will also continually be losing money to my guess of the mean.
Not sure were you went there, but the point is that you can overdetail in a way that abstracts from reality.
It depends on your definition of "very common" of course, but I have come across people using "average" to mean both "typical" and "most common" and "middle of the range" and "my preference" and a whole slew of other things.
