Arithmetic Mean Linear? Is a function that finds the arithmetic mean of a set of real numbers a linear function?
So is $\left(X_1 +  X_2 + \cdots +  X_n\right)/n$ linear or not?
I'm not sure because so long as the set stays the same size $n$ could be defined as a constant.

To talk about linearity, the domain and range of a function must be vector spaces, in this case over the real numbers. So your first question should be, what vector space do you take the arithmetic mean to be defined on? It turns out you must fix the value of $n$ to get any reasonable vector space with the arithmetic mean defined. If you mix various length sequences, they cannot be added (and extending the shorter sequence by zeros in order to perform the addition is not an option, because this changes its mean value). Once you realise $n$ must be fixed, you should have no difficulty seeing that the arithmetic mean is a linear function.
