Sample mean: dependence

Question

I have a question that is possibly more about language than math, but still it concerns me a lot. I understand that this question may irritate many (because it's stupid, and apparently because I am stupid too), but still I ask not to hate me too much.

We all remember the definition of sample mean:

$ {\displaystyle A={\frac {1}{n}}\sum _{i=1}^{n}a_{i}.} $

It is sum divided by size. Okay. Note that we're not talking about population mean, we're talking about sample mean.

So, today a colleague of mine uttered the following sentence:

Sample mean does not depend on sample size.

Now I'll try to explain my hesitation. As I see that there's some $ n $ that is not constant in this formula, I want to say that sample mean depends on $ n $. But as this $ n $ is fully defined by the sample itself, there is, seemingly, a sense in which sample mean does not depend on $ n $.

So the question is: is my colleague's statement true? And what's that important thing that I don't get about sample mean that makes me so confused about such a basic thing?

Answer 1

A clear answer would be "NO". It depends on sample size and as the number of samples goes to infinity, your sample mean will approach to the population mean.

Answer 2

I think that, without a more specific qualification of that statement, it is difficult to say what was really meant.

For example, one interpretation could be that, for a simple random sample, the sample mean is an unbiased estimator of the population mean, and this property is independent of the sample size. But this interpretation goes well beyond what was literally said.