Although I have my own interpretation about representing a random sample using random variables, I am not really sure about it, and I would like to learn others' opinion about this.
The question is this: in many books on statistical inference, data or observed sample is represented as $X_1, X_2, \cdots, X_n$ which are $n$ IID random variables. However, it is clear that in real-world, data are numbers with fixed value, not random variables. For example, if I measure heights of $n$ people, then I get $n$ real numbers $h_1, h_2,\cdots, h_n$, not $n$ random variables. Why don't we simply say that these numbers have come from some particular distribution, instead of representing them using random variables? On the same note, how can we talk about things like $\mathbb{V}(X_i) = \sigma$ when clearly a single number has zero variance?
Most book seem to ignore these naive questions which students ask, and only define procedures to handle a "sample" given in the form of $n$ random variables.
