Here there is my (difficult to be phrased without looking silly) question: is there an actual difference between a random variable, a random sample, and an observation?
[For what below, the books I am reading are De Groot and Schervish's "Probability and Statistics" and Spiegel's "Statistics" (Schaum's)]
Before jumping to the conclusion that I am missing some very basic stuff (which could be, of course), give me a second to explain my problem and read below the example, which could lead to a slightly more precise formulation of the question.
Example
Given a population of 100 students in a campus, we want to gather some information concerning their heights, such as mean, variance, etc. Thus, on Monday we get a sample of 5 students, on Tuesday we get a sample of other 5 students, and so on until Sunday: we end up having 7 samples, with 5 observations each.
In the example above, for what would we use the notation $X_i$?
I know that notation is just notation, but my point is that $X_i$ is used for many things in statistics and sometime I feel I lose sight of the actual object behind the notation.
What I mean is that, by taking the example, my first feeling is that the r.v. is the height, every student is an observation, and we have the 7 samples (this would be my feeling after reading Spiegel). But then, after De Groot and Schervish, I get the feeling that a student is a r.v. in its own right, call it, the $X_i$ random variable.
What above becomes even muddier to me when we move to flipping coins (a favourite example of books): if we flip one coin $n$ times and we do it $k$ times to get if it is fair, what is the r.v.?
I would say that we have just one r.v., which is the behavior of the coin; also, we have $k$ samples of the behavior of this one coin, with $n$ observations each. Thus, $X_i$ would be reserved for the sample?
However, it seems to me that quite often (always), one observation (say the $i$th) of a head or tail of that one coin is considered a r.v. in itself, thus getting the $X_i$ notation.
If you are wondering why this question, simply, I want to get this, because I have the feeling the confusion I have is hampering my possibility of gettings some things right in statistics.
Below, some links that - unfortunately - did not help me in getting this right:
Random Variable vs data vs random sample
https://stats.stackexchange.com/questions/239500/what-is-the-difference-between-random-variable-and-random-sample
Concerning the second link, the comment of hanugm to the accepted answer (which has been left unanswered) really summarizes what my problem is all about.