Difference between population, sample and sample value. I was going through a book and reached a point where the author is comparing a Population, Sample and Sample Values. I don't seem to understand the difference at all.

(Caps are Random Variables, small font are values/data points)

• What is the role of Random Variables here?
• What does the numbering in X imply? ($X_i \forall i\in \{1,2,3,\cdots,n\}$)
• What does the vertical line from $X_1$ to $x_1$ suggest?

The population refers to whatever it is you want to study (let's say the heights of men in Timbuktu). Since you can't examine all of them, you design an experiment that involves taking a random sample of $n$ men and measuring their heights. $X_1$ will be the height of the first man in the sample, $X_2$ the height of the second, etc. These are random variables: if you do the experiment several times, you'll get different results. The theoretical analysis will mostly involve these random variables, e.g. you might construct some function of $X_1,\ldots,X_n$ and say something about its probability distribution. Typically you might do this analysis before you actually do the experiment. When you do the experiment, you get numbers (the sample values): $x_1$ is the measured value of $X_1$, $x_2$ the measured value of $X_2$, etc.