Can a Random Variable be a Sample or just a Single Value?

Question

In my introductory statistics course, I've seen statements like 'Y1 . . . , Yn is a simple random sample without replacement, of size n, from the population.' and that the CLT applies if 'Y1, . . . , Yn are independent and identically distributed random quantities'.

The first suggests that when you say you've 'drawn a random variable from a population', your variable represents a single member of that population, randomly chosen. The second, and other definitions of the CLT or E[X] for example, suggest that a random variable drawn from a population is a random whose members are randomly chosen.

When someone talks of taking a 'random variable' from a population, is the exact meaning of what the r.v. is flexible, and depends on the context (e.g. on the particular formula) the variable is used in? Or is it more fixed? It's just been confusing me because as I try to understand the meaning of E[X], the CLT etc, I keep getting confused as to  from a statistical population each formula applies to.

Many thanks indeed, really appreciate your help.

Answer 1

A random variable is a quantity having a numerical value for each member of a group, especially one whose values occur according to a frequency distribution. If you take a sample of say 30 people and measure their height, then height would be the random variable.