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Like the title says, does $X$ (a random variable) usually refer to the population or the sample?

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I think the question is ill-posed.

A random variable, by definition, is a misurable function. If you estract a sample from a population, they follow the same distribution, so the random variable can describe both. When you use the term "Random Variable" you refer to a function from a set to another, the first being called "sample space" and the second $\mathbb{R}^n$.

I could write you the proper definition, but I don't think that would help in this case, moreover you can find it everywhere (starting from Wikipedia).

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