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In the stats book I am studying, while explaining bias and mean, the author goes like this:

"... Suppose a sample $$S=(X_1, X_2,...,X_n)$$ has been collected. ... Sample mean $\bar{X}$ estimates $\mu$ unbiasedly because its expectation is $$E(\bar{X})=E(\frac{X_1+...+X_n}{n})=\frac{EX_1+...+EX_n}{n}=\frac{n\mu}{n}=\mu $$."

What confuses me is this: The author treats each data item in the sample as if it is representing a different random variable $X_i$. How come? Why?

Zargo
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    You may be mentally conflating the idea of a random variable with the idea of the *distribution* of that random variable. It's likely (I'm guessing because you don't give sufficient context) that the $X_i$ all share the same distribution, but they're still distinct random variables. – Glen_b Dec 14 '14 at 21:36
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    To illustrate @Glen_b's point, if you suppose that a sample could be modeled as an $n$-tuple of *the same* random variable $X$, so that $S=(X,X,\ldots,X)$, then the only samples that would conform to this model would be those that consist of $n$ replicates of the same number! If you're good at critical reading, you might find it useful to spend some time with our [thread on random variables](http://stats.stackexchange.com/questions/50). Although IMHO the accepted answer is worthless as a definition and some comments are confused, collectively the other answers might help your intuition. – whuber Dec 14 '14 at 21:48

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