Equations involving both population and sample means are especially confusing for students. One of them is unbiasedness of the sample mean . In the Econometrics context there are many relations of this type. They need to be emphasized and explained many times until everybody understands the difference.

On the practical side, the first thing to understand is that the population mean uses all population elements and the population distribution, which are usually unknown. On the other hand, the sample mean uses only the sample and is known, as long as the sample is known.

On the theoretical side, we know that 1) as the sample size increases, the sample mean tends to the population mean (law of large numbers), 2) the population mean of the sample mean equals the population mean (unbiasedness), 3) for a discrete uniformly distributed variable with a finite number of elements, the population mean equals the sample mean (see equation (4) in that post) if the sample is the whole population, 4) if the population mean equals , that does not mean that any sample from that population has the same sample mean.

For the preliminary material on properties of means see this post.

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