# the sampling distribution of sample means, why are samples taken with replacement?

I have what might be a silly question. I am tutoring a family member in stats and I am trying to help her understand the sampling distribution of sample means, but I don't quite understand why it's derived the way it is myself.

The way I understand this particular distribution is the distribution of sample means for all possible samples of size n. Her notes contain a simple example the teacher gave:

Suppose we have a small population: {2, 4, 6, 8}. Let's find all possible samples of size n = 2.

I would think all possible samples of size two would be:

{2, 4}, {2, 6}, {2, 8}, {4, 6}, {4, 8} and {6,8}

But according to the notes, there are 16 possible samples of size two, and they are taken with replacement. For example, one of the possible samples is {2, 2} and {2, 4} is counted as a distinct sample from {4, 2}. I don't understand why these are counted. The sampling distribution of sample means is used in hypothesis testing. Say you are testing a claim about the mean of adult male heights and you want to select a sample of size n = 10, you would not count the same individual 10 times. What am I missing?

Thank you.

A sample, by definition is a sequence of draws from the population. The sample $$\{2,4\}$$ implies that $$2$$ is drawn before $$4$$, whereas $$\{4,2\}$$ implies the opposite. In problems where the draws are not independent, the order matter as it has implication for the probability distribution of the remaining elements to be drawn.