[LogicalRisk]

"Where does n come in?"

I *think* this might help to answer your question for where n comes from. It helps me at least think about it.

The definition of the variance of the standard error V[\bar{x}] = V[X]/n. You can back this out from the definition of variance, the property that V[aX] = a^2V[X], and that variances are additive with independent draws. Take the square root of that and you have the standard error.

Why this "feels" right to me via an example.

Suppose we want to know the average height of the US population. Intuitively, we think that (assuming a representative sample) we'll do "better" in the sense of a tighter distribution around our best guess (mean of sample) of the population value if we sample 1000 people as opposed to sampling 10.

This is related to the distribution and would function as our "best guess" about the dispersion of a variable in the same units as the original. Both of them are sampling to try and guess the average. Since \bar{X} is itself a random variable, it has a distribution, and that distribution should probably include something about the sampling process we used to characterize it.

Mean absolute error would be E[|X-mu|] since the true mean of the distribution is a constant.