What is the difference between µ when being the population mean, and µ when being the mean or the expected value?
What confuses me is that the same letter is being used to describe two different metrics- or are they? I am solely basing this question on their equation which isn't the same, so where am I missing the point?
On
The question you ask is not a trivial one but in fact it is very profound. The fact that you're confused means that you are the kind of person who thinks deeply, which is very commendable.
The source of your confusion is that the population mean,µ, can be understood in two very distinct ways. The simple definition of it is based on an elementary calculation:
where N is the size of the population and Xi are it's elements (or observations).
The idea of expected value, however, is a bit more subtle. Imagine drawing an observation from this population repeatedly with replacement i.e. you draw an observation, then place it back in the population, randomize and draw a second observation, and so on. If you make infinitely many draws, then the mean of these infinitely many draws is defined as their expected value. And it can be shown that this expected value exactly equals µ, the population mean.
Note that in the discussion so far I haven't used the idea of the sample mean at all, which shows that there is no need to use the concept of the sample mean to understand the concept of mathematical expectation.
However, it can additionally be shown that the expectation of infinitely many means, each based on a fixed sample size n, is also equal to µ. In other words, you can show that in a repeated sampling sense the sample mean, on average, equals the population mean, which is why it is considered to be an unbiased estimator of the population mean.
Please let me know if you have any follow up questions.
There is a difference between the population mean and the sample mean.
Suppose you have a population of a hundred birds. The weight of the birds is normally distributed about a mean of 10kg with a standard deviation of 1kg.
Now suppose you take a random sample of these hundred birds, and you select ten. Unfortunately, your random sample was not very representative of the population, and you ended up picking the lighter ones. The weight of the sampled birds is normally distributed about a mean of 8kg with a standard deviation of 1kg.
Thus, the population of birds has a mean of 10kg. But the sample we have taken from the population of birds has a mean of 8kg.
In terms of notation, we usually denote a population mean with $\mu$, and a sample mean with $\bar{x}$.
The expected value is the population mean. See the Wikipedia article, which is quite informative. The idea here is that when you randomly select a sample, you "expect" the mean of the sample to approximate the population mean as the size of the sample becomes arbitrarily large.