I understand that if you take a sample from the population that a single data point cannot freely vary if $\bar{x}$ is known and you have the remaining sample items. However, I do not understand why "a sample of size $n$ retains $n$ degrees of freedom if the population mean $μ$ is known". The answer is sort of provided in this paper: Degrees of Freedom (Eisenhauer, J.G.), where it states:
'Note that a sample of size n retains n degrees of freedom if the population mean μ is known, since this does not determine $x_i$ for $i = 1 ... n$ if the other $(n–1)$ values are known. The concept is of importance in statistical inference since it defines the effective size of a sample.'
Which is actually a quote from "A Dictionary of Mathematics".
Why though, does knowing $μ$ "not determine $x_i$ for $i = 1 ... n$ if the other $(n–1)$ values are known"?
Using the same logic, I can calculate the remaining value if I know $n-1$ values and $μ$. So why are the degrees of freedom still $n$?
My intuitive guess is that by knowing $μ$ I must know the entire population of data points. However by the same logic, if I know $\bar{x}$ I must know my entire sample set of data. So I do not understand why the -1 rule does not apply to the population.
The rule seems arbitrary to me, since if the data set is 1000, and I have 999 points, my degrees of freedom are 998, but if my sample is the entire population (1000), then my degrees of freedom are 1000.
My discussion is a bit long winded, so please let me restate my question:
Why does knowing $μ$ "not determine $x_i$ for $i = 1 ... n$ if the other $(n–1)$ values are known"?