Expecteation and population mean

Can someone out there assist me to understand if there exist any difference between Expectation EX and Population mean "mu". I have completely lost myself and I have failed to proceed.
If X is a random variable, then its expected value is its mean (population mean). They are interchangeable. However, expected values are used for much more than that. For example, the expected value:

E[(X-mu)^2] is the variance (sigma^2) of the random variable X.

If we let C = ((X-mu)^2) / sigma^2) then C is also a random variable. It turns out that if X has a normal distribution (N(mu, sigma^2)) then C has a chi-square distribution with r degrees of freedom.


E[((X-mu)^2) / sigma^2) ] = E[C] = the mean of a chi-square distribution with r degrees of freedom. = r.

Not quite. mu does not equal to (1/N SUM xi) <- this quantity is the sample mean, which is a random variable. Since mu is not a random variable (unless you consider Bayesian statistics), (1/N SUM xi) cannot equal mu. However, the sample mean is a good estimator of the population mean (mu). And, as the sample size increases, the sample mean gets closer and closer to mu.