I have some questions regarding the statistical notion of "population'', the mean, standard deviation and the distribution of a population characteristic.
According to Wikipedia, ``A statistical population can be a group of actually existing objects". Let us consider the distribution of heights of a population consisting of 100 people. Assume the population height has the normal distribution with mean $\mu=150$ and standard deviation $\sigma=30$. Let us generate possible actual heights of the individuals in the population using R:
n=100 mu=150 sigma=30 set.seed(5) example_data=rnorm(n,mu,sigma)
It is then possible to make a plot of the cdf:
which gives the folowing plot:
The theoretical cumulative distribution function is given in the following plot
We can calculate the populations mean and variance as
The population standard deviation is then
1) we see that the plot of the empirical cdf and the plot of the cdf are not exactly the same. They have similar shape but do not coincide at all points along the abscissa. How can we then say that the population height is distributed as the specified normal distribution?
2) When we calculate the population mean and standard deviation, we do not get the true values 150 and 30. Why are then the population mean and variance defined as they are? Is it only in an infinite population the expression for the population mean, variance etc. give the true values?
3) According to the weak law of large numbers the sample mean converges to the true mean as the number of observations approaches infinity. How can then the expression for the population mean give the true value of the mean?
4) Can then a population consist of a finite amount of individuals or are finite sets of individuals just samples?