Why does the finite population variance requires the (N-1) factor in the literature? Recently I've encountered the formula for finite population variance in Davison's and Hinkley's book "Bootstrap Methods and their Application" (see screenshot below). The formula includes $(N - 1)^{-1}$ rather than $N^{-1}$ (as it should be according to wikipedia). I understand that the $(n - 1)^{-1}$ factor appears in the sample variance estimate (i.e. sample taken from the finite population), but here we deal with the entire population.
The source of bias in the sample based estimate is the uncertainty introduced by the mean estimate which by itself is a random variable, but if we're talking about entire population the mean obtained from the population is a fixed quantity and therefore should not effect the variance.
I checked a couple of articles online and I noticed the same definition in some of them as well. For example

*

*The variance of the variance of samples from a
finite population

*Estimating the Variance of a Variable in a Finite
Population
Some explanation I found in another question which was posted in 2013: What is the difference between N and N-1 in calculating population variance?. It seems like one reasonable explanation is that using $(N-1)^{-1}$ factor is computationally more convenient (although I don't see any good examples that support the claim) which means that definition of the variance was changed for this particular application since it's no longer the variance in a normal sense, but rather some variation of the discrepancy measurement which gives quite often answers that are very close to the actual variance. The problem is that now the sample variance is biased with respect to the new variance definition, since it's missing $N(N-1)^{-1}$ scaling factor.
I would appreciate if somebody can help me to clarify the points


When we are sampling with replacement then the population size is irrelevant.
Simple counterexample: Imagine an urn model with $k$ red balls, and $k$ blue balls. The probability to draw a red/blue ball is 0.5 independent from the number of total balls $N = 2k$ among the population. So the variance of any sample from that urn should be independent from the size of the population.
It is difficult to find the origin of the error in the text that quoted. It is easy to make little mistakes with $n$ versus $n-1$.
