# sample variance correction without sample size?

Does Bessel's correction have some generalization to if we do not know the sample size itself but instead do have some rate of occurrence estimate, or a probability estimate or similar?

I was contemplating evolution and how "survival of the fittest" doesn't mean that anything less than fittest just immediately dies out. But how does one accurately estimate what "the next generation" would look like?

Say we have an experiment with $$K$$ individuals from some imaginary species, all identical except for some property $$x$$. Mean and variance in $$x$$ are straightforward here, but we measure in identical conditions how often each individual reproduces, i.e. $$\rho_k$$ times.

Now I want to estimate the variance $$\sigma^2$$ in $$x$$ for the next generation, so I thought it reasonable to do $$\mu = \frac{\sum_k \rho_k \cdot x_k}{\sum_k \rho_k}$$ $$\sigma^2 = \frac{\sum_k \rho_k \cdot (x_k - \mu)^2}{\sum_k \rho_k}$$

But I feel like I would now underestimate the variance, because we only sampled $$K$$ individuals, and we also don't know population reproduction rates. Should the correction simply be a factor $$\frac{K}{K-1}$$, or perhaps $$\frac{\sum_k \rho_k}{\sum_k{\rho_k} -1}$$? Or their product? Or... ?