Modeling population growth - Variance
If $W$ is a parameter, I'd expect (in the simplest case) that $W$ itself is a constant (i.e. it doesn't have a distribution). (By the way, I would normally use $T_{1/2}$ to denote the doubling time, $\ln 2/\ln W$, and would use $\lambda = \ln W$ for the exponential rate, i.e. $W^t= \exp(\ln W)^t = \exp(t \ln W) = \exp(\lambda t)$. I don't normally see people express rates scaled by $\ln 2$.)

In the case of a continuous-time birth-death process (the technical jargon for your model), I believe the analytical answer will be that the standard deviation of the population size at time $t$ will grow at the same exponential rate as the mean population size, and that the variance will grow twice as fast (because it has units of $\textrm{population size}^2$), i.e. $\propto \exp(2 \lambda t)$)

I don't have a derivation/reference handy, will dig around for one. This is fairly basic stochastic-process math, but stochastic-process math also gets pretty hairy-looking (to a biologist) quickly. Bartlett 1960 (Stochastic population models in ecology and epidemiology) is one place to look for basic results, although I'm sure there are many others. This recent PhD thesis gives a thorough introduction to the math of birth-death processes ... although, not to my disappointment, a canned derivation of the variance result I'm too lazy to derive for you.