Variance of allele frequencies between generations

For a binomial distribution

$$P(K=k) = {{N \choose k}p^k(1-p)^{N-k}}$$

$$\hbox{Var}(K) = Np(1-p)$$

$$\hbox{Var}(p) = \hbox{Var}(K/N)$$

$$= \frac{1}{N^2}\hbox{Var}(K)$$

$$= \frac{p(1-p)}{N}$$

Applying this to our situation,

$$\hbox{Var}(p_{t+1}) = \frac{p_t(1-p_t)}{2N}$$

Var$(p_{t+1})$ measures the amount of uncertainty about allele frequencies in the next generation, given the current allele frequency. Not surprising, the amount of uncertainty is inversely proportional to population size. The larger the population, the smaller the uncertainty.

If you think about this a bit, you might expect that a smaller variance would ``slow down'' the process of genetic drift - and you'd be right. It takes some pretty advanced mathematics to say how much the process slows down as a function of population size,¹⁰ but we can summarize the result in the following equation:

$$\bar t \approx -4N\left(p\log p + (1-p)\log(1-p)\right) \quad ,$$

where $\bar t$ is the average time to fixation of one allele or the other and $p$ is the current allele frequency.¹¹ So the average time to fixation of one allele or the other increases approximately linearly with increases in the population size.