I am currently reading this article. On page 301 it says:
About a century and a half ago, it was observed by Poisson (1837) (see also Edwards (I960)) that $$\text{Var}(S_n) = n \bar{p} \, (1-\bar{p}) - n s_p^2,$$ where $s_p^2 = \frac{1}{n}\sum_{i=1}^n (p_i-\bar{p})^2$ is the “variance” within $\{p_1, \ldots ,p_n\}$. Therefore, the variance of $S_n$ increases as the set of probabilities $\{p_1, \ldots, p_n\}$ tends to be more and more homogeneous and attains its maximum as they become identical.
Where $S_n$ is the number of successes after $n$ independent trials, the probability of success at the $k$th trial being $p_k$. I am having trouble seeing where this is coming from: can anyone point me in the right direction?