Conditional bootstrap variance of a sample mean (${\mathrm{Var}}_{\ast }({\overline{a}}^{\ast })=n^{-2}{\sum }_i(a_i-\overline{a}{)}^2$)

Conditionally on the observed values, bootstrap draws are iid from their empirical distribution. Each draw has variance $n^{-1}{\sum }_i(a_i-\overline{a}{)}^2$, so averaging $n$ independent draws gives the displayed variance. The variance across $B$ independent replicate means, using denominator $B-1$, is unbiased for this conditional quantity. Its expectation over the original iid data is $(n-1)\mathrm{Var}(a_1)/n^2$, which is slightly smaller than the actual sampling variance $\mathrm{Var}(a_1)/n$.