Let $D_i$ be indicators, $i\in\{1, 2 ... k\}$.
I am interested in calculating variance of $$Y = \frac {\sum_{i=1}^k x_iD_i}{\sum_{i=1}^k n_iD_i}$$ where $x_i$ and $n_i$ are given real numbers, $i \in \{1, 2 ... k\}$.
Exactly m of k indicators is equal to 1 and all possible combinations are equally possible.
Also, probability for each combination of choosing these m indicators is $\frac {1}{k \choose m}$
\begin{align*} \textrm{Var} \left( \frac{\sum x_i D_i}{\sum n_i D_i}\right) &= \frac{1}{{k\choose m}}\sum_{\sigma} \left\{\mathbb{E}\left[ \frac{\sum x_i D_i}{\sum n_i D_i} \right] - \frac{\sum x_i D_i^{(\sigma)}}{\sum n_i D_i^{(\sigma)}} \right\}^2 \\\\ &= \frac{1}{{k\choose m}}\sum_{\sigma} \left\{\frac{1}{{k\choose m}} \sum_{\sigma'} \left[ \frac{\sum x_i D_i^{(\sigma')}}{\sum n_i D_i^{(\sigma')}} \right] - \frac{\sum x_i D_i^{(\sigma)}}{\sum n_i D_i^{(\sigma)}} \right\}^2 \end{align*}
where the $\sigma$ and $\sigma'$ sum over all valid combinations of values of indicators.