Supposte I have a list of numbers $\{x_1, \ldots, x_N\}$, not necessarily ordered, and I divide it in subsets $\{ \{x_1,\ldots,x_{d_1}\}, \{x_{d_1+1},\ldots,x_{d_2}\}, \ldots \}$, where $d_n$ is the number of elements in each subset, then the mean over the entire set $\langle x\rangle$ is the weighted mean of the mean in each subset:
$$ \langle x \rangle = \sum_j \langle x_j\rangle \frac{d_j}{N} $$
where $d_j$ is the number of elements in each subset and the sum is over the subsets. If I want to compute the median of the entire set and I know the median in each subset, does an analogous formula exists?
It does not, since there’s no information in the medians about how far the other elements in the subsets deviate from the medians. For instance, for $\{1,2,3\}$, $\{4,5,6\}$ and $\{7,8,9\}$ the medians are $2$, $5$ and $8$ and the overall median is $5$, whereas for $\{1,2,9\}$, $\{4,5,9\}$ and $\{7,8,9\}$, the individual medians are the same but the overall median is $7$.