Z test on blocked/partitioned data

Question

I am trying to perform a Z test on a dataset which is partitioned/blocked. Partitioned in a sense that each partition/block has some dissimilarities. The straightforward way can be: get the pooled variance estimate and do the Z test on the whole data. But, I am interested to do the Z test on individual block/partition and combine the Z scores into a single Z score. Suppose, I have $1 \cdots n$ blocks. Now I'm interested to get the Z score in the following form: $$Z = k(Z_1+\cdots +Z_n)$$ where $k$ is a constant required due to the combination. Is there any way to do this? Any literature discussing this or similar problem to this will be really helpful for me.