Set $Z$ contains 1000 numeric values.
It is split in 4 subsets $A, B, C, D$, respectively of 100, 200, 350, 350 elements. For each subset we calculate $min, max, mean$, variance $Var, quantiles Q_{10...90}$.
Some relationship between $Z$ and the subsets are obvious: $$min(Z) = min(min(A), min(B), min(C), min(D))$$ $$max(Z) = max(max(A), max(B), max(C), max(D))$$ $$mean(Z) = \frac{mean(A)*100 + mean(B)*200 + mean(C)*350 + mean(D)*350}{1000}$$
Talking about variance and quantiles, is there any relationship between the value calculated for $Z$ and the respective ones calculated for $A, B, C$ and $D$?
for example $$\exists? f_{Var}(x) \vert Var(Z)\simeq f_{Var}(Var(A), Var(B), Var(C), Var(D))$$
Do other functions $f_{Q_{10}}...f_{Q_x}$ exist to express relationships for quantiles?
Being the 1000 values randomly distributed in the 4 subsets, I assume the answer, if any, is not precise but should contain a confidence interval instead.
You can calculate the variance for the full set from the means and variances of the subsets, for example using the law of total variance (the mean of the variances plus the variance of the means)
So $$\text{Var}(Z) = \frac{100\,\text{Var}(A) + 200 \, \text{Var}(B) + 350 \,\text{Var}(C) + 350 \,\text{Var}(D)}{1000} \\+ \frac{100\,\text{mean}(A)^2 + 200 \, \text{mean}(B)^2 + 350 \,\text{mean}(C)^2 + 350 \,\text{mean}(D)^2}{1000} \\- \left(\frac{100\,\text{mean}(A) + 200 \, \text{mean}(B) + 350 \,\text{mean}(C) + 350 \,\text{mean}(D)}{1000}\right)^2$$
There is no such equations for the quantiles without the full data; there may be weak and complicated inequalities but I suspect you will find they are unlikely to be useful in practice.