I have multiple groups of numbers (Arrays):
Array 1 = {X1,X2,X3,...,Xn}
Array 2 = {Y1,Y2,Y3,...,Yn}
Array 3 = {Z1,Z2,Z3,...,Zn}
Where the values are unknown, and the size of each array might be different and unknown KNOWN.
You are given the MAX, MIN and the median values of each array.
How can you estimate with accuracy, the global median value of all the values of the arrays?
In general you can't estimate this with good accuracy. Suppose you are given that:
The min, max, and median of array 1 is $0$. Array 1 is size $1$.
The min, max, and median of array 2 is $0$. Array 2 is size $1$.
For array $3$ the min is $1$, the max is $10^{100}$, and the median is $10^{100}$. Array 3 is size $5$.
Then, we know array 3 is $\{1, X, 10^{100}, 10^{100}, 10^{100}\}$ for some $X$ between $1$ and $10^{100}$. The combined array would be $\{0, 0, 1, X,10^{100}, 10^{100}, 10^{100}\}$
The global median would be $X$, which could be any number between $1$ and a googol, and with the information we have we can't make any better guess!