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Where can I learn more about mean, median, mode, S.D? I just want to know all the properties/theories behind. My math knowledge is not from American education system. Back then I didn't get to know about those and probability until I moved to the US. So there is my weakness.
I don't have any algebric formula to explain this but I definitely agree with HongHu. The global median has to be inside the set of the 2 others median, I put some drawing attached with this post, it doesn't explain anything but it just help me to figure it out...
Let U(A)=the smallest x so that at least half the values are <x, and
V(A)=the largest x so that at least half the values are >x, then that
makes M(A)=(U(A)+V(A))/2: i.e. no need to distinguish between even or
odd number of elements.
We can assume without loss of generality that U(A)<=U(B): otherwise we
just swap A and B. This gives three different possibilities:
In case a, for x<V(A), no more than half the values of A are <x; since
x<U(B), less than half the values are <x. Hence, less than half the
values of A+B are <x, and we must have U(A+B)>x. As this holds for all
x<V(A), this makes V(A+B)>=V(A). Similarly, U(A+B)<=U(B). Hence,
If U(A)<=U(B)<=V(A), for x<U(B) less than half the values in B are <x,
whereas x<V(A) makes at most half the values of A <x; thus less than
half the values of A+B are <x, which makes x<U(A+B). However, for
x=U(B), at least half than values of both A and B are <x, so this is
also true for A+B. This proves that U(A+B)=U(B).
Similarly, if U(B)<=V(A)<=V(B), then V(A+B)=V(A). Case b now follows
as 2*M(A)=U(A)+V(A)<=U(B)+V(A)=U(A+B)+V(A+B)=2*M(A+B) etc.
In case c, the above arguments gives U(A+B)=U(B) and V(A+B)=V(B),
which makes M(A+B)=M(B).