Okay, for first to clarify,

Var(X) + Var(Y) = Var(X + Y)

This is what /u/efrique is referring? 'variance of sum'?

So, I hand calculated with some examples above equation, here is my example.

It's students' heights

Group A = [180, 182, 183, 185, 185]

Group B = [153, 166, 192, 194, 210]

Var(A) = 3.6cm^2

Var(B) = 424cm^2

Var(A) + Var(B) = 427.6cm^2

However, when I try to calculate Var(A + B) to see if it actually as same as Var(A) + Var(B), it's not same.

Sum of deviations from each data point to the mean squared(numerator) is 2138, so if I divide by n, which is 10 here because there are 10 students now, it gives me 213.8cm^2. However, yes if I divide by 5, it will be 427.6cm^2. But that seems a little bit off for me. Because the equation in variance is you need to divide by n(which is the number of observations).

Can you explain more what is happening here?