Is the mean of four integers equal to the mean of means?

Question

It appears to me that the mean of four numbers is equivalent to the mean of the means of two pairs of those numbers:

$\text{mean}(a,b,c,d) = \frac{a+b+c+d}{4}$

$\text{mean}(\text{mean}(a,b),\text{mean}(c,d)) = \frac{\text{mean}(a,b)+\text{mean}(c,d)}{2} = \frac{\frac{a+b}{2}+\frac{c+d}{2}}{2} \cdot \frac{2}{2} = \frac{a+b+c+d}{4}$

I am confused after reading Simpson's paradox though. It seems like this shouldn't necessarily be the case? This leads be to a number of questions:

1. Is this derivation correct for 4 numbers?
2. How does this differ from Simpson's paradox?
3. If it only differs from Simpson's paradox under certain conditions, do those conditions change if we only use integers?
4. If it only differs from Simpson's paradox under certain conditions, do those conditions change if we specify the exact number (i.e., 4) of numbers we use?