Is the average of the averages equal to the average of all the numbers originally averaged?

Question

I am tempted to say yes because of the following pseudo-proof (I say pseudo-proof because I am not convinced):

$$ \frac{\frac{w+x}{2}+\frac{y+z}{2}}{2}=\frac{w+x}{4}+\frac{y+z}{4}=\frac{w+x+y+z}{4} $$

Is this proof enough or am I completely wrong? If I am not wrong but this is not proof, what would be a good proof?

Edit:

I guess the following proves otherwise:

$$ \frac{w+x+y+z}{4} \neq \frac{\frac{w+x+y}{3}+z}{2} $$

That would be proof against my original statement by contradiction.

Answer 1

$1,1,1,2,2$

Their average is $\frac{7}{5}$.

But if you take it as $1,1,1$ and $2,2$, and average the averages, you get a different result.

But, what you said works if the number of numbers is a power of $2$ and you split into two equal sized sets. Interestingly, this observation was used by Cauchy to give an inductive proof of the $\text{AM} \ge \text{GM}$ inequality!