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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. |
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$1,1,1,2,2$ 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! |
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It depends on how you group the elements, so the answer in general is no. It worked in your first example because the groups were of uniform size, while they were not in the second example. |
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