What kind of mean is this?

Question

I recently came across the following method of calculating, from what I understand, a special kind of mean.

For 3 items: $\frac{\frac{a}{2} + b + \frac{c}{2}}{2}$

For 5 items: $\frac{\frac{a}{2} + b + c + d + \frac{e}{2}}{4}$

What kind of mean is this, and is there a general formula to calculate it? What is the advantage of using this mean over the normal one?

Answer 1

Both of those are weighted arithmetic means of the form $\;\displaystyle{\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}}{\sum \limits _{i=1}^{n}w_{i}}}\,$ with $\,w_i \gt 0\,$, for example:

$$ \frac{\cfrac{a}{2} + b + \cfrac{c}{2}}{2} = \frac{\cfrac{1}{2} \cdot a + 1 \cdot b + \cfrac{1}{2} \cdot c}{\cfrac{1}{2} + 1 + \cfrac{1}{2}} $$

Answer 2

It is simply the mean of $(\text{mean}(a,c),b)$ or the mean of $(\text{mean}(a,e),b,c,d)$, etc. Another way you can see this is as the grand mean of $(a,c)$ and $(b,b)$ or $(a,e)$, $(b,b)$, $(c,c)$, $(d,d)$.