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?

What is the advantage of using this mean over the normal one? That can't be answered without (a lot) more context about where you encountered these weighted means. — dxiv, Jun 30 '18 at 00:46
This sort of mean appears in Trapezoidal rule for approximating an definite integral. — achille hui, Jun 30 '18 at 01:21

score 1 · Answer 1 · answered Jun 30 '18 at 00:43

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 $(a,e)$, $(b,b)$, $(c,c)$, $(d,d)$.

score 0 · Answer 2 · answered Jun 30 '18 at 00:42

0

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:

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

answered Jun 30 '18 at 00:42

dxiv

76,497

1

Interesting, I never thought about it but weighted means are really equivalent to grand means. Neat having a different answer to see this. – Jun 30 '18 at 00:48
Maybe only in the rational case, actually. – Jun 30 '18 at 00:50
@ZacharySelk Indeed, and as it happens the basic example on the wikipedia page is a grand mean. – dxiv Jun 30 '18 at 00:58

What kind of mean is this?

2 Answers2