I understand some use cases for arithmetic (standard), geometric (average growth of two successive discrete growth rates), harmonic (average velocity when consecutively traveling the same distance with two different velocities) and heronian (calculating it between the top and bottom areas of a pyramid stump gives you the base area of a prism with the same height and volume) means.
There are many more means though, and I have no idea in what situations they would be useful, such as
- logarithmic mean
- root mean square and other power means I haven't mentioned yet
- centroidal mean $\left(\frac{2}{3}\frac{a^2+ab+b^2}{a+b}\right)$
- possibly others that I don't about or that I haven't found names for$^{[1]}$
This is by no means to discredit work on those means, btw, I'm just curious where they might come into play.
$^{[1]}$ For example, the function $f(t,a,b)=\frac{\int_a^b x^{t+1} dx}{\int_a^b x^t dx}$ gives us the centroidal, arithmetic, heronian, logarithmic, geometric and harmonic mean between $a$ and $b$ if we set $t$ to $1, 0, -0.5, -1, -1.5$ or $-3$ respectively. Setting $t$ to $-2$ gives us $\frac{\ln(|b|)-\ln(|a|)}{\frac{1}{a}-\frac{1}{b}}$, which I haven't found out what name it has, if any.