When would we use the mediant?
Clarifying the problem: I was working on some numbers. De data had duration of a job, the number of steps to complete a job and the starting date of a job. All rows had similar data. I wanted to know the average duration per step. Somehow I got in the process of overthinking.
What I was doing, is what I would normally do when computing an average, so in my case: $$ \frac{1}{n}\sum_{i=1}^{n} \frac{d_i}{s_i} $$ In which $d$ is the duration of a job, $s$ the number of steps to complete a job and $i$ the job (I am neglecting the optional time binning here). But why would I not just compute the average duration and the average number of steps, divide those and use that number, so: $$ \frac{\frac{1}{n}\sum_{i=1}^n d_i}{\frac{1}{n}\sum_{j=1}^n s_i} = \frac{\sum_{i=1}^n d_i}{\sum_{j=1}^n s_i} $$ also called the mediant.
I understand math in the way that I know these are not computing to the same value in general. So the question is more fundamental I guess? Why can't we use the last one, just simply because an average divided by an average is not an average? When would we use the last expression (mediant)?