I want to find a good metric to measure the overtime of appointments in a hospital across different days. For example, there are only 2 kinds of appointments A and B. A is supposed to take 30min and B is supposed to take $40$min (ideally).
In reality:
On Monday, there are 3 A scheduled actually taking 35, 40, 28 min. there are 2 B scheduled actually taking 45, 50 min.
On Tuesday, there are 1 A scheduled actually taking 36min. there are 2 B scheduled actually taking 37, 42min.
I want to find a good metric to measure the overtime. At first, I used average overtime.
For Monday, it is $[(35-30)+(40-30)+(28-30)+(45-40)+(50-40)]/5$
For Tuesday, it is $[(36-30)+(37-40)+(42-40)]/3$
But then I realized I count type A and B equally. Think an actual overtime $10$min for A is definitely worse than $10$min overtime for B (since ideally A should take 30min and B 40min). That's my point. I am thinking to do a weighted average but could not get the best (reasonable) weight. Please help how to get the weight or other good metric. Thanks.
Maybe you could just divide the overtimes by the ideal time (almost kind of like percent error). So for Monday, it would be
$[(35-30)/30 +(40-30)/30$ +$(28-30)/30$ +$(30-40)/40 +(45-40)/40 +(50-40)/40]/5$. For Tuesday, it would be $[(36-30)/30 +(37-40)/40 +(42-40)/40]/3$.
I think this might solve your problem.