What is the correct way of calculating the average percentage in this scenario?

Assume that there two people working in a call centre: PersonA and PersonB.

They are both waiting for calls in separate locations.

I want to calculate the average amount of time spent waiting overall.

The scenario is this:

PersonA spends 50 minutes out of 100 (50.00%) waiting for calls to come in (the other 50 minutes were spent on calls).

PersonB spends 900 minutes out of 1000 (90.00%) waiting for calls to come in (the other 100 minutes were spent on calls).

My question is this: what is the correct way to calculate the average waiting time across both people?

Should I take an average of their respective time-spent-waiting percentages or should I take into account their collective total available time when working this out?

I have tried taking the average of PersonA's and PersonB's time-spent-waiting percentages, which seems plausible:

AVERAGE(0.5, 0.9) = 0.7 (70.00%)

I understand this to mean that, on average, the time spent waiting was 70.00% across both people.

But then, taking into consideration PersonA and PersonB's total available times when calculating this also seems plausible:

950 minutes (waiting) / 1100 minutes (total available time) = 86.36%

Which I understand to mean that, out of 1100 minutes of total availability, the average amount of time spent waiting across both people was 86.36%.

What is the difference, and which is the correct way to calculate this in this scenario (and why)?

• This isn't really a math question. Both averages make sense, but they measure different things. If you are interested in getting a base line on employee efficiency, you should average the two averages. If, on the other hand, you want to get a baseline on how much wait time is involved in a call, you should look at each call. – lulu Oct 3 '18 at 12:13
• That makes sense. Also, I wondered if this belonged here or on a Stats SE site, so opted for here. – SnookerFan Oct 3 '18 at 12:15
• Oh, I think this is a fine place for it. – lulu Oct 3 '18 at 12:15
• Great. So assuming that each employee is available for a different amount of time, then is the first calculation (average of averages) the correct one to use? If so, why? – SnookerFan Oct 3 '18 at 12:34
• As I say, "correct" depends on what you are trying to measure. If I have a lot of students taking Calculus, with a lot of tests, I can compare the averages of two students, even if they have taken different numbers of tests (or different tests). Makes perfect sense. Or, if I am trying to assess how difficult my tests are, I can look at the scores on each test and average them. This also makes perfect sense, though the two averages are different. – lulu Oct 3 '18 at 12:37