is my concept of averaging averages under this scenario wrong?

Question

I'm confused on the concept of whether or not to average the averages under this scenario, when the data relating to a particular technique are collected from different individuals:

say there are 3 patients all having radiotherapy treatment technique A. Each of them have different no. of fractions in their treatment courses. One fraction is delivered per day. ( Fraction is just terms in radiotherapy, you can interpret it as no. of treatment. In the example, patient 1,2 and 3 has a radiotherapy course of 3 fractions , 4 fractions and 5 fractions respectively.) Their treatment room occupancy are as follows:

patient 1: Day 1: 15 mins; Day 2 :13 mins; Day 3: 10 mins ( avg = 12.7 mins)

patient 2: Day1: 8 mins; Day2: 7 mins; Day3: 8 mins; Day4: 6 mins (avg=7.25mins)

patient 3: Day1: 12 mins; Day2: 10 mins; Day3: 8 mins; Day4: 8 mins; Day5: 9 mins (avg=9.4mins)

if i want to conclude the average treatment room occupancy per fraction of those patients receiving technique A, should i do like averaging the daily treatment room occupancy per patient and then average their averages i.e. (12.7+7.3+9.4) / 3 = 9.8 mins per fraction?

or the right way should be ignoring the fractions are from different patients and just sum all the fraction time up and get an average as a whole, getting 114mins/12fractions, so as to conclude that the average treatment room occupancy for patient using technique A is 9.5 mins per fraction?

Answer 1

As you've stated the question --- to calculate "the average treatment room occupancy for patient using technique A" --- you should use the second technique you describe, ignoring that the observations are from different patients and just sum all of the time and divide by the total number of observations.

If you instead calculate the average of the averages, then you're weighting the observations ("fractions") by the number of observations for each patient. In effect, the observations from patient 1 will "count more" in your final average than the observations from patient 3 because patient 1 has only three measurements whereas patient 3 has five.