An average of averages problem, or two valid results?

Question

From an event log I have derived the days between each user's logins. This leaves me with a dataframe like this:

I then took the average of column 'days_between_logins' which gave me a result of 15.9 days.

But then, if I calculate the average number of days between logins on an individual user basis and then average that across all users, the result is 34 days.

I think I have an average of averages problem? Then again, I think both results are valid.

I would really welcome any thoughts or supporting/opposing conclusions.

Answer 1

In you first approach you take into account the number of logins for each user, which you do not do in the second approach. Take the following example with two users with a different amount of logins:

$ u_1 = {7, 5, 10, 10, 8} \\ u_2 = {1} $

Using the first approach the average would be 6.8 days whereas the second approach would give an average of 4.5 days. The second approach does not take into account that we have more observations for the first user (with a higher number of days between logins) than for the first user. Which approach is the best would depend on what question you are trying to answer, but I think generally the first approach would be more accurate in getting the average time between logins.