On Saturday June 3, while others were having fun, I timed the departures of buses from Jamaica for 90 minutes. I then calculated actual times between the departures of buses to the same park.
For AK and MK the average interval between departures was 14 minutes; HS saw 15 minutes between departures; Epcot was 17.
Since you could arrive anytime in the departure interval, on average during this period your expected wait to departure would be half the average interval, or 7-9 minutes.
These averages mask, however, variability in actual departure intervals. The longest waits were 26 minutes for HS, 22 for Epcot, 23 for MK, and 20 minutes for AK. So a family that by chance picked the longest-interval buses, and arrived right after the prior bus departed, would have a very different experience than the “average.”
To illuminate this, I graphed “minutes to next bus to same park” vs percentage of all intervals, and got the following:
View attachment 242970
What you can see is that the 50th percentile interval (marked in red) was 15 minutes—so half of users should have an expected wait of half of that. The 80th percentile interval (gold) was 20 minutes, suggesting that even in the worst case of arriving right after a park bus left, 80% of trips would see a wait of 20 minutes or less.
These conclusions strictly speaking of course apply only to the time I observed, (June 3, 3.30p to 5p), but sometimes data helps interpret anecdotes.