Buses arrive randomly (Poisson process). Your average wait equals the average interval between buses -- not half! Longer gaps "catch" more random passengers.
If buses arrive every 10 minutes on average, you'd expect to wait 5 minutes. But with random arrivals, your average wait equals the full interval: 10 minutes!
This is the inspection paradox: a random passenger is more likely to land in a long gap than a short one. The gaps you experience are biased toward the longer ones -- so you wait longer than half.