Why does the bus you take always feel more crowded than average?
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Why does the bus you take always feel more crowded than average?
The short answer
Your bus feels crowded because you experience the people-weighted average, not the plain bus-average. Most riders are sitting on the few packed buses, so a randomly picked rider almost always lands on a crowded one, even when the average bus is half-empty.
How it works
There are two ways to average crowding. Counting the buses gives every bus one vote, so an empty bus weighs the same as a packed one, and you get the fleet's plain average. Counting the people gives every rider one vote, so a packed bus is counted many times and an empty bus barely at all. Because people pile onto the full buses, the second average runs higher, and that second average is the one any rider actually feels.
What people get wrong
People assume the crowding they personally feel must equal the plain average, so a packed bus means buses really are crowded. In fact a random rider over-samples the full buses, because that is where most people are, so their experience runs above the fleet average whenever buses vary in how full they are.
The catch
Counting the buses is honest: it correctly answers 'how is the whole fleet doing?' and the city is not lying when it says many buses are roomy. Counting the people is honest too: it correctly answers 'what does a typical rider feel?' and that answer really is 'crowded.' Neither number is wrong; trouble only comes from mixing up which question you are asking.
Questions kids ask
Is the city lying when it says the average bus is half-empty?
No. The fleet really does average out to half-empty when you give every bus one vote. But you ride as a person, not as a bus, and most people are crammed onto the few full buses, so the average a rider feels is much higher. Both numbers are true; they answer different questions.
Would this still happen if every bus carried the same number?
No. If every bus is equally full, the bus-average and the rider-average match exactly. The gap only opens up when buses vary, and it grows the more uneven they are.
Where else does this trick show up?
Everywhere people or things bunch together. Your class feels bigger than the school's average class size, popular restaurants always seem packed, and most people have fewer friends than their friends do. In each case you are sampling by the crowded thing, so your experience runs above the plain average.
Why does my bus also seem to come right after a long wait?
A related reason: buses tend to bunch up. A slightly late bus collects extra riders, which slows it down further, so gaps grow. You are more likely to arrive during a long gap than a short one, so your wait feels longer than the schedule promises.
For grown-ups
This is the inspection paradox, or size-biased sampling. Sampling by rider weights each bus by its own occupancy, so the rider-experienced mean equals the plain mean plus variance divided by mean. It is always at least the plain mean, with equality only when every bus carries the same number (zero variance). The same maths explains why a class feels bigger than the school's average class size, why most people have fewer friends than their friends do (the friendship paradox), and why waits feel long when buses bunch.