Why is your bus always so crowded?

The city swears the average bus is half-empty. But the one you ride is always jammed. Is the city lying — or is something sneaky hiding in the word "average"? Let's catch it.

1There are two "averages" hiding here

Count the buses, or count the people?

"Average" sounds like one number. But you can build it two different ways — and they don't have to agree. Watch what each one pays attention to:

Counting the buses

Every bus counts once — the packed one and the empty one weigh exactly the same. Add up the riders, share evenly across all the buses.

Counting the people

Every person counts once — so a packed bus gets counted lots of times, an empty bus barely at all. This is the bus a typical rider sits on.

2The same buses, two questions

Who gets a vote?

Here are three buses, exactly the same in both pictures. The only thing that changes is who does the counting.

the bus's question

"How full is a typical bus?"

Each bus raises one hand. Three buses, one vote each → average is about 5 riders.

the rider's question

"How full is a typical rider's bus?"

Each rider raises a hand. Most riders are on the packed bus → the rider’s-eye average climbs to 6 riders.

3Your turn — reshape the fleet

Make the buses even or lumpy

Same number of buses, same total riders the whole time — so the bus-average can't budge. Drag the slider and watch the riders slosh from "spread out" to "piled up."

Average per bus
6.0
Total riders · buses
60 · 10
How lumpy are the buses?perfectly even
ALL EVENA FEW PACKED

4Now pick a rider at random

Whose bus do you land on?

Make the buses really lumpy above, then come down here. We'll close our eyes and tap one random rider in the whole city. But first — a guess.

Guess before you find out

The plain bus-average stays the same no matter how lumpy you make it. You pick one rider out of everyone, totally at random. What does their bus usually look like?

5So which average is right?

Both! They answer different questions

Counting the buses is honest

It's the true answer to "how is the whole fleet doing?" The city isn't lying — half the buses really are roomy.

The catch: it counts an empty bus the same as a packed one, so it's not what any rider actually feels.
Counting the people is honest too

It's the true answer to "what does a typical rider feel?" — and that answer really is "crowded."

The catch: it over-weights the packed buses, so it always runs higher than the fleet average.

When you ask "why is my bus crowded," you're counting people, not buses — and people pile onto full buses, so a random rider almost always lands on a crowded one. The fleet average and your experience are both true; they're just answering different questions.

Psst, grown-ups: this is the inspection paradox (size-biased sampling). Sampling by rider weights each bus by its own occupancy, so the rider-experienced mean equals mean + variance/mean — always ≥ the plain mean, equal only when every bus is identical (zero variance). The same maths explains why your 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. The gap is exactly the spread of the distribution, not a measurement error.