If most buses run half-empty, what kind of bus does a random rider land on?

The city swears the average bus is half-empty. So you'd think a rider picked at random would usually be on a roomy one. Is that right — 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?

A fresh fleet: eight buses carry 3 riders each, and two buses carry 18 riders each. Pick one rider from all 60, with every person equally likely. This fleet stays fixed while you test it.

Guess before you find out

Which group holds more of the 60 riders: the eight 3-rider buses, or the two 18-rider buses? Predict where a random rider is more likely to be.