the wall  /  0022

The bus you wait for is not the average bus

The buses come every ten minutes on average. Turn up at a moment of your own choosing, and the gap you land in is longer than ten — and on a route with no timetable at all it averages twenty, because a twenty-minute gap has twice as many moments in it to land in as a ten. Nothing is late, nobody is lying, and the timetable is exactly what it says. The same fraction says your friends have more friends than you do, that the college whose average class holds 32 seats the average student in a class of 73, and that sixty out and twenty back is not forty. This page runs all four, and checks each one against the arithmetic rather than against the story.

The stop

how ragged the timetable is

gaps 10.0 ± 5.0 min
the gap on the timetable the gap a rider lands in
the timetable's mean gap—every gap, counted once
the gap riders land in—every gap, counted once per minute
and so the wait is—half of the gap you are in

Both numbers come off the same week of buses. The left one asks the timetable how long its gaps are. The right one asks the riders, and there are more riders standing in the long gaps, because a gap that is twice as long has twice as much standing in it. That is the whole mechanism, and it is not a fact about buses: sample by the thing, and you get one answer; sample by the room the thing takes up, and you get the other.

Drag the dial to 0 and the two numbers meet: when every gap is ten minutes there is no long gap to be caught by, and the wait is five. Push it to 1 — buses arriving at random, with no timetable — and the gap you land in is twice the mean gap, so your expected wait is the whole ten minutes, however cleverly you time your arrival. That is the one everybody has felt.

gap you land in = mean gap × (1 + cv2)   ·   wait = half of it

Your friends

Take the same fraction to a group of people. A person with many friends is, by definition, on many people's friend lists — so picking a random friend is not the same as picking a random person, and the popular are over-drawn in exactly the way a long gap is. Almost everybody's friends have more friends than they do, and nobody is doing anything wrong.

friends per person—the honest average
friends per friend—as your friends see it
worse off than their friends—of the room

The gap between the two is exactly the variance divided by the mean, so it closes only when everyone has the same number of friends — press the second button and the room where everyone knows everyone puts both numbers on the same value, with nobody below average. Every other room has the paradox in it. It is not a fact about people either.

The same fraction, four times

counted bythe thingthe room it takes upratio

Classes. A college with thirty classes — twenty of 12, six of 40, four of 120 — advertises an average class of 32, and it is telling the truth. Ask the students instead and the average class has 73 people in it, and they are telling the truth as well. The four big lectures are 13% of the classes and 50% of the seats.

The drive. Sixty kilometres out at 60 km/h and sixty back at 20 km/h is not an average of 40. It is 30: one hour out, three hours back, 120 km in four hours. The 40 is the average kilometre, and you do not live through kilometres. Three speeds come out of one drive — 40 for the average kilometre, 30 for the journey, and 24 for the average minute, which is slower than the trip because three minutes in four are spent in the slow half. It is the last step that is this page's fraction, exactly: in minutes per kilometre the drive is 2.00 counted by the kilometre and 2.50 counted by the minute, and 2.50 is 2.00 × (1 + cv²) to the last bit. The famous 40-against-30 is the same bias wearing its reciprocal.

The family. Not on the table because it needs a second page, but it is the same object: ask people how many children were in the family they grew up in and the answer is too big, because a family of six hands you six informants and a family of one hands you one. Every survey that samples people to learn about groups pays this tax, and it is the reason a census asks the household.

What this page checked before showing you

The interesting ones are the exact ones. A random moment inside a known timetable lands in a gap of Σg² / Σg — that is not a statistical claim, it is an integral, and the simulation above has to hit it. So does the friendship number: the mean degree of a random end of a random edge is Σd² / Σd, and the page computes it both ways round and requires them to agree to the last bit.