The average you experience is not the average that exists.

The operator computes an average over gaps: thirteen buses, twelve gaps, each counted once. You compute an average over moments of arrival — and a 24-minute gap contains six times as many moments as a 4-minute one. Weighted by where a random arrival actually lands, the long gaps dominate and the experienced average climbs from 10 minutes to 16.

Statisticians call this length-biased sampling, and the more irregular the service, the worse it bites: for fully random (Poisson) arrivals the gap you land in averages exactly twice the scheduled one. Perfectly even service is the only schedule where the two averages agree — which is why "buses every 10 minutes on average" and "you'll wait 5 minutes on average" are different promises, and only one of them is usually kept.

Once you see the mechanism — big units of anything are more likely to contain you — you start finding it everywhere. The rest of this page is a short tour.

A department runs ten classes: nine seminars and one giant lecture. Average the class sizes and the brochure's claim is perfectly honest — about 31 students per class. But ask the students, and most of them are sitting in the 200-seat lecture, because that's where most of the students are. The class experienced by the average student holds about 134.

Both numbers are computed from the same ten classes. One weights each class equally; the other weights each class by the number of people inside it reporting back. Universities advertise the first. Students live the second.

In 1991 the sociologist Scott Feld proved a fact that sounds like an insult and is actually arithmetic: in any friendship network, the average number of friends of a friend exceeds the average number of friends of a person. The reason is the inspection paradox in disguise — when you pick a friend, you're sampling people through their friendships, and sociable people own more friendships to be sampled through. The hub of every social circle is counted once as a person but eight times as somebody's friend.

The bias has serious uses. During outbreaks, asking random people to name a friend and monitoring the friends finds the well-connected — exactly the people infections reach first — which researchers including Nicholas Christakis and James Fowler have used as an early-warning system for flu. The same trick guides vaccination when doses are scarce: immunising friends-of-random-people protects the network's hubs without ever mapping the network.

Crowds. Airlines can truthfully report that their average flight is 80% full while the average passenger sits on one that's rammed — full flights carry more of the passengers doing the reporting. The same arithmetic explains why the gym, the motorway and the restaurant all seem busier than their published averages: you're disproportionately there when everyone else is.

Waiting on hold. "Average call answered in 90 seconds" is a per-call average. The callers stuck in the rare ten-minute waits accumulate far more aggrieved minutes per person — so the average experienced hold, and certainly the average remembered one, is much worse.

A visit to a prison. Survey inmates on one day and long sentences are massively overrepresented — a ten-year sentence is present for ten years' worth of surveys, a ten-day one almost never. Cross-sectional snapshots of hospital stays, unemployment spells and customer subscriptions all inherit the same tilt toward the long-lived.

The diagnostic question is short: is my chance of observing this thing proportional to its size? If yes, you're holding a length-biased sample, and the honest correction is to down-weight the big ones — or at least to stop being surprised that the world feels larger, longer and more crowded than its averages claim.