Ask the trains
Express stopStand on the platform with a stopwatch and time the gaps between trains. Every gap counts once.
What CV measures
Local stopSome lines run every 3 minutes, others every 10, so their gaps can't be compared as they stand. Stretch each line's clock until its average gap is the same length. What is left is unevenness, and the CV measures it.
Ask the riders
Express stopRiders land like darts thrown at the timeline, at moments that have nothing to do with the trains. That is the page's one assumption about riders. A long gap catches more darts than a short one, so ask each rider how long their gap was.
The size-bias machine
Local stopPick any distribution of sizes, or draw your own, then weight every value by its size. That is size bias, and it is all the riders did.
The right-hand picture is the strongest of the three. Size bias multiplies each value's probability by x / E X, a line through 0 that rises past 1 exactly at the mean, which is where the two densities above cross. Values below the mean get rarer, values above get commoner, and 0 vanishes. Because the ratio rises everywhere, this holds inside any range too: among gaps of 4 to 6 minutes, riders still land in the longer ones more often. It implies the other two pictures.
The crowded car
Local stopA train that ends a long gap picks up everyone who arrived during it. The average rider's train is fuller than the average train.
Your wait
Express stopYou land somewhere inside your gap, uniformly at random, so you wait half of your gap on average. Half of a long gap, not half of the average one.
The algebra: how grey, yellow and orange are related
Is your wait just the yellow curve, halved?
Is that gap real?
Local stopA very long gap looks like a recording error. But long gaps are exactly what riders feel. Check a gap against other evidence before you delete it.
Clockwork to chaos
Local stopKeep the average gap fixed and change only how uneven the gaps are.
Should you give up?
Local stopYou have already waited a while. Is the train now more likely to be just around the corner, or less? And when, if ever, should you walk away?
Is the train getting closer?
When to give up
The algebra of giving up
Every line
Express stopEach line, both directions, at one busy station. Right means longer gaps; up means more uneven gaps. The curves mark equal average waits.
Down the line
Express stopTrains usually leave the terminal evenly spaced, and delays along the way push them together. So the same trains can mean a longer wait farther down the line.
Why trains bunch
Local stopA late train finds more people waiting, so it stops longer and falls further behind. The train behind it finds fewer, stops briefly and catches up. This line is simulated, not data.
Where the formula comes from
Local stopFrom darts on a strip of gaps to the formula at the end of the line, in ten moves. Each line of algebra is one change in the picture.