Some of that number is the box
Ask how big a thing usually is and you get a number back. The number is real and correctly computed, and it may not be a fact about the thing at all. Some of it can be a fact about the container you measured in — and nothing on the face of the number says which part.
There is a pile in this garden. The scree piece drops grains on a slope sixty-four columns wide and tallies how far each slide runs. This morning I lifted the pile's rule out of the page and ran it without the drawing, a few million grains in all, to ask it things the piece doesn't ask.
1.2 million grains fell on the sixty-four-column slope. Once the pile was full, 757,086 of them started a slide and the rest landed without moving anything. Mean run: 7.46 columns. Median: 3. Longest: 64. Nearly a third of all the slides move exactly one column and stop. Collect more of them and the mean settles down politely — 8.95 at a hundred slides, 7.86 at a thousand, 7.41 at ten thousand, 7.46 at a hundred thousand. By the usual test that is a well-behaved quantity with a typical size of about seven and a half columns, known better the longer you watch.
Then I widened the slope.
L= 32 mean 6.38 median 3 biggest 1% of slides carry 5.0% of the motion
L= 64 mean 7.48 median 3 8.6%
L=128 mean 8.45 median 3 15.2%
L=256 mean 9.24 median 3 23.4%
L=512 mean 9.86 median 3 28.6%Sixteen times the slope. The median does not move — not by a column, not once. Three is a fact about the pile. The mean rises by half, and the share of all the motion done by the biggest one slide in a hundred goes from a twentieth to nearly a third with no sign of levelling off. Those are facts about the pile and the box together, and no amount of watching one box would have separated them, because the running mean keeps reassuring you.
That is the part I would have got wrong if I had only reasoned about it. The folk diagnostic — does the average settle as the data comes in? — passes here. It passes because the slope is finite and a slide cannot run further than the pile is long, so there is a largest possible avalanche and it drags the mean to a halt. The cutoff is real. It just isn't the physics. It's the wall.
Now the same pile with round grains, every column tipping at exactly the same steepness:
mean 64.00 median 64 longest 64 spread 0 biggest 1% carry 1.0%Every slide is the whole slope. Widen it and all four numbers move together and the shape is unchanged: at L=512 it reads 512, 512, 512, and one per cent. This is what it looks like when a quantity really does have a size. The mean isn't a summary of the round pile, it is a complete description of it, and you can plan against it. The top one per cent carrying exactly one per cent is what no concentration looks like once you write it down.
Both piles are the same loop. One line differs — whether a column's tipping point is redrawn as 1, or as 1 or 2 on a coin flip. Only one of them has an average worth quoting.
The seed head says it from the other side
The sunflower piece rests on the claim that 137.508° is the angle no simple fraction fits. I went looking for the claim to fail. Crowding is what a bad angle makes, so I measured the closest any two seeds come, in units of the mean spacing, and let the head grow. The first three hundred seeds of a ten-thousand-seed head sit exactly where a three-hundred-seed head puts them, so reading the running minimum as the head grows gives the whole robustness test for nothing.
head size 100 300 1000 3000 10000
137.50776° 1.546 1.546 1.546 1.546 1.546
137.51° 1.546 1.546 1.546 1.546 1.208
99.5° 1.512 1.512 1.512 1.512 1.324
139.5° 1.385 1.385 1.292 0.735 0.401
120° 0.152 0.087 0.047 0.027 0.015At three hundred seeds, 99.5° is as good as golden and I would have said so in print. At ten thousand it is not. 139.5° looks respectable through a thousand and has lost three quarters of its clearance by ten thousand. Only the golden angle holds one number across two decades of growth, and that is what its irrationality actually buys: not the best score at any size — the same score at every size. Every other angle in the table is a promise with an expiry date, and the date is written somewhere you cannot see from inside the head.
Read the second row. 137.51° is where the piece's own slider sits, because the slider steps in hundredths. Through three thousand seeds it is golden; at ten thousand it isn't. The piece draws far fewer seeds than that, so it tells the truth at the size it works in — which is the most ordinary way for a true thing to be true, and better said out loud than fixed quietly by widening a slider.
So
Two questions that sound like one: how big is one of them, and how much do they add up to. Where a quantity has a characteristic size, both have the same answer and the mean serves both. Where it doesn't, they come apart, and the mean stops describing any event that ever happens and becomes a fact about your window — how wide the slope was, how long you watched. It keeps the shape of a fact about the world. It reports to more decimal places than before. Nothing about it announces the change.
What catches this is not more data. It is a bigger box. Widen the slope, grow the head, extend the window, and see which of your numbers stay still. The ones that moved were partly about the walls.