Age heaping
A Socratic walk-through of age heaping — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why do old census returns report far more people aged exactly thirty, forty and fifty than the years either side?
Tabulate an old census return by single year of age and the histogram does something no real population does. It spikes at 30, 40 and 50, spikes a little less at 25, 35 and 45, and sags between. Nothing in births or deaths could produce that: a cohort born thirty years ago is not systematically larger than those born twenty-nine and thirty-one years ago.
So the pattern is not in the population but in the reporting. Which raises the better question: is that damage to be cleaned up, or is it telling us something the census never set out to record?
Reasoning it through
REASONING #Start with the enumerator at the door asking how old someone is. For a correct answer to come back, the person must know their birth date and be able to subtract it from the current year. Without compulsory birth registration or general schooling, neither holds reliably: many genuinely did not know the year they were born, only that it was the hard winter, or the year of the flood, or a bit before a neighbour.
So what does someone do when asked a question they cannot answer exactly? They estimate — and an estimate comes out at whatever resolution the estimator feels entitled to, which is where round numbers come from. Nobody carrying five years of uncertainty volunteers "thirty-seven"; they say thirty, or forty. The heap is not a slip. It is a coarse estimate correctly reported at the precision the person has.
The ordering that follows is the account's first testable consequence: heaping worst on multiples of ten, next on the intervening fives, then evens, least on 7 and 9 — roughly the observed pattern, falling out of nothing but the salience of the numbers.
Now the step that turns a defect into an instrument. If heaping arises from an inability to reckon exactly with numbers, its severity measures something: how numerate the reporting population was. That is the reasoning behind the Whipple index — take reported ages over a middle span of adult life, count how many end in 0 or 5, and compare with a fifth of the total, which is what a heap-free distribution gives, scaled so 100 means no heaping and 500 means every age reported on a 0 or 5. Ages under about 23 and over 62 are excluded, heaping there being tangled with other reporting effects. A'Hearn, Baten and Crayen built on this a numeracy proxy computable for populations centuries before any literacy test existed — extracting a measure of human capital from records kept for wholly unrelated purposes.
But is heaping really a property of the person answering? Three parties stand between a true age and a written figure. The respondent rounds. The enumerator may round too — filling many forms, estimating the age of someone absent, writing what looks plausible. And the office imposes structure: some schedules asked age last birthday, some completed years, some bands, and some enumerators took a household head's report of everyone under the roof. Heaping in the returns confounds all three.
The confound has a checkable signature. If it were mostly the office's doing, heaping would vary sharply between districts with different enumeration practice and between years when the form changed, while looking flat across individuals within a district. If mostly the respondent's, it should vary with individual traits — signing a marriage register with a name rather than a mark, occupation, schooling — and hold steady across the clerks recording them. Individual-level correlations with literacy do turn up, the strongest evidence for the numeracy reading, but the sources are not cleanly separable in any single record I know.
Heaping is also not the only age distortion here. There is directional misstatement — overstatement at the top of the distribution, understatement around ages where a tax, conscription liability or marriage norm made one age advantageous. That has a different cause and does not wash out of a Whipple index, which counts terminal digits and cannot see a systematic shift. The wider point is the one census undercount makes about coverage: what matters is never that a return is wrong but that it is wrong unevenly.
The analogy
THE ANALOGY #Think of a room of people asked how far they live from the station. Almost no one says 1.3 kilometres; they say one, or two, or half a mile. The clustering is not lying — each answer is true at the precision the speaker has. And here is the useful part: if some of the room answer in tenths of a kilometre and the rest only in whole miles, you have learned something about the two groups without asking them anything about measurement.
Distance can be checked against a map, so a rounded answer is correctable, whereas for a historical age there is usually no independent record at all — which is why heaping can be measured but the individual ages behind it cannot be repaired.
Clarifying the model
THE MODEL #Two refinements.
First, heaping is not noise. Noise averages out; this does not. It moves people systematically toward particular years, so any statistic computed on a single-year age — an age-specific mortality rate, a mean age at marriage, a cohort size — inherits a bias, not merely extra variance. Demographers smooth the distribution or work in five-year bands, suppressing the artefact by discarding the resolution that made it visible.
Second, read as numeracy the Whipple index does not say a person could not count. It says they could not, or did not, locate themselves precisely on a numbered scale — which needs a birth date recorded somewhere and the habit of reckoning in years. That is as much a fact about institutions as about individuals, and a state that begins registering births drives heaping down without anyone becoming cleverer.
What would refute the numeracy reading? It predicts heaping falls with schooling and with birth registration, correlates with independent literacy measures at the individual level within one enumeration, and stays roughly stable when the same population is re-enumerated by different clerks. So the refuting observations are specific: heaping that tracks the enumerator rather than the enumerated once individuals are matched across sources; heaping unrelated to literacy where both are recorded for the same person; or heaping that fails to fall where birth registration is introduced but schooling is not.
A picture of it
THE PICTURE #How to readThe widths are schematic — they show the mechanism's shape, not counts from any census. Read left to right: each band on the left is a set of people of one true age, each on the right the age written on the form. Follow any true age across and it splits, most draining into the round year while a thin stream is reported correctly. The point is the right-hand side: 30 arrives fat because it collected from every neighbouring year, 31 starved because most who belonged there were absorbed. Nobody is added or removed — the population is identical on both sides — and the spike comes entirely from which way the uncertain cases flow.
What became clearer
WHAT CLEARED #The spikes are not a flaw in the population and not really a lie. They are what happens when people report an estimate at the precision they honestly have, and round numbers are where imprecise estimates land. That makes the heap a fossil of the reporting conditions — and because placing yourself exactly on a numbered scale depends on records and schooling, its size reads as a rough measure of numeracy in populations that left no other trace of it. The caution is that the heap is produced jointly by person, enumerator and form, and the index cannot tell those apart.
Where to go next
ONWARD #- How demographers smooth heaped distributions, and what that quietly assumes about the true shape.
- Why the same digit preference appears in self-reported income and duration data today.
Key terms
TERMS #| Term | What it means |
|---|---|
| Whipple index | a measure of that concentration over a middle adult age span, scaled so 100 is no heaping and 500 is total. |
Every term the collection defines is gathered in the glossary.