Modifiable areal units

Why can redrawing the districts on a map reverse what the very same data appear to show?

A health authority maps disease by neighbourhood and one neighbourhood glows. Money follows the glow. Then the boundaries are redrawn for an unrelated administrative reason, the same cases are mapped again, and the glow is gone. Nobody moved house and nobody recovered.

So what was the first map reporting — the disease, or the boundaries? And if we cannot answer that, what is any zone-level statistic actually a statement about?

Let us shrink it until you can check it with a pencil. Six wards in a row, each with exactly 100 residents. Their case counts, west to east, are 2, 10, 3, 9, 4 and 8: thirty-six cases among six hundred people, an overall rate of 6 per 100.

Now group those six wards into three zones. There are several contiguous ways to do it, all equally legal.

Pair off neighbours — {2,10}, {3,9}, {4,8} — and each zone holds 12 cases in 200 people. The rates are 6.0, 6.0 and 6.0. The map is perfectly flat: no hotspot anywhere.

Cut instead after the first ward — {2}, {10,3}, {9,4,8} — and the rates are 2.0, 6.5 and 7.0. The worst zone now shows three and a half times the rate of the best.

Cut a third way — {2,10}, {3,9,4}, {8} — and you get 6.0, about 5.3, and 8.0, a ratio of roughly 1.5.

Same people. Same cases. Same number of zones. Three maps saying no disparity, a 3.5-fold disparity, and a 1.5-fold disparity. Which is the true one?

Look at what the arithmetic did. A zone rate is a weighted mean of the wards inside it, so a boundary is nothing but a decision about which wards get averaged with which. Averaging destroys the variation inside a zone and preserves only the variation between zones. Put the extremes together and they hide in each other; split them apart and they stand exposed. The published number is therefore not a property of the six hundred people. It is a property of the six hundred people and the partition — and the partition is a human artefact that could have been drawn otherwise. That "could have been otherwise" is the whole force of the word modifiable.

That gives two handles worth keeping apart. Change how many zones there are: merge the six wards into two, {2,10,3} and {9,4,8}, and the rates are 5.0 and 7.0, a ratio of 1.4; merge into one and the disparity is zero by construction. Coarser units systematically absorb variation and flatten differences — the scale effect. But the three-zone maps above all had the same count and still disagreed, purely because the cuts fell elsewhere. That is the zoning effect, and it is the one people forget, because it does not go away by asking for finer data.

What would show this account to be wrong? It predicts a specific dependence: the spread of answers across alternative maps should scale with how heterogeneous the wards are within the zones you draw. So take real point-level data, generate a few thousand random contiguous partitions at a fixed zone count, and record the statistic each time. Where within-zone variance is near zero the distribution should collapse to a spike and no zoning should escape the individual-level relationship; where it is high the distribution should be wide. Find heterogeneous data whose statistic sits still across partitions, or homogeneous data whose statistic swings anyway, and the mechanism I have just described is not the one operating.

Three neighbours sit close, so let me say where each parts company.

district-boundaries.md is about a mapmaker choosing a partition to settle an election — this same arithmetic driven by intent. The uncomfortable finding here is the reverse: no intent is required. Boundaries drawn by an indifferent clerk in 1974 produce a spread of answers too, and the effect is just as strong in variables nobody has a stake in, such as rainfall totals or soil classes.

coastline-paradox.md also has a measurement that moves with scale, but its object is a physical curve whose length genuinely diverges as the ruler shrinks. Here the underlying quantity is well defined at the individual level; only the summary moves. And a coastline has no analogue of the zoning effect — at a fixed ruler length there is one answer, whereas at a fixed zone count there are many.

functional-versus-administrative-regions.md is this piece's mirror. It asks which partition matches how people actually live; this one insists that even a well-shaped partition is one choice among many, and the number will move if you take another.

The literature's famous demonstrations — Gehlke and Biehl finding correlations climbing as census tracts were merged, Robinson's reversal between individual and state-level relationships — I am recalling from memory and have not re-derived; nothing above rests on them. The six wards are the argument.

Two corrections. The fix is not "always use the smallest units": fine units carry more noise, unstable rates over small populations, and disclosure limits, and the zoning effect survives at every level. And this is not the ecological fallacy, though they are relatives — that is the error of reading a zone-level relationship as an individual-level one. Modifiable units are the prior problem: the zone-level number is not unique before anyone misreads it.

• How the ecological fallacy compounds this once zone-level relationships are read back onto individuals.
• Whether zone designs built from the data itself trade one arbitrariness for another.

Every term the collection defines is gathered in the glossary.