Which Average? Ward Density Before and After a Boundary Split
A metropolitan authority is revising its density-linked building rules, and its census wing places five wards on the table. Ward P1 holds 24,000 persons over 60 ha; P2 holds 36,000 over 40 ha; P3 holds 18,000 over 90 ha; P4 holds 30,000 over 50 ha; P5 holds 12,000 over 60 ha. The chairperson reads out that the average ward density is about 460 persons per hectare, and the councillor from P3 objects that the figure flatters his ward. Separately, P2 - the smallest and densest ward - is to be split into two wards of exactly equal population and equal area for electoral reasons. The panel statistician warns that the headline average will shift even though no person and no hectare has changed.
1. Compute each ward's density, the city-wide density, the unweighted mean of ward densities and the median. Explain the gap between the figures.
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Density = population ÷ area (persons per hectare, ppha):
- P1: 24,000/60 = 400
- P2: 36,000/40 = 900
- P3: 18,000/90 = 200
- P4: 30,000/50 = 600
- P5: 12,000/60 = 200
City-wide density = total persons ÷ total area = 120,000/300 = 400 ppha. Unweighted mean of the five ward densities = (400 + 900 + 200 + 600 + 200)/5 = 2300/5 = 460 ppha. Median = 400 ppha (sorted: 200, 200, 400, 600, 900).
The 460 is an arithmetic result, not a description of the city. It gives a 40 ha ward at 900 ppha the same weight as a 90 ha ward at 200 ppha. Only the weighted figure describes the settlement; the unweighted mean describes 'the typical ward' and is inflated by one small, very dense ward. The P3 councillor is right: his ward sits at half the city density, and the 460 headline hides that.
2. P2 is split into two wards of equal population and equal area. Recompute the three measures and state which one a regulatory threshold should use.
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Each half of P2: 18,000 persons on 20 ha → 900 ppha.
New density list: 400, 900, 900, 200, 600, 200.
- Weighted: total persons still 120,000, total area still 300 ha → 400 ppha, unchanged.
- Unweighted mean: 3200/6 = 533.3 ppha (was 460, up about 16 per cent).
- Median: sorted 200, 200, 400, 600, 900, 900 → (400 + 600)/2 = 500 ppha (was 400, up 25 per cent).
Nothing physical changed, yet two of the three statistics moved substantially. Only the population-weighted density is invariant to where the boundary is drawn.
Judgement: a density threshold in a development control rule must be applied as population ÷ area over the zone being regulated. A mean of ward densities or a ward median turns a boundary revision into a change in building rights, which is indefensible in appeal.
3. A draft rule gives a lower FAR to any ward above 600 ppha. Which wards trigger it, and what should the drafting authority watch for?
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Before the split: P1 400 (no), P2 900 (yes), P3 200 (no), P4 600 (borderline), P5 200 (no).
The whole question turns on P4. 'Above 600' excludes it; '600 or above' includes it. A 600 ppha ward's entire FAR entitlement would swing on one preposition, and a ward computed at 599.6 or 600.4 would swing on rounding as well. The rule must be drafted explicitly as either '≥ 600 ppha' or '> 600 ppha', with a stated rounding convention and a stated reference date for the population figure.
After the split, both halves of P2 remain at 900 and still trigger - the split is neutral for P2's own entitlement. The exposure is P4 and any ward sitting within rounding distance of the cut-off.
Judgement: either fix the threshold with a margin (say 650) so that no realistic boundary or rounding change flips it, or apply the test to a composite planning zone rather than a single ward. Do not leave the comparator ambiguous in a rule that grants building rights.