Hi Graeme, I’ve managed to get a copy of Max’s book. It must have been quite a feat to do calculations of density in those days. He’s got charts of residential, urban, and population density by distance from the GPO for Sydney, Perth, and Adelaide. My published charts have used population-weighted density, which is quite different to residential density. It would be possible to calculate residential density by distance from the CBD for the 2016 and 2011 censuses using mesh blocks for a definition of residential land. However there’s no guarantee that would be a comparable approach to what was used for those calculations in the 1960s and 70s (you’d have to hunt down Max’s sources), but you could at least compare the shape of the profiles.

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]]>I am a historian interested in plotting changes in density over the long-term. Your excellent population gradient graphs and maps could be compared with those generated long ago by Max Neutze in his Urban Development in Australia 1977. I wonder if your overlaid your graphs on his, assuming that the statistical bases are common? You have taken some trouble to remove the effects of factors such as water (in Sydney particularly) so I’m not sure whether your density per hectare figures are directly comparable to Max’s.

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]]>I’m actually coming to the conclusion that population-weighted density might better be thought of as density-weighted population. That is, instead of taking the density of each parcel and weighting by share of the population, take the population of each parcel and weight it by its density relative to the overall density of the entire urban area you’re measuring. Then, mathematically, you have the rule

PWD = DWP / GA

where PWD is population-weighted density, DWP is density-weighted population, and GA is gross area (the total area of all the land parcels put together).

What this immediately tells us is that while PWD has the units of persons per hectare, the ‘persons’ are actually density-weighted.

This explains why PWD changes when you take the same gross area and divide it into different sized parcels. For example, if you split an area into just two parcels A and B it’s possible that A could have twice the density of the overall area and B just half the density; then for PWD each resident of A counts as 2 while each resident of B counts as 1/2. If A and B actually have the same population, then DWP is 1.25 times the original population; if A has twice the population of B then DWP is 1.5 times the original population. Since the gross area is unchanged, these increases in DWP pass straight through to increases in PWD.

In general, the more variation in density there is across a specific region, the greater will be the increase in DWP and hence PWD when you split the region into smaller parcels. Conversely, if a region is absolutely uniform in density then DWP, hence PWD, will not increase at all under subdivision.

The calculation that gives the above rule is as follows: let P1, P2, …. be the population of each parcel and A1, A2, …. the corresponding areas: then GA is the sum of A1, A2, …. while the sum of P1, P2, …. is GP, the gross population. The overall density of the region is GP / GA. The population-weighted density calculation is

PWD = (P1 / GP) (P1 / A1) + (P2 / GP) (P2 / A2) + ….

The density-weighted population on the other hand is

DWP = P1 (P1 / A1) / (GP / GA) + P2 (P2 / A2) / (GP / GA) + ….

= P1 (P1 / A1) (GA / GP) + P2 (P2 / A2) (GA / GP) + ….

If one divides each term in DWP by GA, one gets

DWP / GA

= P1 (P1 / A1) / GP + P2 (P2 / A2) / GP + ….

= (P1 / GP) (P1 / A1) + (P2 / GP) (P2 / A2) + ….

= PWD.

This shows in particular that the ratio of PWD to the ordinary overall density (GP / GA) is identical to the ratio of DWP to the ordinary population GP.

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