Dispersion of Scotland's Population

In his comments on our interim report, Alan Arnott noted our statement that,

“Over 10% of the population of Scotland lives in population densities greater than 10,000 people per km², while the least accessible 15% live at densities of less than 1000 people per km²."

He commented that,

the UK has an overall population density of around 350 people per km² with Scotland around 65/km² so the above figures seem somewhat inflated.

We based our analysis on the Census 2001 data for "output areas" (OA). Our interim report attempted to summarise the contrast between the dense cities and sparse hinterland. The figures we gave are correct, but rounded.

In fact over 11.0% (but less than 11.1%) live at densities over 10,000 p/km². The least accessible 14.6% live at densities < 1,000 p/km². The least accessible 15% live at densities < 1,070 p/km². Instead of rounding the percentage, we should have rounded the population count and said, "while the least accessible 15% live at densities of less than 1100 people per km²".

However, the comment prompted us to return to make a finer comparison with England, with a surprising result.
The population densities averaged over each country are Scotland 65 p/km²; England 377 p/km². One might naturally expect this to imply that, in general, people in Scotland live at much lower population densities than people in England. This is not the case.

At a local level, 90% of the population of Scotland lives at population densities at least as high as those in England!

There are 46,604 output areas in Scotland, and 165,665 in England. The "Key Statistics" table gives area (A) and population (P) for each output area.
We calculate population density for each output area (as the quotient P/A), and list the output areas in order of decreasing population density. The cumulative percentage of population included in initial segments of this list is used as abscissa (x-axis) in the graphs below (click on the graphs for larger images).
Our interim report included this plot: It shows, for each x, the percentage of the area of Scotland occupied by the most densely housed x% of the population, with the area shown on a log scale.
To compare the population densities in Scotland and England, we plot two new graphs (below) of population density. In each case we use a log scale, and units of people per hectare (for p/km², multiply by 100). The data for England (green) are compared with those for Scotland (blue).
The upper graph plots the population density for each output area against the cumulative centile within which its population falls. For the first 90 centiles of population from each country, those in Scotland live at marginal densities at least as high those in England. Scotland has many small communities, often packed more densely than their counterparts in England. So "last mile" solutions that work in England should be applicable also to communities in Scotland. The lower graph plots the "cumulative density", computed as the quotient, cumulative population/cumulative area. This shows how the gap between the overall figures of 65/km² for Scotland and 377 p/km² for England as a whole arises because the last 10% of the population is much more sparsely distributed in Scotland than in England.
What these graphs do not show is that many of Scotland's communities are widely separated. The "middle mile" connections required to bring broadband to these communities are a particular concern.