The Stourbridge projectMean centre and standard deviational ellipse: the shape of a town in five numbers
A balance point, two axes and a bearing, from 186 centre points.
The ONS publishes a population-weighted centre point for every Census output area, as an easting and a northing in metres. The Stourbridge built-up area has 186 of them, holding 56,961 residents when we add up the areas ourselves. The learner loads the points into Python and takes the average easting and the average northing, counting each area as many times as it has residents. That is the mean centre: 390495 east, 283903 north, or grid reference SO 904 839. Ignoring population and treating every area equally moves it by only 73 metres.
Next comes spread. The program builds a two-by-two covariance matrix, which records how much the points vary east to west, how much north to south, and how far the two go together. Its eigenvectors are the directions of greatest and least spread, and the square roots of its eigenvalues are the spreads themselves. Python's numpy does this in one call. For Stourbridge the long axis of the ellipse is 1,409 metres either side of the centre and the short axis 904 metres, and the long axis points along a bearing of 110.3 degrees from grid north, a little south of due east.
Standard deviational ellipses for five built-up areas, from population-weighted output area centres inside Dudley borough (our Python run on Census 2021 data)| Built-up area | Long and short axis | Long divided by short | Bearing of long axis |
|---|
| Stourbridge | 1,409 m and 904 m | 1.56 | 110.3 degrees |
| Halesowen | 1,733 m and 974 m | 1.78 | 69.3 degrees |
| Sedgley | 1,296 m and 621 m | 2.09 | 4.5 degrees |
| Brierley Hill | 909 m and 769 m | 1.18 | not meaningful |
| Kingswinford | 1,202 m and 1,051 m | 1.14 | not meaningful |
Two lessons hide in that table. First, a direction only means something when the shape is stretched. Kingswinford and Brierley Hill are close to round, so their long axis could swing a long way with a small change in the data, and we decline to report it. Stourbridge is stretched enough to be stable: removing the five areas farthest from the centre changes the axes to 1,370 and 905 metres and the bearing to 110.4 degrees. Second, "one standard deviation" does not hold what most people expect. On a number line, about 68% of bell-shaped data lies within one standard deviation. On a map the matching figure is only about 39%, and Stourbridge's one-standard-deviation ellipse actually contains 28.8% of residents. Enlarging it by the square root of two, as some mapping software does, raises that to 52.0%. Two tools can draw different ovals from the same data and both call them the standard ellipse.
Ages 8 to 11
Find the balance point of coins on a ruler, then of counters on squared paper, by averaging their positions.
Ages 11 to 15
Compute a weighted mean centre for 186 points in Python and plot it on a scatter of the town.
Ages 15 and up
Build the covariance matrix, find its eigenvectors with numpy, draw the ellipse and count who is inside.
Sources and ownership
Resident counts are Office for National Statistics Census 2021 data from Nomis. Centre points and the area lookup come from the ONS Open Geography Portal. Both are used under the Open Government Licence. The centres, ellipses and percentages are calculated by us, and the ellipse is a statistical summary, not a boundary of any kind.
