The North Yorkshire projectDoes commuting obey a law of gravity?
Real census flows between ten areas, a formula from physics, and the places where it fails.
The learner downloads the 2011 Census table WU01UK from Nomis, which counts people by where they lived and where they worked. For our ten areas that gives 90 journeys between different areas: 62,358 people commuted across a boundary between two of them, while 262,282 lived and worked in the same one. For each area the learner totals the commuters leaving and the jobs filled, finds a centre point from ONS population-weighted centroids, and measures the distance between every pair of centres.
Newton's gravity falls with the square of distance. The commuting version is flow = k × (size of origin)ᵃ × (size of destination)ᵇ ÷ distanceᶜ, and taking logarithms turns it into a straight line that ordinary least squares can fit. Fitted to the 89 pairs with at least one commuter, it gives a distance power of 3.3 and explains 77.6 per cent of the variation in the logged flows, an R² of 0.776. Forcing Newton's square law instead drops the fit to 0.677: commuting falls away with distance much faster than gravity does.
Where our fitted model misses by most, 2011 Census commuters| From | To | Actual | Model | What it tells you |
|---|
| York | Hambleton | 2,915 | 552 | Centres 40 km apart; journeys near the edges are much shorter. |
| Selby | York | 5,093 | 5,403 | A near hit, for comparison. |
| Redcar and Cleveland | Middlesbrough | 10,569 | 32,225 | Centres only 11 km apart, and a steep power law overshoots. |
| Middlesbrough | Redcar and Cleveland | 5,111 | 26,499 | Half the flow of the reverse trip, yet predicted almost as high. |
| Harrogate | Leeds | 8,481 | not in model | Leeds lies outside the ten areas, so the model never sees it. |
Each big miss is a lesson in its own right. York to Hambleton flows more than five times what the model expects, and the likely reason is the crude distance: we measured from centre to centre of two large districts, while real journeys start and end all over them. Between Middlesbrough and Redcar and Cleveland the model overshoots in both directions, predicting 32,225 and 26,499 where the census found 10,569 and 5,111: a distance raised to the power 3.3 becomes tiny for centres only 11 km apart, so the formula explodes. It also barely separates the two directions, although one flow is double the other. And the largest flow from Harrogate in the whole table goes somewhere our model does not include: 8,481 people commuted to Leeds, more than four times Harrogate's biggest flow inside the ten areas, 1,920 to Hambleton.
Ages 10 to 13
Put pins in a paper map for five towns, guess the busiest commute between them, then check the real census number.
Ages 13 to 16
Load the flow table into Python, compute distances, and plot logged flow against logged distance to see the straight line appear.
Ages 16 and up
Fit the full model with least squares, compare it with Newton's square law, and write up the five largest residuals with an explanation for each.
Whose numbers these are
The commuting counts are from the 2011 Census, published by the ONS on Nomis, and the centre points from ONS centroids. The fitted formula, its powers, R² values and every model prediction are ours, computed for teaching. Commuting in 2011 may differ from commuting now.
