Mayotte is a collection of islands about 250 miles north of Madagascar and the same distance East of Mozambique. It is also a départment of France and an NUTS2 region of Europe. It has just under 300,000 residents. Its economy is stronger than its neighbours thanks to French government spending, high-value agriculture, and tourism, but it is extremely weak by European standards. It is the weakest regional economy in France.
Île-de-France, home to Greater Paris and its suburbs, is the EU's largest city economy and along with Greater London it is one of Europe's two undisputed world cities. With over 12 million residents, its economic strengths cover almost all sectors and it is one of the most productive cities in the world. It is the strongest regional economy in France.
If we draw Mayotte, Île-de-France, and all the other NUTS2 regions of France on a graph of GDP/resident it looks like this.
One way of measuring regional inequality within a country is to measure the range. The difference between Mayotte and Île-de-France's GDP/resident is quick to calculate and easy to understand. The Economist most infamously did this in 2018. Many others have done the same. It has value, but we can do better if we have more time. The biggest problem with the range is that it only considers two regions of a country, throwing away a huge amount of information and leaving it at the mercy of outlying datapoints.
The variance is an improvement on the range. It considers every region, which is good. But things get worse quickly, especially if it doesn't consider the population of each region. In this case the mean which is used to calculate the variance is not the GDP/resident of the whole country. Also, since the variance uses the square of the difference between this (wrong) mean and the GDP/resident of each region it assigns enormous weight to outliers. So in calculating the variance of regional GDP/resident for France almost all the contributions once again come from Mayotte and Île-de-France. Île-de-France's contribution is exaggerated still further by the misplaced mean and Mayotte's contribution is vastly exaggerated since its tiny population is not taken into account. Using the coefficient of variance (dividing by the mean) does nothing to fix this. Using the standard deviation (square root of the variance) does little to improve things.
My understanding is that this approach is what The Resolution Foundation used to measure regional inequality in their Mapping Gaps report. I think we can do better.
Dispersion is the measure of regional inequality preferred by Eurostat. I like it too. It considers the population of regions throughout, measures the distance from the GDP/resident of each region from the national GDP/resident, and does not square the distance.This is a good, though far from perfect, representation of how dispersion is calculated. The area of the rectangles is summed.
As a measure of regional inequality, dispersion is better than range or variance at dealing with outliers and taking different sized geographies into account. But this doesn't mean that we can expect a sensible number to come out of the formula if we use non-sensible data.
A mistake that has caught me and many others out when thinking about regional inequality is to use GDP/resident data for regions with significant numbers of inward commuters. The most extreme example of this is the NUTS2 region of the UK called "Inner London - West". This region has a population of 1.2 million, similar to South Yorkshire. But unlike South Yorkshire whose workforce is largely self-contained, Inner London - West has a very large number of inward commuters working at highly productive and profitable firms, especially in the City of London.Inner London - West is not a meaningful geography for understanding regional inequality within the UK. Furthermore, since Spain (Madrid and Barcelona) and France (Paris) do not split their large cities in a similar way this makes the UK look considerably more regionally unequal in comparison with them than it really is. By replacing London's five NUTS2 regions with its NUTS1 region we solve most of this problem. I call this new geography UK*.
Dispersion of GDP/resident is my preferred single measure of regional inequality. But others have good reason for preferring dispersion of income instead. Dispersion of GDP/resident shows the extent to which different parts of a country can produce the wealth needed to pay for the standard of living they enjoy. Dispersion of income shows the extent to which different parts of a country enjoy the same standard of living. Both are valuable.
You probably want a graph showing that the UK is the most regionally unequal country in Europe right here. And about 500 words of my opinions on what it shows. I've written this piece so that I can write that one.