The rank-size rule is a pattern in geography and economics stating that the second-largest city in a country will be about half the population of the largest, the third-largest will be about one-third, the fourth about one-quarter, and so on. In formula terms, a city’s expected population equals the largest city’s population divided by that city’s rank. The pattern is surprisingly consistent across many countries and time periods, making it one of the most reliable regularities in the social sciences.
How the Formula Works
The basic formula is: Population of city n = Population of the largest city รท n, where n is the city’s rank. So if the largest city has 8 million people, the rule predicts the second city will have about 4 million, the third about 2.67 million, and the tenth about 800,000.
You can test this with real U.S. Census data from 2024. New York, the largest city, has about 8.48 million people. The rule predicts Los Angeles (rank 2) should have roughly 4.24 million. Its actual population is 3.88 million, which is reasonably close. Chicago (rank 3) should be about 2.83 million; it’s actually 2.72 million. Houston (rank 4) should be about 2.12 million; it’s 2.39 million. By rank 10, Jacksonville should be about 848,000; its actual population is 1.01 million. The fit isn’t perfect, but the general pattern holds remarkably well.
Researchers express how closely a country follows this rule using an exponent (often written as alpha). When alpha equals exactly 1, the country follows the rank-size rule perfectly. The U.S. comes very close, with an estimated alpha of 1.005 in one study of urban zones. Germany’s urban zones produce an alpha of about 0.93, still in the ballpark. When alpha deviates significantly from 1, the distribution tells a different story about how population concentrates in that country.
Who Discovered It
The rule is most commonly associated with George Zipf, the Harvard linguist who popularized it in the 1940s, which is why it’s often called Zipf’s law. But the underlying pattern was noticed much earlier. Felix Auerbach, a German physicist, first described power-law city size distributions in 1913 in a paper on population concentration. Alfred Lotka, a mathematician better known for his work in ecology, was the first to describe the power-law rank-size relationship in its modern analytical form in 1925. Zipf brought the idea to a much wider audience, and his name stuck.
Why the Pattern Exists
The most widely cited explanation traces back to a principle called proportionate growth: cities of all sizes tend to grow at roughly the same percentage rate over time. A city of 5 million and a city of 50,000 both grow by, say, 2% in a given period, but the absolute numbers added are vastly different. Over decades and centuries, this process of random proportionate growth naturally produces the steep, curved distribution the rank-size rule describes.
The driving forces behind this are random fluctuations in local economic productivity and the movement of workers toward opportunity. When a city’s local economy gets a productivity boost, workers migrate there, increasing its size. When productivity dips, growth stalls or people leave. Because these shocks are random and roughly proportionate to city size, the end result is a distribution where a few cities are very large and many cities are small, with a smooth curve connecting them.
What’s striking is that this pattern appears across countries with very different economic structures and histories. Researchers have noted that the lack of a truly sufficient explanation for why it works so universally remains one of the open questions in urban economics.
When Countries Don’t Follow the Rule
Not every country fits. Some are dominated by a single, oversized capital or economic center that dwarfs every other city. Geographer Mark Jefferson described this phenomenon in 1939 as the “law of the primate city.” A primate city is disproportionately large compared to what the rank-size rule would predict. Jefferson pointed to London, which at the time was more than seven times the size of Britain’s second city, Liverpool.
Modern examples include Bangkok in Thailand, Seoul in South Korea, and Buenos Aires in Argentina. In these countries, the largest city concentrates political power, economic activity, and infrastructure to such a degree that no other city comes close. The gap between rank 1 and rank 2 is far larger than the rule would predict.
Researchers have found that the slope of the rank-size distribution tends to vary in keeping with economic development. Countries with more integrated, diversified economies and multiple strong regional centers tend to follow the rule more closely. Countries where investment, transportation networks, and government functions concentrate in a single hub tend to show primate city patterns instead. Colonial history plays a role too: many former colonies developed a single dominant port or administrative capital that continued to attract disproportionate growth after independence.
The Role of Geography and Proximity
One question researchers have explored is whether the physical distance between cities affects how well the rank-size rule holds. In the U.S., there is a statistically significant spatial effect, but it’s very small. Germany shows a similar pattern, with a minor distance effect that slightly adjusts the distribution. In the United Kingdom, distance effects are insignificant entirely. Slovenia, a much smaller and younger country, shows mixed results depending on whether you measure official municipalities or functional urban areas identified through satellite imagery. In that case, accounting for spatial spillovers between nearby urban areas actually made the fit worse, pushing the distribution further from the expected pattern.
The takeaway is that proximity between cities can matter, but its influence on the overall rank-size distribution is modest in most countries that have been studied.
Why It Matters in Practice
The rank-size rule isn’t just a curiosity. Urban planners and policymakers use it as a benchmark to understand whether a country’s urban system is balanced or top-heavy. A strong primate city pattern can signal that infrastructure, jobs, and services are overly concentrated, which creates pressure on housing, transportation, and public services in the dominant city while smaller cities struggle to attract investment.
For economic development, a rank-size distribution closer to the rule suggests that multiple cities serve as regional centers, spreading economic opportunity more evenly across a country. When a nation’s urban hierarchy deviates sharply, it can prompt policy discussions about decentralization, regional investment, and whether secondary cities need more support to grow into their potential roles.
The rule also has applications well beyond cities. Zipf’s law, the broader version of the same mathematical relationship, appears in the frequency of words in a language, the size of firms in an economy, the distribution of wealth, and even the number of links to websites. The rank-size rule for cities is just the most visible example of a pattern that seems to emerge wherever growth is proportionate and competition is open-ended.