# Rank Size Rule

Why is it that so many people move to or live in a city? What makes a city's growth slow down once it reaches a certain size? The Rank-Size Rule is a principle that can explain the distribution of some cities. This principle explains a pattern found in many countries on the limits and sizes cities will reach based on their rank and relative population. In this explanation, we will explore more about the rank-size rule.

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## Rank Size Rule Definition

The rank-size rule is a rule about an inverse size to rank, often described as the size of cities in a country. The rank-size rule says that the second-largest city will have half the population as the largest. The third largest city will have one-third the size, and the fourth will be one-quarter the size of the largest, and so on and so forth. In other words, you can estimate a city's population size based on its rank relative to the largest city in the country.

The rank-size rule has been inspired by Zipf's Law, which is a law used in natural and social sciences to reflect inverse proportionality between things relative to their ranks.

## Rank Size Rule Formula

The specific formula for the rank size rule is 1/nth, where n equals the ranking of the size of the city in the country. For example, Los Angeles, California is the second largest city in the United States. Therefore, its ranking would be two, and in the rank-size formula, n would equal two.

If we know what rank a city is in terms of the size of the population compared with other cities in the country, we could then see if the largest city corresponds to the ratio to see if the country follows the rank-size rule. Therefore, city sizes are inversely proportional to their rank.

There are arguments that the rank-size rule is more of a statistical phenomenon than a law or universal concept because the rule is sometimes present but certainly not consistently when looking at the distribution of populations among cities in different countries.

Although we usually talk about cities with the rank-size rule, it can be more widely applicable. The figure below shows the populations of countries that follow a pattern of exponential inverse regression as would be expected according to their rank based on the rank-size rule. China and India are anomalies, but every other country's population follows the expected population closely.

Fig. 1 - Rank-Size distribution

## Rank Size Rule Example

Let’s look at the size of different cities in the United States. New York City is the largest city in the United States, with a population of around 8.5 million people.

Los Angeles is the second-largest city in the United States. In our formula, n = 2, and the formula would be 1/2. We would expect the population of Los Angeles to be roughly half, or 50%, of the population of New York. The population of Los Angeles is 3.8 million people, which is about 44.7% of the population of New York City. This is pretty close to half but still a little off. In this example, we could say that the rank-size rule applies as it still gives a rough estimate.2

$NYC=8.5LA=3.88.5}{2=4.25}4.25=50%ofNYC3.8}{8.5=}0.447×100=44.7%$

Let us see if the trend continues in the United States.

Chicago, the third largest city in the United States, has a population of about 2.7 million people. Following our rank-size formula, n would equal three, so we expect Chicago to have a population of around one-third of 33% of the largest city in the country, New York, at 8.5 million. 2.7 million is about 32% of 8.5 million, almost in line with the rank-size rule.2

Fig. 2 - Chicago skyline

Houston, Texas, is the next most populous city in the United States, with an estimated population of around 2.3 million people. As the fourth-largest city in the United States, we should expect the population of Houston to be one-fourth or 25% of that of New York if it follows the rank-size rule. Houston is approximately 27% the size of New York, again falling close to what the rank size rule would predict.2

Last one: the fifth largest city in the United States is Phoenix, Arizona.

Phoenix has a population of 1.6 million people. By now, you should know that the fifth-largest city in the United States should be around one-fifth or 20% the size of New York. Phoenix is about 19% the size of New York, again following the rank-size rule pretty closely.2

There are some additional considerations to this rule for why it is best to interpret loosely rather than strictly.

There can be some controversy over what constitutes the boundaries of a city. What if we compare not just the cities but the greater metropolitan areas to look at different measurements of city population? The metropolitan area of a city is much larger, including the suburbs and communities in close proximity to the city that have a strong dependent relationship with the city. The metropolitan area population of New York City is about 19.8 million people, more than twice the amount that lives within the actual city limits. The metropolitan area of Los Angeles is roughly 13 million people. The Los Angeles metropolitan area is almost 65% the size of the New York metropolitan area. What can this tell us? Well, the rank-size rule doesn’t apply as much here, but also Los Angeles may define their metropolitan area in a different way than New York. Los Angeles famously doesn’t have a metro system, its downtown is not as large, and its population is overall more spread out across more land. Perhaps this leads to a broader definition of a metropolitan area in Los Angeles than in New York City.

## Rank Size Rule Model

The rank-size rule can tell us a lot about a country. It may show us that a country has a higher level of development and inclusive institutions because power and wealth are fairly spread out compared to other models. Rapid growth, as is happening in many countries in Asia, can make following the rank-size rule difficult as a lot of power and investment is in one city, and not enough time has passed for urbanization and development to spread to the entire country.

The rank-size rule tends to work better in countries that have had large urban populations for many centuries, as this gives a lot of time for urbanization to spread out.

Check out our explanation on the Central Place Theory!

## Rank Size Rule vs Primate city

The rank-size rule describes a descending order of progressively smaller but independently functional cities, whereas a primate city is overwhelmingly the largest city in a country and the center of most industry, power, and societal trends. If a country has just one major primate city, rather than a collection of cities that follow the rank-size rule, it may indicate that the country is less resilient; the primate city could have a detrimental impact on the rest of the country, whereas power and wealth are more spread out in countries following the rank-size rule.

An example of a country with a primate city would be Thailand, as Bangkok is by far the largest metropolitan area, with the next largest urban area being more than 30 times smaller. Primate cities are often a less desirable model than the rank-size rule, as primate cities typically are a reflection or cause of inequality, uneven development, and a lack of equity. The provinces around Bangkok may have as much as 8-10 times higher GDP per capita than many rural provinces in Thailand.4

Fig. 3 - Bangkok skyline

Primate cities tend to be in countries that are developing and experiencing rapid economic growth or countries that have had a large history of inequality and authoritarian rule that have concentrated wealth in the hands of a few, often in the center of political power. However, this is not always the case, and authoritarian countries may follow the rank-size rule as well.

## Rank Size Rule Strengths and Weaknesses

The strengths of the rank-size rule are numerous. Most countries that follow the rank-size rule are overall stronger and more developed countries with a long history of urbanization, more even development, and less inequality. A country will be more resilient and secure with a diversity of large cities as it does not all put a majority of its resources and wealth into a single city.

Some weaknesses may be that there is no unified definition of where exactly a city should end and begin, almost making it possible to adjust city boundaries to fit the rule. Another weakness would be that it is a rough estimation of city sizes, and when dealing with large countries, this can mean that the measurement would be off by several hundred thousand people. Lastly, the rank-size rule is only sometimes applicable to some countries, as many countries have primate cities instead; therefore, it would be inaccurate to assume the size of other cities in any given country just because you know the rank and size of one city.

## Rank Size Rule - Key takeaways

• The rank-size rule is not an exact or universal measurement of population distribution in a country but is a principle that displays a pattern that can be seen in many countries.
• The higher number the rank of a city is, the smaller the population is expected to be.
• The rank-size rule is one of several theories that describe the distribution of populations.
• The rank-size rule is a pattern of proportionality.

## References

1. Fig. 1: Country Population Rank (https://commons.wikimedia.org/wiki/File:Rank_order_countries.png) by Loodong (https://commons.wikimedia.org/wiki/User:Loodog) is licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/deed.en)
2. United States Census Bureau.”City and Town Population Totals: 2020-2021.“ https://www.census.gov/data/tables/time-series/demo/popest/2020s-total-cities-and-towns.html 16, May 2022.
3. Fig. 2: Chicago Skyline (https://commons.wikimedia.org/wiki/File:Chicago_Skyline_Oct_2022_2.jpg) by Sea Cow (https://commons.wikimedia.org/wiki/User:Sea_Cow) is licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
4. Office of the National Economic and Social Development Council. “Gross Regional and Provincial Product.” https://www.nesdc.go.th/ewt_dl_link.php?nid=5628&filename=gross_regional 2018.
5. Fig. 3: Bangkok Skyline (https://commons.wikimedia.org/wiki/File:0008871_-_Krung_Thep_Bridge_008.jpg) by Preecha.MJ (https://commons.wikimedia.org/wiki/User:Preecha.MJ) is licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)

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##### Frequently Asked Questions about Rank Size Rule

What is the rank-size rule?

A principle that says that the rank of a city's population within a country will be approximately the largest city's population divided by the rank of the city in question.

What cities follow the rank-size rule?

Several American cities such as Chicago and Pheonix are good examples of cities that follow the rank-size rule.

Where does the rank-size rule not apply?

In many countries, especially less developed countries, countries that have experienced rapid growth in a short amount of time, and countries that have not had a long history of urbanization cities may not follow the rank-size rule.

How does the US follow the rank-size rule?

Los Angeles, the second largest city is roughly half the population of New York City, the largest city in the US. Chicago is the third largest city with around one-third the population of New York City. Houston, the fourth largest city, has approximately one-fourth the population of New York. This trend continues.

How do you calculate the rank-size rule?

The Rank Size Rule is calculated by by first obtaining the population of the largest city in the country. After that the population rank and overall population of the city in question. Then divide the population of the largest city by the population rank of the city in question to determine approximately what size the city would be if it follows the rank-size rule.

## Test your knowledge with multiple choice flashcards

Would the example follow the rank-size rule or primate city pattern?Largest city population: 1 Million.Second largest city population: 500 Thousand.

Would the following be in line with the rank-size rule?Largest city population: 1 Million.Fourth largest city population: 250 Thousand.

Would the following be in line with the rank-size rule?Largest city population: 1 Million.Tenth largest city population: 100 Thousand.

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