Zipf’s Law, City Size and Walking Speed

by | May 8, 2018


The linguist George Zipf discovered something very interesting in 1949: a small number of words are used all the time and most words are used relatively rarely. Further, he found that if words are ranked by their frequency of use the top used word in a language is used about twice the second word, three times the third used word.* And so on. This is Zipf’s law and is also referred to as a rank vs. frequency rule and is a type of power law. Related IFODs: Exponential Thinking Pt. 1 and Exponential Thinking Pt. 2

Zipf’s law also works to describe income distributions in a country: the person who makes the most money makes about 2x the number two income earner and so on. It has been found to apply to the income distributions of companies, network effects on the internet, gene expression and many other situations.

Interestingly, Zipf’s law also applies with respect to large cities with the most populous city in a country being about twice the second ranked city and three times the third ranked city. Even with chaotic city growth in some countries, the Zipf power law relationship has held generally true for the cities in all countries for about the past 100 years. From Gizmodo: “In the 2010 census, the biggest city in the U.S., New York, had a population of 8,175,133. Los Angeles, ranked number 2, had a population of 3,792,621. And the cities in the next three ranks, Chicago, Houston and Philadelphia, clock in at 2,695,598, 2,100,263 and 1,526,006 respectively. You can see that obviously the numbers aren’t exact, but looked at statistically, they are remarkably consistent with Zipf’s predictions.”

Switching gears a bit, there are a number of economic and social factors that are closely correlated with the size of cities. In the paper Growth, innovation, scaling, and the pace of life in cities in the journal PNAS researchers explored “scaling relations for cities . . . [among] a wide range of characteristics, including energy consumption, economic activity, demographics, infrastructure, innovation, employment, and patterns of human behavior.”

The researchers found that many things follow a power law with respect to city size, including the wealth of billionaires, new patent applications, “supercreative” employment, wages, and the length of electrical cables. Interestingly,  the pace of urban life increases with size  as innovation, wealth creation and even the walking speed of resident pedestrians follow power law scaling with city size. This may explain why residents of big cities often walk much faster than the tourists visiting that city.


Log chart of walking speed and population


Log chart of total wages and city population

*Here’s a link to the top 1,000 most frequently used words (English): Top 1000 words

Finally, Zipf’s Law is very similar to the Pareto Principle – the topic of a future IFOD.


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