Applying Zipf's Law

 I asked Gemini the following:

Can you compile a table of the top ten Indonesian cities and their populations ranked from highest to lowest. Use this table to determine how closely populations and rank conform to Zipf’s Law.

Here was its response:

Rank	City	Actual  Population	Zipf's Law Predicted Population	Difference
1	Jakarta	10,684,946	10,684,946	0
2	Surabaya	2,921,996	5,342,473	-2,420,477
3	Bekasi	2,644,058	3,561,649	-917,591
4	Bandung	2,528,160	2,671,236	-143,076
5	Medan	2,494,512	2,136,989	+357,523
6	Depok	2,163,635	1,780,824	+382,811
7	Tangerang	1,963,970	1,526,421	+437,549
8	Palembang	1,718,440	1,335,618	+382,822
9	Semarang	1,702,379	1,187,216	+515,163
10	Makassar	1,474,393	1,068,495	+405,898

Population data reflects mid-2024/2025 official estimates.

Zipf's Law posits that the population of the n-th ranked city in a given country should be roughly equal to the population of the largest city divided by n. When applied to Indonesia's top ten cities, the distribution reveals a notable divergence from the theoretical model:

The Primate City Effect 

Jakarta acts as a classic "primate city," meaning its population is disproportionately massive compared to the rest of the urban hierarchy. Because Jakarta is so large, Zipf's Law predicts that the second-largest city (Surabaya) should have over 5.3 million residents. In reality, Surabaya falls nearly 2.4 million people short of this mark, creating a sheer drop-off at the top of the distribution rather than a smooth proportional curve.

Mid-Rank Convergence 

The urban distribution briefly aligns with Zipf's Law around the fourth rank. Bandung's actual population of 2.52 million closely brushes against the model's prediction of 2.67 million. This is the only point in the top ten where the theoretical distribution and actual settlement sizes tightly converge.

Overperformance in the Lower Ranks 

From the fifth rank (Medan) downward, Indonesian cities consistently exceed the population sizes predicted by the model. Rather than trailing off rapidly as the fractions get smaller, cities ranked six through ten maintain robust populations between 1.4 and 2.1 million, resulting in a "fat tail" on the data curve.

The Metropolitan Distortion 

A fundamental geographical factor skews this list: three of the top ten cities—Bekasi, Depok, and Tangerang—are not independent regional hubs but rather satellite cities directly bordering Jakarta within the greater Jabodetabek metropolitan area. Their massive growth inflates the middle ranks of the hierarchy, concentrating immense population density into a single contiguous megalopolis rather than distributing it evenly across the archipelago.

Ultimately, Indonesia's urban population distribution does not closely conform to Zipf's Law. The sheer dominance of Jakarta creates an initial deficit for the secondary cities, while the massive sprawl of Jakarta's satellite cities props up the lower ranks, resulting in an uneven urban hierarchy shaped more by metropolitan gravity than strict mathematical proportionality.

I then asked:

Can you repeat this analysis for Australian cities.

Here was the response:

Rank	City (Significant Urban Area)	Actual Population	Zipf's Law Predicted Population	Difference
1	Melbourne	5,103,528	5,103,528	0
2	Sydney	5,041,275	2,551,764	+2,489,511
3	Brisbane	2,622,585	1,701,176	+921,409
4	Perth	2,289,366	1,275,882	+1,013,484
5	Adelaide	1,426,803	1,020,705	+406,098
6	Gold Coast–Tweed Heads	735,213	850,588	-115,375
7	Newcastle–Maitland	526,515	729,075	-202,560
8	Canberra–Queanbeyan	503,402	637,941	-134,539
9	Sunshine Coast	407,859	567,058	-159,199
10	Central Coast	348,435	510,352	-161,917

Population data reflects Australian Bureau of Statistics (ABS) 2023 Estimated Resident Population for Significant Urban Areas.

When applied to Australia's top ten cities, the distribution reveals a stark contrast to both the theoretical Zipf's Law model and the primate city effect seen in Indonesia.

The Twin-Primate Anomaly 

Australia exhibits a bimodal or "twin-primate" distribution at the top of its urban hierarchy. Melbourne and Sydney are virtually identical in size, functioning as a duopoly. Zipf's Law dictates that the second-ranked city should be half the size of the first. Because Sydney is nearly 99% the size of Melbourne, it overshoots the model's prediction by nearly 2.5 million residents, thoroughly breaking the curve at the very first step.

The State Capital Inflation 

Because Melbourne sets a relatively "low" mathematical baseline compared to the combined dominance of Australia's two largest cities, the next tier of state capitals massively overperforms. Brisbane, Perth, and Adelaide operate as vast, isolated economic hubs for enormous geographical states. Without nearby rival cities to siphon off growth, they concentrate their state's populations, exceeding Zipf's predictions by 400,000 to over 1 million residents each.

The Steep Regional Drop-Off 

The distribution flips entirely at the sixth rank. From the Gold Coast down to the Central Coast, every city falls significantly short of the model's predictions. Unlike the state capitals, these cities are secondary regional centers lacking dominant political and economic gravity. Rather than a smooth curve, Australia's hierarchy demonstrates a "missing middle"—a sheer drop of almost 700,000 people between fifth-ranked Adelaide and sixth-ranked Gold Coast.

Corridor Megalopolises 

Geographical proximity heavily skews the lower half of the list. Six of the top ten cities are concentrated within just two massive coastal conurbations: the Sydney-Newcastle-Central Coast web in New South Wales, and the Brisbane-Gold Coast-Sunshine Coast corridor in Queensland. This highlights that Australia's lower-ranked top-ten cities are largely sprawling coastal satellites anchored to the major state capitals, rather than independent urban centers distributed evenly across the continent.

Australia's urban geography rejects strict mathematical proportionality, favoring a highly concentrated model defined by two mega-cities, a few hyper-dominant state capitals, and a steep drop-off into coastal satellite cities.

I then asked:

Clearly neither Indonesia nor Australia has cities that conform to Zipf’s Law. Is there a country that does conform closely to Zipf’s Law?

Here was the response:

Rank	City	Actual Population	Zipf's Law Predicted Population	Difference
1	New York	8,546,038	8,546,038	0
2	Los Angeles	3,863,148	4,273,019	-409,871
3	Chicago	2,727,758	2,848,679	-120,921
4	Houston	2,419,191	2,136,510	+282,681
5	Phoenix	1,677,947	1,709,208	-31,261
6	San Antonio	1,572,360	1,424,340	+148,020
7	Philadelphia	1,567,280	1,220,863	+346,417
8	San Diego	1,410,522	1,068,255	+342,267
9	Dallas	1,336,440	949,560	+386,880
10	Fort Worth	1,049,355	854,604	+194,751

Population data reflects 2026 demographic estimates for city limits.

Top-Tier Convergence 

Unlike the massive primate city distributions or bimodal anomalies seen in other regions, the top of this urban hierarchy aligns tightly with mathematical expectations. Chicago falls short of the prediction by a mere 4%, and Phoenix sits a fraction of a percent away from its exact mathematical target.

The Sunbelt Expansion 

The most notable deviations occur in the lower half of the top ten, driven by the rapid modern expansion of cities in Texas and the Sunbelt. Houston, San Antonio, Dallas, and Fort Worth all overperform their theoretical predictions. This reflects a contemporary migration pattern where expansive, less dense southern cities have grown rapidly, creating a slight bulge in the mid-tier ranks.

Absence of a Primate City 

New York is undeniably the largest settlement, but it does not exert the suffocating gravitational pull of a primate city. Its size is balanced by a robust network of secondary and tertiary economic hubs distributed across a massive continental landmass, allowing cities like Los Angeles and Chicago to scale naturally according to the law's proportions.

The urban geography of the United States conforms remarkably closely to Zipf's Law, serving as the classic mathematical model for proportional urban population distributions.

Top 20 Largest US Metropolitan Areas 2023 This video visualizes the population scale and rankings of the largest urban centers in the United States, providing a clear illustration of how the demographic distribution roughly follows the predicted curve.

Another Mid-Millenium Number

On Saturday 23rd of October 2021, I made a post titled Counting People with Mid-Millennium Numbers in which I examined my diurnal age of 26500 on that date. Now, almost three years later, I've reached another "milestone": 27500.

As I said in that post, mid-millennial numbers are popular rounding numbers for populations of towns and islands or communities with common interests or characteristics. These are to be preferred to millennium numbers such as 27,000 or 28000 that appear a little too "approximate" (having only two significant figures as opposed to the three of 27,500). Such numbers are also popular with a wide variety of quantities such as tonnage, money and distances.

The 30 divisors of 27500 are 1, 2, 4, 5, 10, 11, 20, 22, 25, 44, 50, 55, 100, 110, 125, 220, 250, 275, 500, 550, 625, 1100, 1250, 1375, 2500, 2750, 5500, 6875, 13750 and 27500. It's special in the sense that all the digits (apart from 0) are prime. This won't occur again until 30500. As with the previous mid-millennial number, the number appears frequently in population statistics. Here are some examples:

• Published today, the NHS Workforce Race Equality Standard shows Black and minority ethnic (BME) staff make up almost a quarter of the workforce overall (24.2% or 383,706 staff) – an increase of 27,500 people since 2021 (22.4% of staff).
  
  
• The Portuguese Grand Prix of Formula 1, which will be played between Friday and Sunday, will have a maximum capacity of 27,500 spectators, according to the government dispatch published today, October 21, in Diário da República.
  
  
• More than 27,500 people in Gaza have already been killed over the past four months, according to Gaza’s Ministry of Health. Further fighting in Rafah risks claiming the lives of even more people. It also risks further hampering a humanitarian operation already limited by insecurity, damaged infrastructure and access restrictions.
  
  
• The anonymous online study, ‘The Global Brain Health Survey’, involved more than 27,500 people worldwide and was led by the Norwegian Institute of Public Health in collaboration with the University of Oslo. 
  
  
• Kirchberg’s resident population is expected to grow almost six-fold over the next 20 years, according to projections from the Fonds Kirchberg.The body responsible for coordinating the development of one of the capital’s business districts forecasts that there will be 23,700 people living in the area by 2040, up from 4,000 in 2020. It suggests that the district’s maximum capacity would be capped at 27,500 beyond 2040.
  
  
• The Kenyan government says it has set up more than 100 camps to house over 27,500 people displaced by flooding.According to government data, more than 190,000 people have so far been affected by the floods and at least 210 are known to have died.
  

The aliquot sequence for 27500 has a length of 206 steps which are:

[27500, 38104, 40016, 40708, 30538, 15272, 14968, 13112, 13888, 18624, 31160, 44440, 65720, 89800, 119450, 102820, 119444, 105760, 144476, 121804, 97380, 198552, 297888, 518592, 909904, 998456, 889384, 795416, 774784, 768986, 444454, 261146, 141274, 100934, 52186, 27194, 13600, 21554, 13306, 6656, 7666, 3836, 3892, 3948, 6804, 13580, 19348, 19404, 42840, 125640, 283860, 633420, 1562004, 2535180, 5206260, 9371436, 12495276, 20190804, 26921100, 55087540, 60803732, 56587948, 45117684, 69280236, 116780184, 208518216, 312777384, 469166136, 772745304, 1187955816, 1781933784, 2716157736, 4851795384, 8337024936, 14614443864, 27722614536, 48023931924, 82013245164, 134692975476, 205780934846, 107492498242, 53746249124, 42348933724, 35852825156, 27118586620, 29830445324, 24116652916, 18087489694, 10639699874, 5379726934, 2706299954, 1471210894, 735605450, 632620780, 816656420, 898322104, 809756216, 781610824, 685415096, 602212744, 711287846, 355643926, 246590714, 123608986, 61804496, 72770224, 68943920, 91350880, 128146160, 170324320, 242956688, 264953512, 248965388, 248965444, 290225852, 310243108, 343261436, 345226084, 363478556, 363478612, 383892908, 438979156, 520540076, 520540132, 539131250, 616169806, 498886994, 249443500, 314159540, 346712980, 406492340, 486797260, 537866996, 403400254, 201700130, 166294750, 145009490, 131361862, 68478170, 54782554, 27444794, 17643046, 8821526, 6384874, 3696566, 1888594, 944300, 1555540, 2282924, 2282980, 3442460, 4965604, 5062876, 6042092, 6693988, 8128904, 9877396, 8355308, 7779412, 5834566, 2942234, 1471120, 2600048, 3337072, 3287504, 3661456, 3432646, 2557142, 1826554, 1027814, 519394, 259700, 408226, 345758, 246994, 164846, 111634, 55820, 61444, 46090, 44630, 35722, 19034, 10534, 6026, 3478, 1994, 1000, 1340, 1516, 1144, 1376, 1396, 1054, 674, 340, 416, 466, 236, 184, 176, 196, 203, 37, 1, 0]

With a logarithmic vertical axis, the trajectory has a pleasing mountain-like appearance. See Figure 1.

Figure 1

The Collatz trajectory of 27500 has 90 steps and, with a logarithmic vertical axis, its trajectory has a noticeably jagged appearance (see Figure 2):

[27500, 13750, 6875, 20626, 10313, 30940, 15470, 7735, 23206, 11603, 34810, 17405, 52216, 26108, 13054, 6527, 19582, 9791, 29374, 14687, 44062, 22031, 66094, 33047, 99142, 49571, 148714, 74357, 223072, 111536, 55768, 27884, 13942, 6971, 20914, 10457, 31372, 15686, 7843, 23530, 11765, 35296, 17648, 8824, 4412, 2206, 1103, 3310, 1655, 4966, 2483, 7450, 3725, 11176, 5588, 2794, 1397, 4192, 2096, 1048, 524, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

Figure 2

Here are some other interesting facts about the number:

• The Anti-Divisors of 27500 are [3, 7, 8, 9, 21, 27, 40, 63, 81, 88, 97, 189, 200, 291, 440, 567, 679, 873, 1000, 2037, 2200, 2619, 5000, 6111, 7857, 11000, 18333]
  
  
• The Arithmetic Derivative of 27500 is 52000
  
  
• The Maximum - Minimum Recursive Algorithm for 27500 produces [27500, 74943, 62964, 71973, 83952, 74943]
  
  
• The Minimal Goldbach decomposition of 27500 is 13 and 27487
  
  
• The number of steps required is to reach home prime is 7:
  [27500, 22555511, 1110511951, 3313355021, 31337105733, 3373163729137, 4936768328311, 101312973757451]
  
  
• 27500 has Odds and Evens Trajectory of length 12 and is [27500, 27510, 27521, 27530, 27543, 27552, 27565, 27574, 27587, 27596, 27609, 27617, 27624, 27617]

For the previous mid-millennial day see Counting People with Mid-Millennium Numbers.

Looking Back at History

Before I started this Mathematics blog in the second half of 2015, I had started making some mathematical posts in my Pedagogical Posturing blog. I only remembered this when I was looking back over posts that I had made to that blog which I had begun in March of 2009. The first mathematical post I made was on August 5th of 2013 and was titled Reflections on 23500.

The main point that emerged from this post was that, while rounding off to the nearest thousand is common enough, it's also the case that rounding off to the nearest multiple of 500 is also popular, especially when the numbers involved are not large. This is why the number popped up in reference to the populations of various towns and cities, both currently and historically.

The other interesting fact about this number was that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source). Thus:$$\text{radius of Earth }: \text{ radius of Earth's orbit }: : 1:23500$$Of course, this again is an approximation or rounding off of the more accurate figure of 23,460 earth radii or \(1.496 \times 10^{11}\) metres (termed an Astronomical Unit or AU). 

Because of my reliance on the OEIS, I often look at a number in terms of its membership in various sequences but many numbers, like 23500, have a standalone physical significance. Here is the full post made almost 11 years ago now:

Today I'm 23500 days old and I was looking around to see if there was any significance to this number. It's factors are unremarkable, \(2^2 \times 5^3 \times 47 \), but the fact that it's halfway between 23000 and 24000 means that it pops up quite frequently in Internet searches, as would 22500 or 24500 I would imagine. A search reveals that the approximate population of Boston in 1620 was 23500 and there are several towns around the world that are listed as having this population currently e.g. Bishopbriggs in Scotland.

Bishopbriggs grew from a small rural village on the old road from Glasgow to Kirkintilloch and Stirling during the 19th century, eventually growing to incorporate the adjacent villages of Auchinairn, Cadder, Jellyhill and Mavis Valley. It currently has a population of approximately 23,500 people.

Source: http://en.wikipedia.org/wiki/Bishopbriggs

It turns out that Mount Isa has the same population (source):

Mount Isa is located just 200 kilometres from the Northern Territory border and 1,829 kilometres from Queensland’s capital, Brisbane. The nearest major city, Townsville, can be found 883 kilometres from The Isa. Mount Isa covers an area of over 43,310 square kilometres, making it geographically the second largest city in Australia to Kalgoorlie-Boulder, Western Australia ... With a population of approximately 23,500, Mount Isa is a major service centre for north-west Queensland.

Many other examples of towns having populations of about 23500 could be quoted. In addition of populations, the number sometimes comes up as a dollar figure (source):

In its third annual funding cycle, the Black Philanthropy Initiative has pumped $23,500 back into the Winston-Salem area to help African Americans improve their parenting skills.

Interestingly, it turns out that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source).

The sun is far enough away (about 23,500 earth radii) that it took a long time before people knew accurately how far away the sun was. Certainly the ancient Greeks had calculated the distance, but they also knew that their results could be off. 

Many countries in the world have five digit postal codes or zip codes as they are sometimes known. These codes identify particular locations within the country e.g. Muang Prachinburi, Prachinburi, Thailand has a postcode of 23500. The United States uses a five digit system but apparently there is no location corresponding to 23500, although there is for 23499 and 23501.

At the time I made this post, I hadn't discovered the OEIS although I was aware of it in late 2014 or early 2015. Nor had I discovered Numbers Aplenty and several other resources that have proved most helpful in finding out more about numbers. 

Counting People with Mid-Millennium Numbers

Yesterday marked a milestone of sorts. I reached a diurnal age of 26500, a somewhat unremarkable number commemorated with the tweet shown in Figure 1.

Figure 1

I appended two screenshots to the tweet, one from the OEIS (Figure 2) and the other from Numbers Aplenty (Figure 3):

Figure 2

Figure 3

For some reason, I was prompted to think about what the representation of 26500 might be in Roman numerals. It turns out to be as shown in Figure 4.

Figure 4

So what has any of this to do with counting people? Well, nothing really but as I was investigating the number's properties, it occurred to me that this would be a popular rounding number for populations of towns and islands or communities with common interests or characteristics. This can be confirmed from the following data:

• Tick-borne encephalitis, TBE, has been observed in Aland Islands (population 26,500) for more than 60 years (source). Interestingly:

The self-governing province of the Åland Islands lies off the southwest coast of Finland. Åland is an autonomous, demilitarised, Swedish-speaking region of Finland. Åland consists of more than 6,700 islands, but the current population of over 30,000 live on only 60 islands. Source.

• The most recent mid-year population estimates (2020) for the Outer Hebrides gives a population of 26,500. This shows a decrease of 0.8% (220 persons) from mid 2019 to mid 2020. Source. 
  
  
• Unemployment in the first quarter of 2020: Further rise in Gaza’s unemployment rate; 26,500 people lost their jobs even before the pandemic. Source.
  
  
• CDC estimates Salmonella bacteria cause about 1.35 million infections, 26,500 hospitalisations, and 420 deaths in the United States every year. Food is the source for most of these illnesses. Source.
  
  
• On 30 June 2020, the population of Na h-Eileanan Siar was 26,500. This is a decrease of 0.8% from 26,720 in 2019. Over the same period, the population of Scotland increased by 0.0%. Source.

• The Aromanians in Romania are a non-recognised ethnic minority in Romania that numbered around 26,500 people in 2006. Source.
  
  
• At June 2011, there were 1.01 million children under 15 years of age in Victoria, with boys outnumbering girls by 26,500. This age group comprised 18% of the total Victorian population, down from 20% in June 2001. Source.
  
  
• Published estimates of the risk of vaccine-induced thrombotic thrombocytopenia (VITT) from countries with moderate to high data quality range from 1 case per 26,500 to 1 case per 127,300 first doses of AstraZeneca/COVISHIELD administered. Source.
  
  
• The number of dwellings in Camden Council is forecast to grow from 26,500 in 2016 to 79,631 in 2036, with the average household size falling from 3.08 to 3.01 by 2036. Source.
  
  
• Welcome to the official website of Nicholas County -  a safe, rural community located in the heart of central WV.  County Seat - Summersville; Population: 26,500; Home of WV's largest lake  in the county. Source.
  
  
• February 11, 2021 — The number of vaccine doses available for New York City each week received a 26,500-dose boost through a federal program that provides retail pharmacies with the COVID-19 vaccine. Source.
  
  

The list goes on indefinitely. Of course, the same would hold true of other mid-millennium numbers like 25500 and 27500. For example:

The increase in the number of students inside Midland ISD compared to the start of the previous school year will top 1,000. Superintendent Orlando Riddick said Tuesday the district’s student population appears to not only have eclipsed 25,000 but will end up closer to 25,500. September 2017. Source.

So I've coined this special term, mid-millennium numbers, to describe a class of numbers that are commonly used when quantifying data that cannot be precisely determined. These are to be preferred to millennium numbers such as 26,000 which appear a little too "approximate" (having only two significant figures as opposed to the three of 26,500).

For the next mid-millennium number (27500) see Another Mid-Millennium Number.