Zipf's Law Calculator

Predict city population by rank using the rank-size distribution rule.

Last reviewed: June 2026
Built and maintained by · Last updated June 2026
Predicted Population (Rank 2)
4,168,409
Largest city / Rank = 8,336,817 / 2

Rank-Size Distribution (Top 10)

Rank Predicted Population % of Largest
Important Disclaimer: This calculator is a simplified educational model. Population presets are examples; replace them with current census, metro-area, or administrative data when accuracy matters.

How the Zipf's Law Calculator works

This calculator applies the city rank-size form of Zipf's Law. Enter the population of the largest city and a target rank, and the calculator estimates the target city's population as P(rank) = P1 / rank. It also builds a top-10 table so you can see the full benchmark distribution from the largest city through rank 10.

The model is intentionally simple. In the classic rank-size rule, the second-largest city is about half the largest, the third-largest is about one-third, the fourth-largest is about one-fourth, and so on. Real countries rarely match that perfectly, but the rule is a useful baseline for spotting unusually dominant primate cities or unusually balanced urban systems.

When the model is useful

  • Urban geography: Compare actual city populations against a quick rank-size expectation.
  • Market sizing: Estimate rough city tiers when only the largest-city population is known.
  • Classroom checks: Demonstrate power-law distributions without a spreadsheet.
  • Data QA: Flag population tables where a rank looks implausibly high or low.

Common mistakes

  • Mixing definitions: Do not compare city proper, urban area, and metro area populations in the same run.
  • Treating the estimate as current census data: The calculator gives a model benchmark, not an official population count.
  • Ignoring primate-city effects: Some countries have one very dominant city, which can make lower ranks much smaller than the rank-size rule predicts.
  • Using stale largest-city values: Update the largest-city input if you are modeling a specific year.

Example

If the largest city has 8,336,817 people, the rank-size rule estimates rank 2 at 4,168,409, rank 3 at 2,778,939, and rank 10 at 833,682. The table is not saying those are official city populations; it is showing the benchmark curve that actual city data can be compared against.

Frequently Asked Questions

It estimates the population of a city at a target rank by dividing the largest city population by that rank, following the simple rank-size rule P_n = P_1 / n.
No. It is a simple power-law approximation. Real city systems are affected by boundaries, migration, economic concentration, administrative definitions, and historical growth patterns.
In the classic Zipf rank-size model, rank 2 is P_1 / 2, rank 3 is P_1 / 3, and so on. The table shows that simplified benchmark for ranks 1 through 10.
Yes. Choose Custom and enter the largest city or metro population you want to model. Use the same city-definition method for all comparisons.
It is useful for quick geography, urban planning, market-sizing, and classroom checks where you want a benchmark before comparing against real census or metro-area data.