Does Benford's law hold for the populations of the world's towns?

Why it makes a good exploration

In many real datasets, 1 is the leading digit about 30% of the time — Benford's law. It is used to detect fraud. Testing it, and finding when it fails, is a surprising and rigorous statistics IA.

The mathematics you'll need

• Logarithms: P(d) = log₁₀(1 + 1/d)
• Chi-squared goodness-of-fit test
• Understanding why scale-invariance leads to Benford's law
• Comparing datasets with different ranges

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use country and city population tables from the UN or World Bank, and a second dataset with a narrow range (e.g., adult heights) for contrast.

• UN World Population Prospects 2024 — Official UN population estimates 1950–present and projections to 2100, CSV bulk download.
• World Bank Open Data — Development indicators (GDP, life expectancy, mortality, internet use) for every country, usually from 1960.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

1. Explain Benford's law and where the formula comes from.
2. Extract leading digits for each dataset.
3. Test the fit with chi-squared.
4. Compare datasets spanning many or few orders of magnitude.
5. Reflect on why fraud detectors use it and its limits.

Showing personal engagement

• Test it on data from your own life (bank statement amounts, a book's page references).
• Predict which datasets will fail and why.
• Explore the second-digit law.

See Criterion C: personal engagement for what examiners look for.

Taking it further

HL: show that a scale-invariant distribution of log₁₀x leads to the Benford probabilities.