On a slide rule the numbers are spaced by their logarithms, so the band that starts with a 1 takes up log₁₀2 = 30.1 % of the strip while the 9 band takes only 4.6 %. A quantity that drifts smoothly across the strip lands in the wide band most often. Hover a bar in the chart to light up its band.
Where's the maths? This is statistics — data handling shaking hands with logarithms. Take a big pile of real measurements and look only at the first digit of each one. You would guess that 1, 2, 3, … 9 turn up about equally often, one time in nine, but for real-world data they do not: about 30 % of the values start with a 1 and only about 5 % start with a 9. The gold bars on screen are the actual counts from the dataset you picked, drawn as a bar graph; the blue ghosts behind them are what Benford's law predicts. The reason is hidden in the ruler panel: on a logarithmic scale the room given to numbers beginning with 1 really is about six times the room given to numbers beginning with 9. So the lopsided bar graph is not a mistake in the data — it is what "spread evenly across the orders of magnitude" looks like once you read off only the leading digit.
Beyond the syllabus: the log-based explanation of Benford's law is beyond O-Level.
Write any positive number in standard form, A × 10ⁿ with 1 ≤ A < 10. The leading digit is just the whole-number part of A. Benford's law says that for many natural datasets
P(d) = log₁₀(1 + 1/d)giving 30.1 %, 17.6 %, 12.5 %, 9.7 %, 7.9 %, 6.7 %, 5.8 %, 5.1 % and 4.6 % for d = 1 to 9. The nine probabilities add to log₁₀10 = 1, because the log laws collapse the sum: log₁₀2 + log₁₀(3/2) + … + log₁₀(10/9) = log₁₀10.
Press Multiply everything by 3. Every single value changes, yet the bar graph keeps its shape: roughly 30 % still start with a 1 and the fit stays good. Multiplying by 3 slides every number along the log ruler by the same distance log₁₀3, and a pattern that is even along the ruler is still even after a slide. That is why a rule about first digits can survive changing kilometres to miles. Individual bars do jump about a little — with only about 200 values, luck alone moves each bar by a couple of percentage points, which is the second number in the panel. The uniform random control is not spread evenly along the ruler: it sits near 11.1 % for every digit and then changes shape completely when you multiply it by 3, which is exactly what scale invariance looks like when it fails.
Auditors use this. When people invent numbers — fake expense claims, invented invoices, doctored election tallies — they spread the first digits too evenly, or crowd them just under a reporting threshold, and the bar graph stops matching log₁₀(1 + 1/d). The mismatch is measured by the MAD shown in the panel: the average gap between the observed percentage and the predicted percentage across the nine digits. It is only meaningful next to the second number, the gap you would expect from luck alone with this many values — the uniform control misses by more than three times that, the real datasets do not. A large MAD is not proof of fraud, but it tells an auditor which books to open first. It is also a good example of syllabus point 1.4: the same set of numbers can be made to look honest or suspicious depending on which diagram you draw.
The population, area and river figures built into this exhibit are real published values, rounded — they are approximate, which is exactly the point: rounding to two or three significant figures never changes a leading digit.
Drag to orbit · scroll or pinch to zoom · hover or tap a bar to see its count.