In many real collections of numbers, about 30% start with the digit 1 and fewer than 5% start with 9. Count the first digits of a few famous sequences, a simulated economy, or your own data, and see who follows the rule.
Switch to the uniform control and watch every digit settle near 11%. Then try the growing populations: start them all at the same size and they spread across many powers of ten before Benford appears.
The bars show what share of the numbers begin with each digit. The white marks are Benford's prediction: digit d leads with probability log10(1 + 1/d), which is 30.1% for 1, 17.6% for 2, down to 4.6% for 9.
The strip below places every number on a logarithmic scale and folds it into one decade, from 1 to 10. The nine bands are the stretches where each digit leads. On a log scale, the 1 band is six and a half times wider than the 9 band.
You might expect each digit to lead one time in nine. But if a set of numbers has no preferred scale, so that measuring in dollars or yen or doubling everything leaves the pattern unchanged, then the numbers must be spread evenly on a log scale. Benford's law is the only first-digit rule with that property.
It needs data spanning several orders of magnitude. Heights of adults, dice rolls, or prices capped at $10 don't follow it, and that is fine.
People who invent numbers tend to spread first digits too evenly. That makes Benford's law a cheap screening test. It flags data worth a closer look; it never proves wrongdoing on its own.
From the 1990s accountant Mark Nigrini turned Benford's law into a standard audit test. Auditors compare the first digits of invoices, payments or expense claims with the expected curve, then pull the records behind the digits that stick out.
In the demo: the mean absolute deviation and its conformity bands come from Nigrini.Nigrini's 1992 doctoral work applied the test to tax returns, and digit analysis is now part of the toolkit tax and audit agencies use to choose which returns to examine. A spike of amounts just under an approval limit is a classic warning sign.
In the demo: paste a column of amounts and see which digits stand out.Benford tests have been run on vote counts, including after Iran's 2009 election. Statisticians warn that precinct totals often span only one or two powers of ten, so honest results can fail the test and rigged ones can pass. Most now treat it as weak evidence at best.
In the demo: the uniform control fails without any cheating at all.Researchers have used first-digit tests to look for fabricated numbers in published studies, and a 2011 study of EU economic statistics found Greece's figures deviated most from Benford's law. Like any screen, it raises questions rather than answering them.
In the demo: χ² grows with sample size; MAD does not.The coefficients inside a JPEG file follow a Benford-like first-digit pattern. Saving an image twice, which happens when it is edited, disturbs that pattern, so forensic tools can use it as one clue that a picture has been altered.
In the demo: a process that rescales numbers can push them off the log strip's even spread.Numbers inside a computer are stored as a mantissa times a power of two. Richard Hamming showed in 1970 that results of long calculations tend toward the log-even spread behind Benford's law, which matters when engineers estimate how rounding errors build up.
In the demo: repeated multiplication (populations) drifts toward Benford.