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In many real datasets, numbers start with the digit 1 about 30% of the time
Benford's law observes that in many real-world collections of numbers, the leading digit is more often small. The digit 1 leads about 30% of the time, while 9 leads less than 5% of the time, compared with 11.1% if digits were uniform. It appears in electricity bills, street addresses, stock prices, populations, river lengths and many physical constants. The law holds best when values span several orders of magnitude, especially when they come from processes that follow a power law. Simon Newcomb noticed it in 1881 from wear on logarithm tables, and Frank Benford popularised it in 1938.
If values are spread evenly across a logarithmic scale, the leading digit is not uniform. Numbers starting with 1 cover a wider slice of that scale, from 1 to 2, than numbers starting with 9, which cover only 9 to 10, so 1 appears far more often.
Because genuine data tends to follow this pattern, accountants and auditors use deviations from it as a red flag for possible fabrication or fraud. It is a surprising reminder that everyday numbers follow hidden structure.
Simon Newcomb noticed the pattern in 1881 because the early pages of logarithm tables were much more worn than the later ones.
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Illustrative photo · Gknor, Public domain, via Wikimedia Commons
