Benford’s law

An illustration of Benford's law, using the population of the countries of the world. The chart depicts the percentage of countries having the corresponding digit as first digit of their population (red bars). For example, 64 countries of 237 (=27%) have 1 as leading digit of the population. Black points indicate what is predicted by Benford's law.

Have you ever thought that a group of statistics look a bit dodgy? Well just apply Benford’s law and it should show up any frauds. We know that the Tax Office use these when checking a business’s accounts to see if the figures haven’t just been made up – DeskaratiĀ –

Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on the logarithmic scale.

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

The graph to the right shows Benford’s law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization to second digits and later digits.

It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881

via Benford’s law

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