Benford’s law, aka first-digit law, states that in lists of numbers of naturally occurring data, the leading digit is distributed in a specific, non-uniform way. In number sequences, most people assume that in a string of numbers sampled randomly all nine numbers would be equally probable for the leading digit. Benford’s Law states otherwise. He found that the number 1 will appear first about 30% of the time and the number 9 will only appear first around 4.5%. Naturally occurring can be anything from check amounts or stock prices to lengths of rivers. Benford’s law is both scale invariant and base invariant. If something is scale invariant that means if you multiplied every number in the list by the same constant, it does not significantly change the distribution. For example, it does not matter whether the numbers are based on the dollar prices of stocks or their prices in Yen or Euros. Mathematicians have found that the larger and more varied the sampling of numbers from different data sets, the more closely the distribution of numbers approaches Benford's Law.
Benford’s Law does have limitations. Benford’s Law does not apply to uniform or non-naturally occurring data sets. Examples of non-naturally occurring data sets are made up of pre-assigned numbers like zip codes or UPC numbers. Benford's law can only be applied to data that are distributed across multiple orders of magnitude. Moreover, if there is any cut-off which excludes a portion of the underlying data above a maximum value or below a minimum value, then the law will not apply.
Benford’s law has many applications. Several countries, states, large corporations and accounting firms use detection software based on Benford's Law. Benford’s Law is used in the accounting profession to detect fraud. It can be a powerful and simple tool for detecting frauds, embezzlers, tax evaders, inaccurate accounting and computer glitches. Dr. Mark Nigrini used...