Benford`s Law Second Digit Formula

Why it works. Using a non-technical description, Benford`s law works regardless of whether you count dollars, acres, inventory, populations, or anything else, because you first have to count 1 before counting 2, 3, or 4. You must first count 10 before counting 20, 30 or 40. You need to count 100 before counting 200, 300 or 400. And so on. Every counting job starts with lower numbers, but not all counting orders go ahead to include the ever-increasing numbers. For example, if you count up to 25, 11 numbers have a principal digit of 1, seven numbers have the main digit 2, and only one number leads with the number 3. Even in this simplified counting exercise, you can see Benford`s law at work – instead of each digit having the same chance of being the first digit, the lower digits are always more likely to be the main digit compared to the higher digits. For example, the probability that a number begins with the numbers 3, 1, 4 is log10(1 + 1/314) ≈ 0.00138, as in the figure on the right. The numbers that meet this criterion are 3.14159…, 314285.7. and 0.00314465. The square roots and reciprocals of successive natural numbers do not obey this law.

[42] Telephone directories violate Benford`s law because (local) numbers generally have a fixed length and do not begin with the remote dialling code (in the North American dial plan, number 1). [43] Benford`s law is violated by the population of all places with at least 2500 residents of five U.S. states according to the 1960 and 1970 censuses, where only 19% started with No. 1 but 20% began with No. 2, for the simple reason that the reduction introduces statistical biases to 2500. [42] The latest figures from pathology reports violate Benford`s law due to rounding. [44] This result can be used to determine the probability that a certain number will occur at a certain position in a number. For example, the probability that a „2“ would be encountered as a second digit.[35] In 1972, Hal Varian suggested that the law could be used to uncover possible fraud in lists of socio-economic data submitted in support of public planning decisions.

Based on the plausible assumption that people who make numbers tend to distribute their numbers fairly evenly, a simple comparison of the frequency distribution of the first digit from the data with the expected distribution under Benford`s law should yield abnormal results. [19] And the probability that d (d = 0, 1, …, 9) is encountered as the nth digit (n > 1) is, for example, the first digit (non-zero) of this list of lengths should have the same distribution, whether the unit of measurement is the feet or the yards. But there are three feet in a yard, so the probability that the first digit of a length in yards is 1 must be the same as the probability that the first digit of a length in feet is 3, 4 or 5; Similarly, the probability that the first digit of a length in yards is 2 must be the same as the probability that the first digit of a length in feet is 6, 7 or 8. If you apply this to all possible measurement scales, you get the logarithmic distribution of Benford`s law. Similarly, some continuous processes conform exactly to Benford`s law (within the asymptotic limit, if the process continues over time). One is an exponential process of growth or decay: if a size increases or decreases exponentially over time, then the percentage of time it has each first digit meets Benford`s law asymptotic (i.e. increase accuracy as the process progresses over time). As another example, an accountant writing fictitious checks may intentionally keep cheque amounts below the company`s authorization thresholds of $500 or $1,000, and so an analysis of those cheque amounts could show that numbers 4 and 9 occur more frequently than the main numbers than Benford`s law predicted. The results obtained using Benford`s legal analysis should not be considered definitive; The process of counting the main figures will never clearly prove the absence or existence of fraud.

The results of this process are only an analytical tool that can help the CPA assess whether further investigative work is warranted. However, if the Benford curve does not materialize, CPAs should intensify their efforts to examine the data as follows. But consider a list of lengths evenly distributed over many orders of magnitude. For example, a list of 1000 lengths mentioned in scientific articles containing measurements of molecules, bacteria, plants and galaxies. If you write all these lengths in meters or all in feet, it is reasonable to expect that the distribution of the first digits on both lists will be the same. As you can see in this graph, the top of the bars produces nothing near a Benford curve, and this simple result tends to repeat itself even if the random numbers are recalculated several times (by pressing the F9 key).

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