benford’s law formulas benford’s law research the basic digit tests access steps prepared by:...
TRANSCRIPT
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CHAPTER 5
Benford’s Law formulas
Benford’s Law research
The basic digit tests
Access steps
Prepared by:Mark J. Nigrini
Copyright © 2012 by Mark J. Nigrini. All rights reserved.
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BENFORD’S ORIGINAL TABLE
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BENFORD’S HYPOTHESIS
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A BIT OF CALCULUS
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BENFORD’S LAW
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THE FORMULAS
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LITERATURE TO 1980 1948, mental numbers 1960, scale invariance (x by a
constant) 1965, can’t use middle 3 digits as
random numbers 1969, article in Scientific American 1971, first Fibonacci numbers paper 1972, use Benford to test forecasts 1976, Raimi’s review paper
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LITERATURE 1980 TO 2000 1988, rounding up of accounting
numbers 1988, invented numbers 1989, U.S. study of accounting
numbers 1992, rounding up of EPS numbers 1995, random samples from random
distributions are Benford 1996, tax evasion application 1997, auditing application 1999, Journal of Accountancy article
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CONFORMITY TO BENFORD
Numbers should represent the sizes of facts or events
No built-in maximum or minimum Numbers should not be
identification numbers (bank account numbers or flight numbers)
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INVOICE AMOUNTS
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FIRST-TWO DIGIT FINDINGS
Spike at 24 when vouchers are needed @ $25
Spikes below psychological cutoffs of 48, 49, 98, and 99
Systematic frauds, e.g., credit card balances
Comparing two inventory sheets Spike at 14 for U.K. revenue numbers Spike at 24 for purchasing cards Spike at 48 for employee
reimbursements
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FIRST-TWO DIGITS: ACCESS
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FIRST-TWO DIGITS: ACCESS
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FIRST-TWO DIGITS: ACCESS
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SUMMARY Benford’s Law gives the expected frequencies of
the digits in tabulated data Large bias towards the low digits Some important papers in the 1940 to 2000
period Applies to data sets that meet the three
conditions Works best on data sets with > 1,000 records Easy to spot excess activity below control
thresholds Access can calculate the digit frequencies Excel can be used to graph the results