Transcript Data Diagnostics Second Order Benford

Data Diagnostics Using Second Order Tests of
Benford’s Law
Mark J. Nigrini, Saint Michael’s
College
Steven J. Miller,
Brown University
Copyright © 2007 by Mark J. Nigrini and Steven J. Miller. All rights reserved.
Frank Benford, Jr. 1883-1948
Benford’s Law
City in North Carolina
Carolina
New Benford’s Law Theorem
The differences between the ordered (ranked)
elements of a data set of N elements
conforming to Benford’s Law gives a second
data set of N-1 elements which also conforms
to Benford’s Law.
Geometric sequence:
Sn  ar n 1
Difference between successive elements:
ar  ar
n
n 1
 a(r  1)  r
n 1
Panel A: Normal Distribution
Panel B: Uniform Distribution
Panel C: Triangular Distribution
Panel D: Gamma Distribution
Panel A: Normal Distribution
Panel B: Uniform Distribution
Panel C: Triangular Distribution
Panel D: Gamma Distribution
DIGIT PATTERNS OF ACCOUNTS PAYABLE NUMBERS
Panel A: Accounts Payable numbers
Panel B: Second order test
Conclusions
• Early Benford’s Law papers have only
advocated its use for data sets that are
expected to follow Benford’s Law.
• The second order test can be used on any
data set.
• Deviations from the Almost Benford
behavior would signal a serious data
integrity issue.