Frank Albert. Benford, Jr. 1912
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Frank A. Benford, Jr.1883-1948
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Lunch with Harry Benford
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Son of Frank Benford, 12/1993
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Home of Frank A. Benford
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Rugby Road, Schenectady, NY
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Photos taken April 22, 1994
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A first digit 1
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Pieter Allaart, Mark Nigrini, Ted
Hill, Vrije Universiteit, 1994
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Prof. Ralph A. Raimi, joint seminar,
Niagara Falls, 11/1996
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Prof. Ralph A. Raimi, joint seminar,
Niagara Falls, 11/1996
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 | Journal of Accountancy: |
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http://www.aicpa.org/pubs/jofa/may1999/nigrini.htm
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| Smithsonian reference to Frank
Benford's discovery of the laser pointer: |
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Smithsonian
Science Service
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| Forensic Accountants Report
Using Benford's Law (April, 2002) |
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For an extract from the
full report please click here. For more |
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information contact Darrell D. Dorrell, darrelld@financialforensics.us.com |
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Nigrini
notes: First-two digits graph, x-axis should run from 10 to
99. First-three digits graph, x-axis should run from 100 to
999. |
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If
you are using Excel, In step 2 of 4 of Chart Wizard, click on
Series tab and then use Category (X) axis labels. |
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Dartmouth
College, December, 2000. Benford's Law presentation titled
"Benford's Law, Its a secret law of numbers."
RealVideo and slides. You need to click on my presentation
title to start the video. Title should be viewable when you go
to the page. |
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| Test the file sizes of the files
on your hard drive against Benford's Law |
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Test
your file sizes against Benford's Law - prepared by Jerry Anderson,
Internal Audit, NIKE, Beaverton, Oregon.
I'll add comments in November, 2002. Thanks for the good work Jerry.
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| Output from round numbers test
at major health insurer |
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Round
Numbers Report
- this is the output from running the round numbers test on health
care providers at a major health insurer. All proportions
higher than 0.90 are listed. |
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When
the multiples of 10 test gives so many subsets with high round
number percentages this suggests that multiples of 10 are quite
normal for this data. My suggestion is to rerun the test to
test for multiples of $100 which should be quite odd for medical
charges. |
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| Analysis of accounts payable
invoice dollar amounts |
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First
Digit and First-Two
digits graphs.
This was an analysis done by a senior internal auditor of accounts
payable data for 2000 and 2001. Note the good fit to
Benford's Law. The spikes were mainly due to recurrences of
low value invoice amounts. |
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The
analysis was done in IDEA using the Benford's Law analysis tool. |
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| An Assessment of the Change in
the Incidence of Earnings Management around the Enron-Andersen
Episode |
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In
2001 Enron filed amended financial statements setting off a chain of
events starting with its bankruptcy filing and including the conviction
of Arthur Andersen for obstruction of justice.
The end of 2001 and the first half of 2002 included a heightened
level of publicity for the accounting practices of listed companies.
This paper addresses whether there was a detectable change in the
incidence of earnings management around this time period.
Earnings
reports released in 2001 and 2002 were analyzed.
The results showed that revenue numbers were subject to upwards
management. Benford’s Law
was used to detect such manipulations.
Earnings Per Share (EPS) numbers showed a marked discontinuity in
the distribution around zero which is consistent with upwards
management. The results also showed a tendency towards neat round EPS
numbers such as 0.10, 0.20, etc. The
overall results are consistent with a small but noticeable increase in
earnings management in 2002. Enron’s
reported numbers are reviewed and these show a strong tendency towards
making financial thresholds.
Published
in The Review of Accounting and Finance, Volume 4, Number 1 (2005) To
obtain a copy of this
paper please contact Helen Trudgill at htrudgill@emeraldinsight.com
and subscribe to the journal.
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Notes on Frank Benford |
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Please
do not copy these images to another website. It took some effort
on my part to find and collect these documents.
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| Analysis of a manufacturing company's own
sales data using Benford's Law |
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Can
Benford's Law be used in Forensic Accounting? |
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This was my first published Benford's Law
article. It was a two-page article published in The Balance
Sheet, the journal of the Investigative and Forensic Accounting Interest Group of the
Canadian Institute of Chartered Accountants. I made quite a bold
prediction about analysis of digital frequencies and the audit of
tabulated data. The article was published in June, 1993 and was
written in early 1993.
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Analysis of Distribution
company's Sales Numbers (40,000 records) for six months to
June 30, 2002 and the Accounts Receivable Numbers (1,000
records) at June 30, 2002. Would you expect a
relationship between the sales numbers that gave rise to the AR
numbers, and the AR numbers? |
The graphs do not match each other indicating
that the Accounts Receivable numbers are not a random sample of the
sales numbers.
The Number
Duplication tables were prepared using DATAS for Excel and then combined
onto a single spreadsheet. The number duplications show that the
accounts receivable is not a random sample of the invoices. It
seems that the high dollar balances are made up of hundreds of small
invoices. Note the high counts of low-value invoices (efficiency
finding). Very few of the invoice numbers can be seen on the
accounts receivable table. Note the gravitation towards invoices
that are multiples of $40. |
| Purchases for resale and other Accounts Payable numbers for a
major US retailer for the first fiscal quarter for year-end
2003. This was a large data set with a total dollar value of
about $10 billion. |
The analyst deleted all numbers below $100 to
reduce the data set to about 3.3 million records. Usually an
artificial minimum causes nonconformity with Benford's Law.
However, a minimum that is an integer power of 10 (100 equals 10 to the
power of 2) is OK and Benford's Law should still hold provided that
other conditions are met. The graphs show that the actual digits
are more skewed towards the low digits than Benford's Law.
This always occurs when the data set has an excessive amount of low
numbers. The graphs tell me that this retailer has a very high
proportion of numbers from $100 to $999 and this is very inefficient for
operations of this scale. Remember that the under $100s were
already deleted.
The numbers were run using
Microsoft Access and the digit extraction and digit count queries were
run in under 5 minutes. |
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This section is now "in
progress."
Employee Travel and Expense claims, NYSE-listed company.
Data Profile
September, 2002,
Benford's Law tests
October, 2002, Benford's Law
tests
Commentary, to follow.
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New!
March 22, 2004. I've loaded a student (or trial) version of the
Excel program that does the Data Profile, and Benford's Law First Digit,
Second Digit, and First-Two Digits test with output statistics in the
DATAS Software section of this website. .The file works with Excel 2000, 2002, or 2003.
A test dataset is also downloadable.
The
Student/Trial version will only process the first 1,000 records in
your data set (unless you find the piece of code where I restrict the
operation to the first 1,000 records and change it to 65,535 records!).
Click
here to go to the DATAS software
section of this site.
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New!
April 3, 2004. I've seen a new paper on Benford's Law that
may be of interest to auditors, accountants, and researchers. The
paper was published in The American Statistician in February,
2004. To see the abstract click
here. |
May,
2004.
Benfordslaw.com has finally come home! This name was registered by
a cybersquatter in 1999 very soon after my Benford's Law paper was
published in The Journal of Accountancy. For a year or two
Benfordslaw.com lived in New Jersey and it then moved to a company with
a registered office on Sahara Avenue in Las Vegas. I visited the
office in October, 2002 to ask about buying benfordslaw.com. To see the
old office in Las Vegas, click
here. |
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CBS
detective drama Numbers (spelled Numb3rs).
The
episode on February 3, 2006 spoke about Benford's
Law. I was flying from Cleveland to Montreal when the show was
aired and as of now I have not yet seen the episode. Comments sent
to me by e-mail are welcome.
For
the Activities write-up that accompanied the show click here.
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My
current work (with Steven J. Miller, Brown University):
We
show that for certain distributions if we take the numbers and rank them
from smallest to largest and then take the differences between the
numbers then those differences will either follow Benford's Law exactly
or with only a small degree of error (which we call Almost Benford
behavior). The paper can be viewed by going to www.arxiv.org
and for searching for Benford's Law or by going directly here. |
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My
current work (with Steven J. Miller, Brown University):
Benford's
Law Applied To Hydrology Data - Results and Relevance to Other
Geophysical Data
Abstract:
Benford’s
Law deals with the expected frequencies of the digits in tabulated data.
These expected digit patterns are quite counterintuitive in that
the lower digits (1, 2, and 3) are expected to occur far more frequently
than the higher digits. The
law has been found to hold true in several studies that analyzed
financial data. Since
Benford’s original study in 1938 there have not been any subsequent
analyses of earth science data, nor have any of the recently published
studies dealt with large data sets.
The objective of this paper is to test whether the law applies to
selected large earth science data sets and to discuss and comment on the
potential usefulness of testing digital frequencies.
The first test analyzed U.S. Geological
Survey data related to streamflow statistics.
The conformity to Benford’s Law was excellent.
The second test analyzed data related to the sizes of lakes and
wetlands. Here conformity
to Benford’s Law was weak. The
nonconformity of the lake data could be explained by the fact that the
data set was incomplete and only included lakes above a certain size.
Also, the nonconformity seemed to be the result of the data
following a power law. It
is only under certain circumstances that data following a power law will
conform to Benford’s Law.
The results of the study suggest that data related
to water bodies should conform to Benford’s Law and that nonconformity
could be indicators of either (a) an incomplete data set, (b) the sample
not being representative of the population, (c) excessive rounding of
the data, (d) data errors, or (e) adherence of the data to a power law
with a “high” a
value. The results of this
study suggest that an analysis of digit frequencies could be useful in
assessing the authenticity and integrity of earth science data.
Keywords:
Digit frequencies, data interrogation, data
authenticity, power law, hydrometry statistics, hydrography statistics.
Notes: This paper should be complete by
February 12, 2006. Contact us for more information. |
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My
current work (with Steven J. Miller, Brown University):
Advanced
Tests Based on Benford's Law to Test the Reliability of and Control Risk
Pertaining to Accounting Data
Abstract:
SAS
No. 99 requires auditors to employ analytical procedures during the
planning phase of the audit to identify the existence of unusual
transactions, events, and trends. This
use of analytical procedures and the effective use of the computer on
transaction level data could present an efficient means for auditors to
partially fulfill their responsibilities with regards to the detection
of fraud and material misstatements.
Benford’s Law gives the expected patterns of the digits in
numerical data and it has been advocated as a test for the authenticity
and reliability of transaction level accounting data.
To date these tests been tests of the first, second, first-two,
and last-two digits of accounting data.
This paper extends the prior work on
Benford’s Law and (a) proposes a test for conformity to Benford’s
Law based on the distribution of the mantissas of the accounting data,
and (b) proposes a new second-order test of Benford’s Law that
provides new insights into the data.
The tests are demonstrated using two accounting databases and the
proposals for future research call for work demonstrating their
applicability as a continuous monitoring tool to assist with compliance
with section 404 of the Sarbanes-Oxley Act.
Keywords:
Internal control, fraud detection, continuous monitoring,
Benford’s Law, accounting data.
Data
availability: The accounts payable data are available from the
author and the author will consider requests for the data used for the
second case study.
Notes:
This paper will be significantly edited and revised in March,
2006. Contact us for more information. |
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Institute of Internal
Auditors: Web-based seminars.
Institute
of Internal Auditors Audit Learning: This web-based training course
will review Benford's Law and some other data analysis tests and will
show how the tests can be done in Microsoft Excel and Microsoft Access. The
IIA is a NASBA approved sponsor for CPE credit.
All
my
seminars have been developed and posted as of September, 2003.
Data
Analysis with Benford's Law and Microsoft Excel
Data
Analysis with Benford's Law and Microsoft Access, first session
Data
Analysis with Benford's Law and Microsoft Access, second session
The above three
seminars are based on chapters 2, 4, and 5 of my Using Microsoft Access
for Data Analysis and Interrogation book. If you have worked through chapters 2 and five in my book then the
courses will feature familiar material. This is a great way to
earn CPE without the stress and costs of travel!
Click
here to link to the site for web-based seminars from the Institute of
Internal Auditors. Feel free to send me an e-mail if you have
questions. |
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Copyright ©
2002-2006 by Mark J.
Nigrini. All rights reserved.
Except as permitted under the United
States Copyright Act of 1976, no part of this web page may be reproduced
or distributed in any form or by any means, or stored in a data base or
retrieval system, without the prior written permission of the publisher.
Requests for such permissions should be addressed to Mark J.
Nigrini.
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