Benford’s law of
anomalous numbers
By: Zac Snively
Which of these data
sets are true
What is Benford’s
law?
• It is a phenomenological law that states the first digit,
or leading digit of numbers will roughly follow the
distribution of:
Digit
Percent
Digit
Percent
1
30.1%
6
6.7%
2
17.6%
7
5.8%
3
12.5%
8
5.1%
4
9.7%
9
4.6%
5
7.9%
How it was founded
• In 1881 an American astronomer Simon Newcomb
noticed that the early pages of log tables seemed to
always be grubby, while the later pages seemed to be
cleaner.
• He sketched out a law stating that, the probability of
the leading digit to be:
• Instead of proving this he simply stated that his rule
was evident, and nothing further was said on the topic
for 57 years until it was brought up again by Frank
Benford
How it was Founded
• In 1938, Frank Benford, a physicist for General
Electric brought together over 20,000 numbers from
different sources, such as readers digest articles, street
addresses, atomic weights, population sizes and many
more.
• He found that all of these numbers followed the
pattern that Newcomb had come up with. Benford
then gave the law its name but also provided no true
explanation of it.
Data
Pool that
Benford
used for
his
findings
When it will work
• Benford’s Law will work on any data set that the
distributions spans over large magnitudes of numbers.
• Examples of when it wont work
1.
2.
Human height in meters.
Pitch speeds of a Major League Baseball game
Uses
• Because numbers generally follow this distribution it
can be used for things such as:
1.Fraud Detection
• Fraudulent Campaign Election Finance
• Fraudulent Toxic Gas Emissions
• Tax Fraud
How does Benford’s law
help with fraud detection
• As you can see numbers that are generated by a Random Number Generator
(RNG) don’t follow Benford’s Law so if an auditor sees a set of numerical
entries that don’t follow Benford’s Law a flag will be raised and thus that set
of taxes will be looked into further.
Other Information
• Looking at the graph you
will see the data values for
numbers that appeared in
US tax returns. An
interesting fact about this is
that if you wanted to
convert all of the US dollars
into Rupees all 14, 414
numbers would be changed,
but it would still follow
Benford’s Law. A system
that remains unchanged
after it is multiplied by a
constant is called a scale
invariant.
Explanation of the
Law
• While there is no true explanation of the reason why
Benford’s Law works there are a few ideas. There are a
few that can be easily explained.
• Another one is the Raffle Ticket example.