Benford's Law

Copyright ©2003 by Paul Niauette. All rights reserved.

What is your explanation?

onsider some increasable number that starts with the cipher 1 followed by zeroes.  To get to the first number that starts with the cipher 2, you have to bring about an increase of 100% in the things being counted -- or by whatever phenomenon your number measures.  Some amount of time or some quantity of other resources will be required for that, depending on the rate at which the number is enabled to grow. 

Now, for continuing on to the first cipher 3, you have to assure a change of only 50%.  If exponential growth is going on at a steady rate, more or less, then less time will go by.  Likewise toward succeeding ciphers, each being reached with equal increments in absolute amounts but declining over time relative to the size of the growing number itself. 

When your number eventually reaches the cipher 9 followed by zeroes, only about 10% more growth will take you over the top, giving you another number that starts with cipher 1, and once again you face a 100% climb to reach cipher 2.

ou will find a simple Monte Carlo model of Benford's Law right here, and a technical treatment of the subject at the Mathworld site, where a derivation is given for the probability that a given cipher will appear in the first position of a number.

pC = log10 (1 + 1/C)

For a non-mathematical explanation, you might want to read "Following Benford's Law, or Looking Out for No. 1" by Malcolm W. Browne.  That article explores a practical application of Benford's Law...

Dr. Mark J. Nigrini, an accounting consultant,...gained recognition a few years ago by applying a system he devised based on Benford's    Law to some fraud cases in Brooklyn. The idea underlying his system is that if the numbers in a set of data like a tax return more or less match the frequencies and ratios predicted by Benford's Law, the data are probably honest. But if a graph of such numbers is markedly different from the one predicted by Benford's Law, he said, "I think I'd call someone in for a detailed audit." 
Some people will find that practical -- but for nepherious reasons.



In another context (Prime Numbers are Odd), correspondents noted that people who spend time studying numbers are themselves "odd" and received a cryptic response...

In the Land of Integers, among sums you can give even odds to us odds, but among products, we odds are outnumberd by evens three-to-one.
...which required some explanation.

An integer operand can be either even or odd, and it takes two operands to make a sum or a product, in four combinations -- even|even, even|odd, odd|even, odd|odd.  The respective sums will be even, odd, odd, even; the respective products will be even, even, even, odd.

Thus, for example, it is an even bet that a pair of dice will turn up an odd number on a given toss, but that's only because craps takes sums not products.
Many of the integers in the world result from measurements or counting, and their non-uniform biases are explored above in Benford's Law. The rest of the integers in circulation are the consequences of arithmetic whereby the evens are apparently crowding out the odds.
Which may explain something.  Is it not Tyranny of the Majority that pejorates odd as unusual/strange/eccentric?

Benford's Law: The Model

The empiricists among us may enjoy demonstrating Benford's Law with a simple Monte Carlo model using a spreadsheet.  The conventions used here include...

  • columns designated by letters A, B,...
  • rows designated by numbers 1, 2...
  • arithmetic operators +, -, *, /
  • SUM(range) returns the sum of the entries in a range
  • RAND() returns a random number between 0 and 1
  • INT(x) returns the integer part of x
  • logical operation IF(condition, do this, else do that)
For the formulas below, you will need to create a table with eleven columns (A...K) and 100 rows, plus some space above and outside for labels and control parameters.
A3...A102 ~~ x1...x100, a sequence of 100 growing numbers, limited to six digits.
B3...B102 ~~ the first non-zero digit of xN
C3...C102 ~~ cells marked with 1 for each appearance of cipher 1 in xN
D3...D102 ~~ cells marked with 1 for each appearance of cipher 2 in xN

K3...K102 ~~ cells marked with 1 for each appearance of cipher 9 in xN

Generating the 100 Increasing Values of xN:
Let N3 = x0 ~~ an arbitrary and changable seed number (1000, say)
Let N5 = INT(N3*RAND()) ~~ a random number between 1 and 1000.
Let N7 = Incrementing Coefficient (1.2, say, for a 20% rate of growth).

A3 = N5 ~~ feeding the table with the seed number as x1.
A4 = IF(A3*N$7>1000000,A3/M$10,A3*N$7) ~~ increments and limits x2

A102 = IF(A101*N$7>1000000,A101/M$10,A101*N$7) ~~ computes x100

Parsing First Non-Zero Digit
For B3, B4, ... B102 ~~ creates a column of first digits.
BN= IF(AN>100000,INT(AN/100000),
Classifying the Ciphers
Let C2 = 1, D2 = 2,... K2 = 9 ~~ column headings for each cipher 1, 2,... 9

CN = IF($B3=C$2,1,0) ~~ mark the table for cipher 1s
DN = IF($B3=D$2,1,0) ~~ mark the table for cipher 2s

KN = IF($B3=K$2,1,0) ~~ mark the table for cipher 9s

Counting the First-Digit Ciphers
C1 = SUM(C3:C102) ~~ counts the cipher 1s
D1 = SUM(D3:D102) ~~ counts the cipher 2s

K1 = SUM(K3:K102) ~~ counts the cipher 9s

B1 = SUM(C1:K1) ~~ tallies all ciphers (should always = 100).

To carry out experiments, simply make entries anywhere outside the table, which operates the random number generator, simulating 100 growing numbers of up to six digits each.

Watch the locations C1, D2, ... K1 and you will see Benford's Law in action.


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