Benford’s Law
Cont’d from page 1
He conjectured a simple formula: nature seems to have a tendency
to arrange numbers so that the proportion starting with the digit
D is equal to “log10 of 1 + (1/D).”
Newcomb`s observations were then virtually ignored until 57 years
later when Frank Benford published his paper. (Benford, F. “The
Law of Anomalous Numbers.” Proc. Amer. Phil. Soc. 78, 551-572,
1938). He rediscovered the phenomena and came up with the same law
as Newcomb. Conducting a monumental research, he analyzed 20229
set of numbers gathered from everything from listings of the areas
of rivers to physical constants and death rates, he showed that
they all adhere to the same law: around 30.1 per cent began with
the digit 1, 17.6 per cent with 2, 12.5 per cent with 3, 9.7 per
cent with 4, 7.9 percent with 5, 6.7 percent with 6, 5.8 per cent
with 7, 5.1 percent with 8 and 4.6 percent with 9.
Benford’s law is scale-invariant (the distribution of digits
is unaffected by changes of units) and base-invariant. In fact in
1995, 114 years after Newcomb’s discovery, Theodore Hill,
proved that any universal law of digit distributions that is base
invariant has to take the form of Benford’s law (“Base
invariance implies Benford’s law”, Proceedings of the
American Math. Society, vol 123, p 887).
In applying Benford’s law three rules should be observed:
first the sample size should be big enough to give the predicted
proportions a chance to show themselves so you will not find Benford’s
law in the ages of your family of 5 people. Second, the numbers
should be free of artificial limits so
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obviously you cannot expect the telephone
numbers in your neighborhood follow Benford’s law. Third,
you don’t want numbers that are truly random. By definition,
in a random number, every digit from 0 to 9 has an equal chance
of appearing in any position in that number.
An excellent fraud-buster
This fascinating mathematical theorem is a powerful and relatively
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GROUP 1
30 suras whose no.of verses start with 1 |
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Sura
No. |
No.
of Verses |
| 1 |
4 |
176 |
| 2 |
5 |
120 |
| 3 |
6 |
165 |
| 4 |
9 |
127 |
| 5 |
10 |
109 |
| 6 |
11 |
123 |
| 7 |
12 |
111 |
| 8 |
16 |
128 |
| 9 |
17 |
111 |
| 10 |
18 |
110 |
| 11 |
20 |
135 |
| 12 |
21 |
112 |
| 13 |
23 |
118 |
| 14 |
37 |
182 |
| 15 |
49 |
18 |
| 16 |
60 |
13 |
| 17 |
61 |
14 |
| 18 |
62 |
11 |
| 19 |
63 |
11 |
| 20 |
64 |
18 |
| 21 |
65 |
12 |
| 22 |
66 |
12 |
| 23 |
82 |
19 |
| 24 |
86 |
17 |
| 25 |
87 |
19 |
| 26 |
91 |
13 |
| 27 |
93 |
11 |
| 28 |
96 |
19 |
| 29 |
100 |
11 |
| 30 |
101 |
11 |
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simple tool for pointing suspicion at
frauds, embezzlers, tax evaders and sloppy accountants.
The income tax agencies of several nations and several states,
have started using detection software based on Benford's Law to
detect fabrication of data in financial documents and income tax
returns.
The idea is that if the numbers in a set of data like sales figures,
buying and selling prices, insurance claim costs and expenses claims,
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, it
arouses suspicion of fraud.
Application to the Quran
The Quran is divided into chapters of unequal length, each of
which is called a sura.
The shortest of the suras has ten words, and the longest sura, which
is placed second in the text, has over 6000 words. From the second
sura onward, the suras gradually get shorter, although this is not
a hard and fast rule. The last sixty suras take up about as much
space as the second sura. This unconventional structure does not
follow people’s expectations as to what a book should be.
However it appears to be a deliberate design on part of the author
of the Quran. Let’s verify the evidence:
Quran consists of 114 suras. Each sura is composed of certain
number of verses, for example sura 1 has 7 verses and sura 96 (the
first sura revealed to prophet Muhammad ) has 19 verses. So we have
a set of 114 data to which we can apply the Benford’s law.
The first table showing Group 1 with 30 suras is shown on the left:
Cont'd on Page 3
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