factorial randomness
The article below is about the question: Is there
factorial randomness in the sequence of natural numbers. The answer is: Yes for
now because the discribed results also indicate that there maybe a mathematical
formula possible from wich you can calulate the factors from a given number. So
with that formula it is possible to gess better than change what the next
number of the factorial sequence will be.
Scientia Araneae Totius Orbis
J.G. van der Galiën ‘Factorial Randomness’ 2.3. (2003)
Communication to the editor
S.A.T.O. Journal of RANDOMICS
Factorial Randomness
The Laws of Benford and Zipf
with respect to the first digit distribution of the factor sequence from the
natural numbers.
By: Johan Gerard van der
Galien (M.Sc. in Chemistry)
(For Comments: galien8@zonnet.nl)
Version 1.1. November 8,
2003
HOME of Scientia Araneae Totius Orbis
Six first digit distributions of the factors of the
natural numbers up to n=10d (d Î
{2,3,4,5,6,7}) were calculated by means of a self-written
The Chi-Square increased rapidly, from 6.91 for n=102
and 47.1 for n=103 to 222435 for n=107 Indicating that
only for n=102 Benford's Law is followed. (It is a well known fact,
and you will find it referenced in this article, that Chi-Square will always
deviate if the sample space is large enough!)
These results are compared to the ones calculated from
20 Benford distributions found on the internet.
But the relative frequencies of the Factorial First
Digit Distribution (fD) correlates well with fD = cD-a
or Zipf's Law. Empirical and mathematical evidence is given that Benford's
formula is an (approximate) special case of the more general formula of Zipf's
Law. The Correlation Coefficient (R) from Linear Regression of scatter plotting
log10(fD) = alog10(D) + b asymptotically goes
to -1 with sample size. from R = -0.996 for n=102 to R = -0.9995 for
n = 107. So a nearly perfect fit. Since the Benford formula also
nearly perfect fits (R = -0.9992) with Zipf's Law it is reasonable to say that
the Factorial First Digit Distribution follows Benford's and Zipf's Law, which
also is confirmed by comparing the Linear Regression results of the Factorial
First Digit Distribution with the regression formula log10(1+1/D) » 0.3135080577D-0.8636655870.
The consequences of these results for our
understanding of randomness are discussed.
It is a widespread opinion that the randomness
in our universe is a consequence of the uncertainty principle of Heisenberg
from Quantum Mechanics.1 It is also the last hurdle mankind
must take to obtain a deterministic and holistic theory of everything. Since
there may be "hidden" variables in Quantum Mechanics.3-7,9,14
The definition of a truly random sequence
used in this paper is: That no known deterministic process can do better than
chance at guessing from a given sequence what the next element of this sequence
will be.2 (Then this is also true for the sequence of factors
of the natural numbers. Although there are a kind of patterns in it. Since
there is no mathematical formula yet17, or in other words no
deterministic process for calculating the [prime] factors of a natural number.
If there was, then nowadays prime factors based encryption like RSA would be
obsolete. There are only several trail divisions and sieve based algorithms
that all are an enormous burden on nowadays computers.) This implies that true
randomness is a temporary phenomenon. As we learn more from nature and science
evolves we might develop deterministic processes which are for instance able to
predict better than change what the next element of a sequence from a Quantum
Mechanical source will be. It looks like randomness is the Achilles heel of
traditional physics, and in my opinion it is the only way at the present day to
tackle the deterministic properties of nature.
The way the
The Output of FACBEN21LR of the Chi-Square
Goodness-Of-Fit8 analysis can be interpreted as follows.
According to this test there is only a fit for the smallest sample n = 102.
The Chi-Square (6.91) then falls within the 5% and 95% Confidence Interval >
2.71 and < 15.51.10,15,22 With increasing sample size the
Chi-Square also increases to 222435 for n = 107. But it is a well
known fact that Chi-Square always will deviate when the sample size is large
enough!25 And the sample sizes of my experiment are very
large indeed, see Table 3. So the deviation does not have to be significant. I
compared these results by calculating the Chi-Squares of other suspected
Benford distributions which can be found on the internet.18
These results are shown in Table 4. The Chi-Squares of these distributions have
a range of 1.27 to 441. There are 2 who are below the 5% criterion and 10 above
the 95% criterion. According to reference 19 when a Chi-Square is below
5% Confidence Level or above the 95%, the distribution is suspect (in other
reject the H0-hypothesis, in the case of reference 19 the
sequence is not random and in my case Benford's Law does not apply). The
average distribution is definitely within this range (3.50). And this makes
these 12 distributions less suspect. Tho comeback on the sample size of the
average from the 20 distributions from reference 18 this is only 20229,
compare this to 162728526 of the Factorial First Digit Distribution for n=107.
Now about the hypothesis that Benford's Law
is in fact a special case of the more general Law of Zipf. There is evidence
from the Linear Regression analysis of the results from Benford's formula (see
Table 1).
|
Log10(1+1/D) |
Log10(D) |
Log10(log10(1+1/D)) |
|
|
1 |
0.3010299957 |
0 |
-0.5213902277 |
|
2 |
0.1760912591 |
0.3010299957 |
-0.7542622013 |
|
3 |
0.1249387366 |
0.4771212547 |
-0.9033028900 |
|
4 |
0.0969100130 |
0.6020599913 |
-1.0136313480 |
|
5 |
0.0791812460 |
0.6989700043 |
-1.1013776680 |
|
6 |
0.0669467896 |
0.7781512504 |
-1.1742702440 |
|
7 |
0.0579919470 |
0.8450980400 |
-1.2366323100 |
|
8 |
0.0511525224 |
0.9030899870 |
-1.2911329450 |
|
9 |
0.0457574906 |
0.9542425094 |
-1.3395378010 |
|
Linear Regression results for log10(log10(1+1/D)) = alog10(D) + b |
|||
|
Correlation Coefficient R = |
-0.9992296195 |
||
|
|Slope| = a = |
0.8636655870 |
||
|
Intersection with Y-axis b = |
-0.5037512926 |
||
Table 1: Evidence that the Benford first digit distribution is a special case of
Zipf's law.
From the good R = -0.9992 of Table 1 can be deduced
that the Benford first digit distribution law must be a special case of Zipf's
law. (A perfect fit has an |R| = 1.20) The approximate
formula of Zipf's law applied for the Benford distribution can be deduced from
the result of Table 1 since:
log10(log10(1+1/D)) = alog10(D) + b Þ log10(1+1/D) » 10bD-a = cD-a (1.)
fD = log10(1+1/D) » 0.3135080577D-0.8636655870 (2.)
Here below is the mathematical proof that
that Benford's Law is an (approximate) special case of Zipf's Law.
Theorema: log10(1+1/D) » cD-a D Î {1,2,3,4,5,6,7,8,9} or 1 £ D £ 9 (3.)
x = (1+1/D) (4.)
log10(x)
= ln(x)/ln(10) (5.)
ln(x) = (x-1)/x+1/2((x-1)/x)2+1/3((x-1)/x)3+
…….. (
ln(x) = (x-1)/x+(x-1)2/2x2+(x-1)3/3x3+
……. Þ
(1/D)/(1+1/D)+1/2((1/D)/(1+1/D))2+1/3((1/D)/(1+1/D))3+
……. Þ
(1/(D+1)+1/2(1/(D+1))2+1/3(1/(D+1))3+
……. (7.)
(1/(D+1) > 1/2(1/(D+1))2 >
1/3(1/(D+1))3 > ……. (8.)
Now some gross approximations:
ln(1+1/D) » 1/(D+1) » 1/D (9.)
log10(1+1/D)
= ln(1+1/D)/ln(10) » 1/(Dln(10)) Þ
fD »
(1/ln(10))D-1 = 0.4342D-1 (10.)
So a = 1 and c » 0.4342 which is of the same magnitude found
by Linear Regression (2.)
I also did Linear Regression on the Factorial
First Digit Distribution. The |Correlation Coefficient| = R goes asymptotically
to 1 with increasing amount of natural numbers tested. This fact is shown in
Fig. 1.
Fig. 1: The development of the Correlation Coefficient (R = |Correlation
Coefficient|) of the Factorial First Digit Distribution with Zipf's Law with
increasing amount of natural numbers (n) tested.
Also the constants derived from the Linear
Regression analysis (11.) of the data from n = 107 are in good
agreement with the ones found for the Linear Regression analysis of Benford's
formula (2.):
fD » 0.3237122232D-0.8996241844 (11.)
|
D |
Log10(1+1/D) |
0.3135080577D-0.8636655870 |
Factorial First Digit Distribution |
|
1 |
0.30 |
0.31 |
0.32 |
|
2 |
0.18 |
0.17 |
0.18 |
|
3 |
0.12 |
0.12 |
0.12 |
|
4 |
0.097 |
0.095 |
0.096 |
|
5 |
0.079 |
0.078 |
0.074 |
|
6 |
0.067 |
0.067 |
0.063 |
|
7 |
0.058 |
0.058 |
0.056 |
|
8 |
0.051 |
0.052 |
0.050 |
|
9 |
0.046 |
0.047 |
0.045 |
Table 2: Comparing the outcome of the Benford formula, the Benford/Zipf
regression formula and the factorial first digit distribution for n = 107.
Rounded to two significant digits.
The results from the Linear Regression of the
average of the 20 distributions from reference 18 also confirm Zipf's
Law. Compare these results a = 0.89874592240, b = -0.4.8940098622 and R =
0.99661451810 with the results from the Factorial First Digit Distribution for
n =
From these results the following conclusions
can be deduced:
- Benford's Law
is an (approximate) special case of Zipf's Law. Proven empirical and
mathematical.
- All Benford
distributions also follow Zipf's Law.
- If there is
good correlation with Zipf's Law of a First Digit Distribution then
Benford's law also applies.
- From the
above conclusions follows that the Factorial First Digit Distribution
follows Zipf's AND Benford's Law.
- It is a
well-known (empirical?) fact that Chi-square will always deviate if the
sample size is large enough. I wonder if this is mathematical proven! Can
someone tell me that and give me a reference. At least my data supports
this ‘hypothesis’. So it is highly likely that the Factorial First Digit
Distribution follows Newcomb-Benford’s Law.
- The factorial
sequence is still a random sequence since there is no mathematical formula
yet to calculate the factors of a given integer.
- Benford's Law
is an approximate law. Zipf's Law is more exact.
- Benford's Law
applies when at random "unbiased" distributions are
selected and that random samples are taken from each of these
distributions.24 The way FACBEN21LR gathers data seems
to fulfil this criterium, in other words the factor sequence from the
natural numbers is random.
It looks like there are at least two kinds of
randomness. Pure randomness from Quantum Mechanical sources which fully concurs
with an equidistribution of digits of the used number system (for instance by
transforming random bits to byte integers [octal number system], or word
integers [hexadecimal number system], etc.) according to a Chi-Square analysis.
And randomness from an interaction between chaos and order for which the first
digit distribution concurs with Newcomb-Benford's and Zipf's Law. But which
does not pass the Chi-Square test at large sample sizes for accepting the H0-hypothesis
that the distribution follows the log10(1+1/D) formula. In other
words such kind of randomness has a deterministic and, a still, by lack of a
better word, "undeterministic" component. It should be possible to
develop experiments with which you can isolate the deterministic and "undeterministic"
part of such a kind of randomness. (The factor sequence of the natural numbers
is in my opinion the most accessible, and a limitless resource for Benford/Zipf
randomness, to do such experiments with.) And also finding the mathematical
formula for the factors of natural numbers will bring us closer to
understanding pure randomness.
Epilogue:
As a matter of fact divising the Factorial
Distribution in to a deterministic and by lack of a better word
"undeterministic" part has been done recently by myself. One can
extract the analytical pure chaos (= "undeterministic" part), which
means that it passes critical Randomness Test Suites like DIEHARD and NIST,
these temporary secret extracting methods are applied in the FRAG Pseudo Random
Number Generator described in this journal and are on this site.26
An example of the deterministic part that leads from order to a
Newcomb-Benford-Zipf distribution is the effect of the power of two calculation
from bits in the formula needed to generate computer reals with different kind
of mantissa bits. Research also described in this journal.27
-o0o-
1a) Walker J. 'Index
Librorum Liberorum’
1b) Anonymous 'Generating random numbers' http://www.randomnumbers.info/content/Generating.htm
1c) Anonymous 'A fast and compact quantum random generator'
http://www.quantum.at/research/photonentangle/rng/
1d) Davies R. 'Hardware random number generators'
http://www.robertnz.net/hwrng.htm
2) Inspired by: Paul P. Budnik 'What is and what will be'
http://www.mtnmath.com/book.html
3) Anonymous 'The system at work' webpage.
4) Not used.
5) Van der Galien J.G. 'Are the prime numbers randomly
distributed?' Scientia Araneae Totius Orbis 1.2. http://home.zonnet.nl/galien8/
6) Eric Weisstein's World of Physics 'Einstein Rosen
Podolsky paradox'
http://scienceworld.wolfram.com/physics/Einstein-Podolsky-RosenParadox.html
7) Anonymous, 'Outline: Bohm,
http://www.drury.edu/ess/philsci/bell.html
8) Knuth, D.E 'The art of computer programming, Volume 2 /
Seminumerical algorithms'
9) Anonymous 'A prime case of chaos'
http://www.ams.org/featurecolumn/archive/prime-chaos.html
10) Kreyzig E. ‘Advanced
engineering mathematics: Table A12 chi-square distribution’ 4th
edition, John Wiley and Sons (1979)
11) Not used.
12) Not used.
13) Not used.
14) Anonymous 'The hidden-variable theory of David Bohm'
http://www.meta-library.net/ghc-obs/hidvar-frame.html
15) Walker J. 'Chi-square calculator'
http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html
16a) Lowry R. 'Chapter 8: Chi-square procedures for the
analysis of categorical frequency data. Part 1'
http://faculty.vassar.edu/lowry/PDF/c8p1.pdf
16b) Lowry R. 'Chapter 8: Chi-square procedures for the
analysis of categorical frequency data. Part 2'
http://faculty.vassar.edu/lowry/PDF/c8p2.pdf
16c) Lowry R. 'Chapter 8: Chi-square procedures for the
analysis of categorical frequency data. Part 3' http://faculty.vassar.edu/lowry/PDF/c8p3.pdf
17) Raiter B. 'Prime number hide-and-seek: How the RSA
cipher works'
http://www.muppetlabs.com/~breadbox/txt/rsa.html
18) Eric Weisstein's Mathworld 'Benford's Law'
http://mathworld.wolfram.com/BenfordsLaw.html
19) Walker J. 'Ent: A pseudorandom number sequence test
program'
http://www.fourmilab.ch/random/
20) Hays W.L. 'Statistics' 5th Edition,
21) Efunda Engineering Fundamentals '
http://www.efunda.com/math/taylor_series/logarithmic.cfm
22) Kreyszig E. 'Advanced engineering mathematics' 4th
Edition John Wiley & Sons (1979)
23) Wikipedia 'Zipf's law' http://en.wikipedia.org/wiki/Zipfs_law
24) Hill T.P. 'A statistical derivation of the
significant-digit law' Stat. Sci. 10 354-363 (1995)
25) P.D. Scott, M. Fasli 'Benford's law: An empirical
investigation and a novel explanation' CSM Technical Report 349
http://cswww.essex.ac.uk/technical-reports/2001/CSM-349.pdf
26) Van der Galien J.G. 'A factorial randomness generator
(FRAG PRNG)' Scientia Araneae Totius Orbis 3.1.
27) Van der Galien J.G. 'Sample space for reals follow
Newcomb-Benford and Zipf' Scientia Araneae Totius Orbis 4.1.
Appendix
Table 3: The output
of
First digit distribution
of all factors of natural numbers
including 1 and n, squares
double. With Linear Regression and Chi-Square analysis.
Cumulative amount of
factors under 100=492
e(n) is the observed
cumulative amount of factors per digit
part1=
3.47560975609756E-0001 e1=171
part2=
1.78861788617886E-0001 e2=88
part3=
1.17886178861788E-0001 e3=58
part4=
9.34959349593496E-0002 e4=46
part5=
6.50406504065041E-0002 e5=32
part6=
5.48780487804878E-0002 e6=27
part7=
5.08130081300813E-0002 e7=25
part8=
4.67479674796748E-0002 e8=23
part9=
4.47154471544715E-0002 e9=22
Total=
1.00000000000000E+0000
Expected cumulative amount
of factors digit 1=148
Expected cumulative amount
of factors digit 2=87
Expected cumulative amount
of factors digit 3=61
Expected cumulative amount
of factors digit 4=48
Expected cumulative amount
of factors digit 5=39
Expected cumulative amount
of factors digit 6=33
Expected cumulative amount
of factors digit 7=29
Expected cumulative amount
of factors digit 8=25
Expected cumulative amount
of factors digit 9=23
CHI h1= 3.57432432432432E+0000
CHI h2=
1.14942528735632E-0002
CHI h3=
1.47540983606557E-0001
CHI h4=
8.33333333333333E-0002
CHI h5=
1.25641025641026E+0000
CHI h6=
1.09090909090909E+0000
CHI h7=
5.51724137931034E-0001
CHI h8=
1.60000000000000E-0001
CHI h9= 4.34782608695652E-0002
CHI-square=
6.91921464025773E+0000
slope
=-9.72906230000316E-0001
intersection
=-4.64032106346377E-0001
Correlation coefficient
=-9.95978305301610E-0001
Cumulative amount of
factors under 1000=7100
e(n) is the observed
cumulative amount of factors per digit
part1=
3.34366197183099E-0001 e1=2374
part2=
1.79295774647887E-0001 e2=1273
part3=
1.20845070422535E-0001 e3=858
part4=
9.46478873239437E-0002 e4=672
part5=
6.77464788732394E-0002 e5=481
part6=
5.87323943661972E-0002 e6=417
part7= 5.23943661971831E-0002
e7=372
part8=
4.78873239436620E-0002 e8=340
part9=
4.40845070422535E-0002 e9=313
Total=
1.00000000000000E+0000
Expected cumulative amount
of factors digit 1=2137
Expected cumulative amount
of factors digit 2=1250
Expected cumulative amount
of factors digit 3=887
Expected cumulative amount
of factors digit 4=688
Expected cumulative amount
of factors digit 5=562
Expected cumulative amount
of factors digit 6=475
Expected cumulative amount
of factors digit 7=412
Expected cumulative amount
of factors digit 8=363
Expected cumulative amount
of factors digit 9=325
CHI h1=
2.62840430510061E+0001
CHI h2=
4.23200000000000E-0001
CHI h3=
9.48139797068771E-0001
CHI h4=
3.72093023255814E-0001
CHI h5=
1.16743772241992E+0001
CHI h6=
7.08210526315789E+0000
CHI h7=
3.88349514563107E+0000
CHI h8=
1.45730027548209E+0000
CHI h9=
4.43076923076923E-0001
CHI-square=
5.25678307028779E+0001
slope
=-9.49139954699959E-0001
intersection
=-4.71479877812393E-0001
Correlation coefficient
=-9.98054367296489E-0001
Cumulative amount of
factors under 10000=93768
e(n) is the observed
cumulative amount of factors per digit
part1=
3.25974746182066E-0001 e1=30566
part2=
1.77704547393567E-0001 e2=16663
part3=
1.21853937377356E-0001 e3=11426
part4=
9.54376759662145E-0002 e4=8949
part5=
7.06104427949834E-0002 e5=6621
part6=
6.09802917839775E-0002 e6=5718
part7=
5.39843016807440E-0002 e7=5062
part8=
4.87906321986179E-0002 e8=4575
part9=
4.46634246224725E-0002 e9=4188
Total=
1.00000000000000E+0000
Expected cumulative amount
of factors digit 1=28227
Expected cumulative amount
of factors digit 2=16512
Expected cumulative amount
of factors digit 3=11715
Expected cumulative amount
of factors digit 4=9087
Expected cumulative amount
of factors digit 5=7425
Expected cumulative amount
of factors digit 6=6277
Expected cumulative amount
of factors digit 7=5438
Expected cumulative amount
of factors digit 8=4796
Expected cumulative amount
of factors digit 9=4291
CHI h1=
1.93818719665568E+0002
CHI h2=
1.38087451550388E+0000
CHI h3= 7.12940674349125E+0000
CHI h4=
2.09574116870254E+0000
CHI h5=
8.70593939393939E+0001
CHI h6=
4.97819021825713E+0001
CHI h7=
2.59977933063626E+0001
CHI h8=
1.01836947456213E+0001
CHI h9=
2.47238405965975E+0000
CHI-square=
3.79919910326875E+0002
slope =-9.25143717743863E-0001
intersection
=-4.80373761375731E-0001
Correlation coefficient
=-9.98895639943961E-0001
Cumulative amount of
factors under 100000=1167066
e(n) is the observed
cumulative amount of factors per digit
part1=
3.21242329054227E-0001 e1=374911
part2=
1.77480108237238E-0001 e2=207131
part3=
1.22468652158489E-0001 e3=142929
part4=
9.56972442004137E-0002 e4=111685
part5=
7.22409872278003E-0002 e5=84310
part6=
6.21241643574571E-0002 e6=72503
part7=
5.47372642164196E-0002 e7=63882
part8= 4.92037296948073E-0002
e8=57424
part9=
4.48055208531480E-0002 e9=52291
Total=
1.00000000000000E+0000
Expected cumulative amount
of factors digit 1=351322
Expected cumulative amount
of factors digit 2=205510
Expected cumulative amount
of factors digit 3=145812
Expected cumulative amount
of factors digit 4=113100
Expected cumulative amount
of factors digit 5=92410
Expected cumulative amount
of factors digit 6=78131
Expected cumulative amount
of factors digit 7=67680
Expected cumulative amount
of factors digit 8=59698
Expected cumulative amount
of factors digit 9=53402
CHI h1=
1.58384877975191E+0003
CHI h2=
1.27859520217994E+0001
CHI h3=
5.70027775491729E+0001
CHI h4=
1.77031388152078E+0001
CHI h5=
7.09988096526350E+0002
CHI h6= 4.05400980404705E+0002
CHI h7=
2.13132446808511E+0002
CHI h8=
8.66205903045328E+0001
CHI h9=
2.31137597842777E+0001
CHI-square=
3.10959652196646E+0003
slope
=-9.13937169064273E-0001
intersection
=-4.84462468715366E-0001
Correlation coefficient
=-9.99231411946420E-0001
Cumulative amount of
factors under 1000000=13971034
e(n) is the observed
cumulative amount of factors per digit
part1=
3.17889141204581E-0001 e1=4441240
part2=
1.77206139502631E-0001 e2=2475753
part3=
1.22882100208187E-0001 e3=1716790
part4= 9.59136596475250E-0002
e4=1340013
part5=
7.33948539528284E-0002 e5=1025402
part6=
6.29282700192412E-0002 e6=879173
part7=
5.52830234326250E-0002 e7=772361
part8=
4.95320532467389E-0002 e8=692014
part9=
4.49707587856418E-0002 e9=628288
Total=
1.00000000000000E+0000
Expected cumulative amount
of factors digit 1=4205700
Expected cumulative amount
of factors digit 2=2460177
Expected cumulative amount
of factors digit 3=1745523
Expected cumulative amount
of factors digit 4=1353933
Expected cumulative amount
of factors digit 5=1106244
Expected cumulative amount
of factors digit 6=935316
Expected cumulative amount
of factors digit 7=810207
Expected cumulative amount
of factors digit 8=714654
Expected cumulative amount
of factors digit 9=639279
CHI h1=
1.31914049028699E+0004
CHI h2=
9.86155776596562E+0001
CHI h3=
4.72973022412194E+0002
CHI h4=
1.43113728670473E+0002
CHI h5=
5.90776443894837E+0003
CHI h6=
3.37002301788914E+0003
CHI h7=
1.76784416328173E+0003
CHI h8=
7.17227637430141E+0002
CHI h9=
1.88966133722522E+0002
CHI-square=
2.58579326228841E+0004
slope
=-9.05478543219127E-0001
intersection
=-4.87634509667732E-0001
Correlation coefficient
=-9.99384864491395E-0001
Cumulative amount of
factors under 10000000=162728526
e(n) is the observed
cumulative amount of factors per digit
part1=
3.15517102391747E-0001 e1=51343633
part2=
1.77055644196027E-0001 e2=28812004
part3=
1.23173173706495E-0001 e3=20043789
part4=
9.60524155426812E-0002 e4=15630468
part5=
7.42102463338235E-0002 e5=12076124
part6= 6.34936925563991E-0002
e6=10332235
part7=
5.56627115272955E-0002 e7=9057911
part8=
4.97573056121703E-0002 e8=8096933
part9=
4.50777081333607E-0002 e9=7335429
Total=
1.00000000000000E+0000
Expected cumulative amount
of factors digit 1=48986167
Expected cumulative amount
of factors digit 2=28655071
Expected cumulative amount
of factors digit 3=20331096
Expected cumulative amount
of factors digit 4=15770024
Expected cumulative amount
of factors digit 5=12885047
Expected cumulative amount
of factors digit 6=10894152
Expected cumulative amount
of factors digit 7=9436944
Expected cumulative amount
of factors digit 8=8323975
Expected cumulative amount
of factors digit 9=7446049
CHI h1=
1.13453374320060E+0005
CHI h2=
8.59462762768935E+0002
CHI h3= 4.06005225930761E+0003
CHI h4=
1.23499350007330E+0003
CHI h5=
5.07841702035701E+0004
CHI h6=
2.89835055439836E+0004
CHI h7=
1.52237859087645E+0004
CHI h8=
6.19272279938371E+0003
CHI h9=
1.64339294570852E+0003
CHI-square=
2.22435460243621E+0005
slope
=-8.99624184409583E-0001
intersection
=-4.89840901540702E-0001
Correlation coefficient
=-9.99457780214643E-0001
Table 4: The Linear Regression and Chi-Square Tests performed on some suspected
Benford distributions found on the internet.18 (a = |slope|, intersection
= b and c = 10b)
Suspected Benford
Distribution = Benford formula (Program Test)
F1 = 3.0102999570E-01
F2 = 1.7609125910E-01
F3 = 1.2493873660E-01
F4 = 9.6910013000E-02
F5 = 7.9181246000E-02
F6 = 6.6946789600E-02
F7 = 5.7991947000E-02
F8 = 5.1152522400E-02
F9 = 4.5757490600E-02
slope =-8.6366558707E-01
intersection
=-5.0375129256E-01
Correlation coefficient
=-9.9922961953E-01
Suspected Benford
Distribution = Rivers, Area
slope =-8.5007232486E-01
intersection
=-5.1356871625E-01
Correlation coefficient
=-9.7179279069E-01
Chi-Square =
4.9617226594E+00
Suspected Benford
Distribution = Population
slope =-1.1823811930E+00
intersection
=-3.7250056401E-01
Correlation coefficient
=-9.7607292076E-01
Chi-Square =
1.1862939846E+02
Suspected Benford Distribution
= Constants
slope =-1.0157812073E+00
intersection
=-5.2066747111E-01
Correlation coefficient
=-6.9676575170E-01
Chi-Square =
2.4440656854E+01
Suspected Benford
Distribution = Specific Heat
slope =-9.5386971754E-01
intersection
=-4.7117207520E-01
Correlation coefficient
=-8.8848255381E-01
Chi-Square =
1.1121293917E+02
Suspected Benford
Distribution = Pressure
slope =-8.8961353901E-01
intersection
=-4.9167746996E-01
Correlation coefficient
=-9.9371644466E-01
Chi-Square =
1.2703741165E+00
Suspected Benford
Distribution = H.P. Lost
slope =-9.2398277231E-01
intersection
=-4.7632655922E-01
Correlation coefficient
=-9.8644255296E-01
Chi-Square =
3.4605628180E+00
Suspected Benford
Distribution = Molecular Weigt
slope =-1.1411996629E+00
intersection
=-3.8973413084E-01
Correlation coefficient
=-9.5494997221E-01
Chi-Square =
1.2575708356E+02
Suspected Benford
Distribution = Drainage
slope =-1.2128593722E+00
intersection
=-3.5741388937E-01
Correlation coefficient
=-9.2944807224E-01
Chi-Square =
1.1142218800E+01
Suspected Benford
Distribution = Atomic Weight
slope =-1.0642120481E+00
intersection
=-4.8758981189E-01
Correlation coefficient
=-8.8867625323E-01
Chi-Square =
1.7245691826E+01
Suspected Benford
Distribution = n^-1 and sqrt(n)
slope =-5.9540500961E-01
intersection
=-6.4345712127E-01
Correlation coefficient
=-8.5045516020E-01
Chi-Square =
4.4076412324E+02
Suspected Benford
Distribution = Design
slope =-6.6096834428E-01
intersection
=-5.9960139086E-01
Correlation coefficient
=-9.6027583379E-01
Chi-Square =
1.9212693139E+01
Suspected Benford
Distribution = Reader's Digest
slope =-9.4238206892E-01
intersection
=-4.7591090015E-01
Correlation coefficient
=-9.9364779402E-01
Chi-Square =
3.2271432117E+00
Suspected Benford Distribution
= Cost Data
slope =-9.8005651446E-01
intersection
=-4.5804015636E-01
Correlation coefficient
=-9.6766307822E-01
Chi-Square =
1.5601254902E+01
Suspected Benford
Distribution = X-Ray
slope =-8.2429874377E-01
intersection
=-5.2065454636E-01
Correlation coefficient
=-9.8620626973E-01
Chi-Square =
5.4256271820E+00
Suspected Benford
Distribution = American League
slope =-9.8581811852E-01
intersection
=-4.5294564119E-01
Correlation coefficient
=-9.8110932937E-01
Chi-Square =
1.4595355311E+01
Suspected Benford
Distribution = Black Body
slope =-8.7963246096E-01
intersection
=-5.0068542909E-01
Correlation coefficient
=-9.8472190189E-01
Chi-Square =
9.5229019643E+00
Suspected Benford
Distribution = Addresses
slope =-8.5696179253E-01
intersection
=-5.0719069791E-01
Correlation coefficient
=-9.9332558555E-01
Chi-Square =
1.2966139294E+00
Suspected Benford
Distribution = n^1,n^2....n!
slope =6.4992448601E-01
intersection
=-6.0023561887E-01
Correlation coefficient
=-9.9052764917E-01
Chi-Square =
2.4993708303E+01
Suspected Benford
Distribution = Death Rate
slope =-8.6699419233E-01
intersection
=-5.0216308616E-01
Correlation coefficient
=-9.7115965914E-01
Chi-Square =
7.5549793791E+00
Suspected Benford
Distribution = Average over 20 distributions
Samplesize = 20229
slope =-8.9874592240E-01
intersection
=-4.8940098622E-01
Correlation coefficient
=-9.9661451810E-01
Chi-Square =
3.5050637915E+01
Suspected Benford
Distribution = Factorial First Digit Distribution
Samplesize = 162728526
F1 = 3.1551710239E-01
F2 = 1.7705564420E-01
F3 = 1.2317317371E-01
F4 = 9.6052415543E-02
F5 = 7.4210246334E-02
F6 = 6.3493692557E-02
F7 = 5.5662711527E-02
F8 = 4.9757305612E-02
F9 = 4.5077708133E-02
slope =-8.9962418441E-01
intersection
=-4.8984090154E-01
Correlation coefficient
=-9.9945778023E-01
Chi-Square =
2.2243549463E+05