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.
SATOCONOR.COM
J.G. van der Galiën ‘Factorial Randomness’ 2.3. (2003)
Communication to the editor
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: johan.van.der.galien@satoconor.com)
Six first digit distributions of the factors of the
natural numbers up to n=10^{d} (d Î
{2,3,4,5,6,7}) were calculated by means of a selfwritten
The ChiSquare increased rapidly, from 6.91 for n=10^{2
}and 47.1 for n=10^{3} to 222435 for n=10^{7} Indicating that
only for n=10^{2} Benford's Law is followed. (It is a well known fact,
and you will find it referenced in this article, that ChiSquare 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 (f_{D}) correlates well with f_{D} = 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
log_{10}(f_{D}) = alog_{10}(D) + b asymptotically goes
to 1 with sample size. from R = 0.996 for n=10^{2} to R = 0.9995 for
n = 10^{7}. 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 log_{10}(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.^{37,9,14}
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 yet^{17}, 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 ChiSquare
GoodnessOfFit^{8} analysis can be interpreted as follows.
According to this test there is only a fit for the smallest sample n = 10^{2}.
The ChiSquare (6.91) then falls within the 5% and 95% Confidence Interval >
2.71 and < 15.51.^{10,15,22} With increasing sample size the
ChiSquare also increases to 222435 for n = 10^{7}. But it is a well
known fact that ChiSquare 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 ChiSquares of other suspected
Benford distributions which can be found on the internet.^{18}
These results are shown in Table 4. The ChiSquares 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 ChiSquare is below
5% Confidence Level or above the 95%, the distribution is suspect (in other
reject the H_{0}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=10^{7}.
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).
Log_{10}(1+1/D) 
Log_{10}(D) 
Log_{10}(log_{10}(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 log_{10}(log_{10}(1+1/D)) = alog_{10}(D) + b 

Correlation Coefficient R = 
0.9992296195 

Slope = a = 
0.8636655870 

Intersection with Yaxis 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:
log_{10}(log_{10}(1+1/D)) = alog_{10}(D) + b Þ log_{10}(1+1/D) » 10^{b}D^{a} = cD^{a} (1.)
f_{D} = log_{10}(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: log_{10}(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.)
log_{10}(x)
= ln(x)/ln(10) (5.)
ln(x) = (x1)/x+1/2((x1)/x)^{2}+1/3((x1)/x)^{3}+
…….. (
ln(x) = (x1)/x+(x1)^{2}/2x^{2}+(x1)^{3}/3x^{3}+
……. 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.)
log_{10}(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 = 10^{7} are in good
agreement with the ones found for the Linear Regression analysis of Benford's
formula (2.):
f_{D} » 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 = 10^{7}.
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:
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 ChiSquare analysis.
And randomness from an interaction between chaos and order for which the first
digit distribution concurs with NewcombBenford's and Zipf's Law. But which
does not pass the ChiSquare test at large sample sizes for accepting the H_{0}hypothesis
that the distribution follows the log_{10}(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
NewcombBenfordZipf 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?' SATOCONOR.COM 1.2. http://www.satoconor.com/
6) Eric Weisstein's World of Physics 'Einstein Rosen
Podolsky paradox'
http://scienceworld.wolfram.com/physics/EinsteinPodolskyRosenParadox.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/primechaos.html
10) Kreyzig E. ‘Advanced
engineering mathematics: Table A12 chisquare distribution’ 4^{th}
edition, John Wiley and Sons (1979)
11) Not used.
12) Not used.
13) Not used.
14) Anonymous 'The hiddenvariable theory of David Bohm'
http://www.metalibrary.net/ghcobs/hidvarframe.html
15) Walker J. 'Chisquare calculator'
http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html
16a) Lowry R. 'Chapter 8: Chisquare procedures for the
analysis of categorical frequency data. Part 1'
http://faculty.vassar.edu/lowry/PDF/c8p1.pdf
16b) Lowry R. 'Chapter 8: Chisquare procedures for the
analysis of categorical frequency data. Part 2'
http://faculty.vassar.edu/lowry/PDF/c8p2.pdf
16c) Lowry R. 'Chapter 8: Chisquare procedures for the
analysis of categorical frequency data. Part 3' http://faculty.vassar.edu/lowry/PDF/c8p3.pdf
17) Raiter B. 'Prime number hideandseek: 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
significantdigit law' Stat. Sci. 10 354363 (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/technicalreports/2001/CSM349.pdf
26) Van der Galien J.G. 'A factorial randomness generator
(FRAG PRNG)' SATOCONOR.COM 3.1.
27) Van der Galien J.G. 'Sample space for reals follow
NewcombBenford and Zipf' SATOCONOR.COM 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 ChiSquare analysis.
Cumulative amount of
factors under 100=492
e(n) is the observed
cumulative amount of factors per digit
part1=
3.47560975609756E0001 e1=171
part2=
1.78861788617886E0001 e2=88
part3=
1.17886178861788E0001 e3=58
part4=
9.34959349593496E0002 e4=46
part5=
6.50406504065041E0002 e5=32
part6=
5.48780487804878E0002 e6=27
part7=
5.08130081300813E0002 e7=25
part8=
4.67479674796748E0002 e8=23
part9=
4.47154471544715E0002 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=
1.14942528735632E0002
CHI h3=
1.47540983606557E0001
CHI h4=
8.33333333333333E0002
CHI h5=
1.25641025641026E+0000
CHI h6=
1.09090909090909E+0000
CHI h7=
5.51724137931034E0001
CHI h8=
1.60000000000000E0001
CHI h9= 4.34782608695652E0002
CHIsquare=
6.91921464025773E+0000
slope
=9.72906230000316E0001
intersection
=4.64032106346377E0001
Correlation coefficient
=9.95978305301610E0001
Cumulative amount of
factors under 1000=7100
e(n) is the observed
cumulative amount of factors per digit
part1=
3.34366197183099E0001 e1=2374
part2=
1.79295774647887E0001 e2=1273
part3=
1.20845070422535E0001 e3=858
part4=
9.46478873239437E0002 e4=672
part5=
6.77464788732394E0002 e5=481
part6=
5.87323943661972E0002 e6=417
part7= 5.23943661971831E0002
e7=372
part8=
4.78873239436620E0002 e8=340
part9=
4.40845070422535E0002 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=
4.23200000000000E0001
CHI h3=
9.48139797068771E0001
CHI h4=
3.72093023255814E0001
CHI h5=
1.16743772241992E+0001
CHI h6=
7.08210526315789E+0000
CHI h7=
3.88349514563107E+0000
CHI h8=
1.45730027548209E+0000
CHI h9=
4.43076923076923E0001
CHIsquare=
5.25678307028779E+0001
slope
=9.49139954699959E0001
intersection
=4.71479877812393E0001
Correlation coefficient
=9.98054367296489E0001
Cumulative amount of
factors under 10000=93768
e(n) is the observed
cumulative amount of factors per digit
part1=
3.25974746182066E0001 e1=30566
part2=
1.77704547393567E0001 e2=16663
part3=
1.21853937377356E0001 e3=11426
part4=
9.54376759662145E0002 e4=8949
part5=
7.06104427949834E0002 e5=6621
part6=
6.09802917839775E0002 e6=5718
part7=
5.39843016807440E0002 e7=5062
part8=
4.87906321986179E0002 e8=4575
part9=
4.46634246224725E0002 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
CHIsquare=
3.79919910326875E+0002
slope =9.25143717743863E0001
intersection
=4.80373761375731E0001
Correlation coefficient
=9.98895639943961E0001
Cumulative amount of
factors under 100000=1167066
e(n) is the observed
cumulative amount of factors per digit
part1=
3.21242329054227E0001 e1=374911
part2=
1.77480108237238E0001 e2=207131
part3=
1.22468652158489E0001 e3=142929
part4=
9.56972442004137E0002 e4=111685
part5=
7.22409872278003E0002 e5=84310
part6=
6.21241643574571E0002 e6=72503
part7=
5.47372642164196E0002 e7=63882
part8= 4.92037296948073E0002
e8=57424
part9=
4.48055208531480E0002 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
CHIsquare=
3.10959652196646E+0003
slope
=9.13937169064273E0001
intersection
=4.84462468715366E0001
Correlation coefficient
=9.99231411946420E0001
Cumulative amount of
factors under 1000000=13971034
e(n) is the observed
cumulative amount of factors per digit
part1=
3.17889141204581E0001 e1=4441240
part2=
1.77206139502631E0001 e2=2475753
part3=
1.22882100208187E0001 e3=1716790
part4= 9.59136596475250E0002
e4=1340013
part5=
7.33948539528284E0002 e5=1025402
part6=
6.29282700192412E0002 e6=879173
part7=
5.52830234326250E0002 e7=772361
part8=
4.95320532467389E0002 e8=692014
part9=
4.49707587856418E0002 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
CHIsquare=
2.58579326228841E+0004
slope
=9.05478543219127E0001
intersection
=4.87634509667732E0001
Correlation coefficient
=9.99384864491395E0001
Cumulative amount of
factors under 10000000=162728526
e(n) is the observed
cumulative amount of factors per digit
part1=
3.15517102391747E0001 e1=51343633
part2=
1.77055644196027E0001 e2=28812004
part3=
1.23173173706495E0001 e3=20043789
part4=
9.60524155426812E0002 e4=15630468
part5=
7.42102463338235E0002 e5=12076124
part6= 6.34936925563991E0002
e6=10332235
part7=
5.56627115272955E0002 e7=9057911
part8=
4.97573056121703E0002 e8=8096933
part9=
4.50777081333607E0002 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
CHIsquare=
2.22435460243621E+0005
slope
=8.99624184409583E0001
intersection
=4.89840901540702E0001
Correlation coefficient
=9.99457780214643E0001
Table 4: The Linear Regression and ChiSquare Tests performed on some suspected
Benford distributions found on the internet.^{18} (a = slope, intersection
= b and c = 10^{b})
Suspected Benford
Distribution = Benford formula (Program Test)
F1 = 3.0102999570E01
F2 = 1.7609125910E01
F3 = 1.2493873660E01
F4 = 9.6910013000E02
F5 = 7.9181246000E02
F6 = 6.6946789600E02
F7 = 5.7991947000E02
F8 = 5.1152522400E02
F9 = 4.5757490600E02
slope =8.6366558707E01
intersection
=5.0375129256E01
Correlation coefficient
=9.9922961953E01
Suspected Benford
Distribution = Rivers, Area
slope =8.5007232486E01
intersection
=5.1356871625E01
Correlation coefficient
=9.7179279069E01
ChiSquare =
4.9617226594E+00
Suspected Benford
Distribution = Population
slope =1.1823811930E+00
intersection
=3.7250056401E01
Correlation coefficient
=9.7607292076E01
ChiSquare =
1.1862939846E+02
Suspected Benford Distribution
= Constants
slope =1.0157812073E+00
intersection
=5.2066747111E01
Correlation coefficient
=6.9676575170E01
ChiSquare =
2.4440656854E+01
Suspected Benford
Distribution = Specific Heat
slope =9.5386971754E01
intersection
=4.7117207520E01
Correlation coefficient
=8.8848255381E01
ChiSquare =
1.1121293917E+02
Suspected Benford
Distribution = Pressure
slope =8.8961353901E01
intersection
=4.9167746996E01
Correlation coefficient
=9.9371644466E01
ChiSquare =
1.2703741165E+00
Suspected Benford
Distribution = H.P. Lost
slope =9.2398277231E01
intersection
=4.7632655922E01
Correlation coefficient
=9.8644255296E01
ChiSquare =
3.4605628180E+00
Suspected Benford
Distribution = Molecular Weigt
slope =1.1411996629E+00
intersection
=3.8973413084E01
Correlation coefficient
=9.5494997221E01
ChiSquare =
1.2575708356E+02
Suspected Benford
Distribution = Drainage
slope =1.2128593722E+00
intersection
=3.5741388937E01
Correlation coefficient
=9.2944807224E01
ChiSquare =
1.1142218800E+01
Suspected Benford
Distribution = Atomic Weight
slope =1.0642120481E+00
intersection
=4.8758981189E01
Correlation coefficient
=8.8867625323E01
ChiSquare =
1.7245691826E+01
Suspected Benford
Distribution = n^1 and sqrt(n)
slope =5.9540500961E01
intersection
=6.4345712127E01
Correlation coefficient
=8.5045516020E01
ChiSquare =
4.4076412324E+02
Suspected Benford
Distribution = Design
slope =6.6096834428E01
intersection
=5.9960139086E01
Correlation coefficient
=9.6027583379E01
ChiSquare =
1.9212693139E+01
Suspected Benford
Distribution = Reader's Digest
slope =9.4238206892E01
intersection
=4.7591090015E01
Correlation coefficient
=9.9364779402E01
ChiSquare =
3.2271432117E+00
Suspected Benford Distribution
= Cost Data
slope =9.8005651446E01
intersection
=4.5804015636E01
Correlation coefficient
=9.6766307822E01
ChiSquare =
1.5601254902E+01
Suspected Benford
Distribution = XRay
slope =8.2429874377E01
intersection
=5.2065454636E01
Correlation coefficient
=9.8620626973E01
ChiSquare =
5.4256271820E+00
Suspected Benford
Distribution = American League
slope =9.8581811852E01
intersection
=4.5294564119E01
Correlation coefficient
=9.8110932937E01
ChiSquare =
1.4595355311E+01
Suspected Benford
Distribution = Black Body
slope =8.7963246096E01
intersection
=5.0068542909E01
Correlation coefficient
=9.8472190189E01
ChiSquare =
9.5229019643E+00
Suspected Benford
Distribution = Addresses
slope =8.5696179253E01
intersection
=5.0719069791E01
Correlation coefficient
=9.9332558555E01
ChiSquare =
1.2966139294E+00
Suspected Benford
Distribution = n^1,n^2....n!
slope =6.4992448601E01
intersection
=6.0023561887E01
Correlation coefficient
=9.9052764917E01
ChiSquare =
2.4993708303E+01
Suspected Benford
Distribution = Death Rate
slope =8.6699419233E01
intersection
=5.0216308616E01
Correlation coefficient
=9.7115965914E01
ChiSquare =
7.5549793791E+00
Suspected Benford
Distribution = Average over 20 distributions
Samplesize = 20229
slope =8.9874592240E01
intersection
=4.8940098622E01
Correlation coefficient
=9.9661451810E01
ChiSquare =
3.5050637915E+01
Suspected Benford
Distribution = Factorial First Digit Distribution
Samplesize = 162728526
F1 = 3.1551710239E01
F2 = 1.7705564420E01
F3 = 1.2317317371E01
F4 = 9.6052415543E02
F5 = 7.4210246334E02
F6 = 6.3493692557E02
F7 = 5.5662711527E02
F8 = 4.9757305612E02
F9 = 4.5077708133E02
slope =8.9962418441E01
intersection
=4.8984090154E01
Correlation coefficient
=9.9945778023E01
ChiSquare =
2.2243549463E+05