Let’s examine the 10 by 10 multiplication table that we were forced to memorize at elementary school.

We view the table as a DATA SET.

This is equivalent to the random game:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} x {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Quantitative Analysis of the

10x10 Multiplication Table

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The entire range of (1, 100) is partitioned

into 10 equitable territorial sections

(1, 10), (11, 20), (21, 30), … (81, 90), (91, 100) and a count is made of the numbers falling within each section - namely grouping them according to quantities.

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Quantitative Territorial Partition of the 10 by 10 Table

Histogram of Relative Quantities

{27%, 19%, 15%, 11%, 9%, 6%, 5%, 4%, 3%, 1%}

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Let’s improve the digital results of our Multiplication Table

We need repeated multiplications of the multiplication table for perfect Benford behavior.

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

times

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

times

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

times

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Equivalent to the game:

(represented by the Lognormal)

{29.9, 17.9, 12.6, 10.5, 7.2, 6.2, 5.6, 5.6, 4.7}

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{29.9, 17.9, 12.6, 10.5, 7.2, 6.2, 5.6, 5.6, 4.7}

Normal(7, 1)*Normal(2, 2)

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Exponential(5)*Exponential(0.08)

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Uniform(0, 3)*Uniform(5, 11)

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Exponential(3)*Uniform(0, 100000)

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Normal(0, 1)*Uniform(0, 1)

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Dice₁*Dice₂*Dice₃*Dice₄

RANDOM MULTIPLICATION PROCESSES

Empirical Statistical Evidence via

Monte Carlo Computer Simulations:

Uniform(0, 1)*Uniform(0, 100): {24.1, 18.0, 14.4, 11.9, 9.6, 7.7, 6.5, 4.3, 3.7}

Uniform(0, 30)*Uniform(0, 60): {28.9, 14.1, 11.3, 10.0, 9.5, 8.3, 7.1, 6.3, 4.6}

Uniform(0, 33)*Uniform(0, 70): {34.6, 12.4, 10.4, 8.5, 8.0, 7.8, 6.9, 6.0, 5.5}

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Normal(2, 9)*Normal(5, 13): {31.7, 17.6, 11.3, 9.7, 7.8, 6.0, 5.8, 5.4, 4.9}

Normal(4, 7)*Normal(2, 3): {28.7, 18.5, 13.1, 10.2, 7.7, 6.6, 5.9, 4.9, 4.3}

Normal(2, 4)*Normal(5, 3): {29.0, 19.3, 13.3, 10.6, 8.1, 6.4, 4.7, 4.9, 3.7}

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Exponential(4)*Exponential(11.0): {30.1, 17.6, 12.8, 9.9, 7.3, 7.1, 6.0, 4.7, 4.5}

Exponential(5)*Exponential(0.07): {30.2, 17.5, 12.7, 9.3, 9.4, 7.3, 5.8, 3.8, 3.9}

Exponential(13)*Exponential(0.2): {30.4, 17.1, 12.9, 9.9, 7.5, 6.9, 5.7, 5.3, 4.3}

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Benford’s Law 1st Leading Digits: {30.1, 17.6, 12.5, 9.7, 7.9, 6.7, 5.8, 5.1, 4.6}

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Conclusion: There exists a strong Benfordian tendency here!

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Multiplication processes produce sets of numbers favoring small quantities.

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Confirming the motto:

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Small is Beautiful

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Central Limit Theorem

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Adding many IID variables leads to the Normal distribution in the limit.

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Sum = X₁+X₂+X₃+… = Normal(m, sd)

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Multiplicative Central Limit Theorem

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Multiplying many variables leads to the Lognormal distribution in the limit.

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Product = X₁*X₂*X₃*… = Lognormal(s, l)

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e^X1+X2+X3+… =(e^x1)(e^x2)(e^x3)…

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e^Normal = Lognormal

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Lognormal =(e^x1)(e^x2)(e^x3)…

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Lognormal with high shape parameter (shape > 1) is perfectly Benford for all practical purposes.

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Why?

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Because the Lognormal is ‘made of’ multiplications !

On a more profound level, the typical multiplicative form of the equations in physics, chemistry, astronomy, and other disciplines, as well as those of their many applications and results, lead to the manifestation of Benford’s Law in the physical world.

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Newton gave us F = M*A,

not F = M + A.

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He gave us F_G = G*M₁*M₂ / R²,

not F_G = G + M₁ + M₂ - R²

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and such is the state of affair in so many other physical expressions.

======================

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Data Aggregation

&

Chains of Distributions

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Data Set A: {2, 3, 5, 7}

Data Set B: {1, 4, 6, 9, 13, 14}

Data Set C: {2, 6, 7, 9, 11, 15, 16, 21}

Data Set D: {1, 2, 6, 8, 13, 14, 19, 23, 25}

Data Set E: {3, 4, 8, 12, 15, 19, 22, 24, 29, 35, 41}

Data Set F: {1, 5, 8, 11, 12, 17, 19, 24, 27, 32, 38, 43, 47}

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The combined data set A, B, C, D, E, F:

{2, 3, 5, 7, 1, 4, 6, 9, 13, 14, 2, 6, 7, 9, 11, 15, 16, 21, 1, 2, 6, 8, 13, 14, 19, 23, 25, 3, 4, 8, 12, 15, 19, 22, 24, 29, 35, 41, 1, 5, 8, 11, 12, 17, 19, 24, 27, 32, 38, 43, 47}

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A good example is house number of address data

{1, 2, 3, 4, 5, 6} - Floral Drive

{1, 2, 3, 4, 5, 6, 7, 8 ,9 ,10 ,11 ,12, 13} - Pine Avenue

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} - Main Street

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} - South Street

{1, 2, 3, 4, 5, 6, 7, 8, 9} - Lodge Street

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Example: Prince Edward Island in Eastern Canada

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http://www.gov.pe.ca/civicaddress/download/

There are 23,633 addresses there:

1 Chestnut Street

1 Wellington Road

9 Pine Drive

15 Duck Pond Lane

27 Brooks Avenue

95 Kelly Drive

121 Beaton Rd - Route 138

318 Beaton Rd - Route 138

2257 Beaton Rd - Route 138

43015 Western Rd - Route 2

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Chains of Distributions

Classic Random Distributions:

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Normal( constant mean, constant s.d.)

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The parameter is a constant number

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Unorthodox Random Distributions:

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Normal( random mean, random s.d.)

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The parameter itself is a random number!

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Uniform(a, b)

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Parameter a = 0

parameter b = Uniform(5, 27)

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Uniform(0, Uniform(5, 27))

A Chain of 2 Uniform Distributions

A Chain of 4 Uniform Distributions

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Uniform(0, Uniform(0, Uniform(0, Uniform(0, 31))))

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More generally:

A Chain of 4 Uniform Distributions

20,000 simulated values

Quantitative Histogram of Chain of

4 Uniform Distributions

Small is beautiful !

The concept is by far more general, not limited to Uniform distributions. The figure below is but one manifestation of such possible constructions:

1st Chain Conjecture:

An infinite chain of parametrical dependencies is Benford.

2nd Chain Conjecture:

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AnyDensity(AnyBenford) = Benford

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Namely, if the parameter is Benford,

then the chain is also Benford, immediately!

Which distributions/parameters are chain-able, and which are not?

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In extreme generality:

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Scale parameters such as λX or X/λ

(divisions & multiplications)

YES!

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Location parameters such as X – μ

(subtractions)

YES!

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Shape parameters such as X^k

(powers)

NO!

MORE PRECISELY:

A parameter that does not continuously involve itself in the expression of centrality [mean, median, midpoint, Tukey’s Biweight, etc.] is not chain-able at all.

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‘not continuously involved’ means that

∂(center) / ∂(parameter) = 0

==========================

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Quantitative Partitions

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5 ← few big parts

4 + 1 ← few big parts

3 + 2

3 + 1 + 1

2 + 2 + 1

2 + 1 + 1 + 1 ← many small parts

1 + 1 + 1 + 1 + 1 ← many small parts

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The Organization of All Possible Integer Partitions of 5

A Histogram of Sorts

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Small is beautiful !

The small is beautiful

in almost ALL partition models!

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Benford’s Law is found

in MANY partition models!

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The next figure depicts one possible random partition in the natural world where:

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approximately 1/3 of the entire oval area consists of big parts (around the left side);

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approximately 1/3 of the entire oval area consists of small parts (around the center);

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approximately 1/3 of the entire oval area consists of medium parts (around the right side);

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namely endowing equal portions of overall quantity fairly to each size without any bias.

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An Equitable Mix of Small, Medium, and Big Yielding ‘Small is Beautiful’

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END

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ROCK BREAKING

Randomly breaking a 1000-kilogram rock repeatedly into small pieces in 12 stages, yielding 2¹² = 4096 pieces.

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The probability distribution Uniform(0, 1) is used to determine how each piece is being broken.

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One Monte Carlo simulation result yielded:

etc.

Resultant percent % digital proportions after each stage:

First digits for the 4096 pieces after the 12^th stage:

Resultant ordered data of all 64 pieces after the 6^th stage

The small clearly outnumbers the big!

Adjusted Histogram of Relative Quantities

Small is beautiful !

This reminds us of the Multiplicative Central Limit Theorem, and hence of the Lognormal, except for the dependencies between the products sharing some identical Us.

[Starting with 1-kilogram rock]

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1

-------------------------------------------------------------------------------------------------------

U1, (1-U1)

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U2*(U1), (1-U2)*U1, U3*(1-U1), (1-U3)*(1-U1)

-------------------------------------------------------------------------------------------------------

U4*(U2*U1), (1-U4)*(U2*U1), U5*(1-U2)*U1, (1-U5)*(1-U2)*U1,

U6*U3*(1-U1), (1-U6)*U3*(1-U1), U7*(1-U3)*(1-U1), (1-U7)*(1-U3)*(1-U1)

-------------------------------------------------------------------------------------------------------

etc. etc.

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=================

SMALL IS BEAUTIFUL

The Small is Numerous but the Big is Rare

Benford’s Law is a consequence of the

larger and more prevalent SMALL IS BEAUTIFUL phenomenon

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We have 3 very distinct explanations, yes they

all lead to the same exact

LOG(1 + 1/d) result!

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Magic!

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Imagine six chefs, French, Italian, Thai, Chinese, Japanese, and African, on different continents, using totally different ingredients, cooking styles, pot sizes, and flame sizes; yet they come up with six dishes totally equal in taste!

Incredible!

They all taste the same!

END