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Benford’s Law: An Overview

Updated Fall 2021

The Basics

Benford’s Law predicts the expected digit frequencies in a list of numbers.

First digit has been emphasized, but it can be used for all digits.

1 2 , 3 6 7

First digit

Benford’s Law can answer the question: What’s the likelihood of 1 being the 1st digit in a set of numbers?

The Basics

History

Frank Benford, a physicist at GE, noted that the pages of logarithm table books* covering numbers with initial digits 1 and 2 were more worn and dirty that for the pages for 7, 8, and 9.

Benford hypothesized that worn and dirty pages were that way because there were more numbers with low first digits.

Benford forumulated the expected frequencies of the various digits in lists of numbers using a mathematical assumption based on geometric sequences.

*log tables were used to do large multiplication problems; since multiplication is converted to addition after taking the log of the factors.

The Basics

His findings: A higher probability of lower digits in the 1st, 2nd, 3rd position, but the probabilities are much lower and closer together for about the 4th position onwards.

POINT: Digits are not necessarily distributed uniformly over 0 to 9 (or 1 to 9 for the first digit).

Benford’s Law Formulae

First digit

P(D1=d1) = log10(1 + (1/d1))

We’ll confine ourselves to the 1st digit in the case.

Benford’s Law Formulae

Second digit

P(D2=d2) =  log10(1 + (1/d1d2))

d1=1

Note that you sum over the 1st digit. If you want P(D2=1), you need to account for: 11, 21, 31, 41, 51, 61, 71, 81, 91.

Benford’s Law Formulae

First 2 digits

P(D1D2=d1d2) = log10(1 + (1/d1d2))

Benford 1st Digit Proportions

First Digit	Benford Proportion
1	0.30103000 [log10(2/1)]
2	0.17609126 [log10(3/2)]
3	0.12493874 [log10(4/3)]
4	0.09691001 [log10(5/4)]
5	0.07918125 [log10(6/5)]
6	0.06694679 [log10(7/6)]
7	0.05799195 [log10(8/7)]
8	0.05115252 [log10(9/8)]
9	0.04574749 [log10(10/9)]

Naïve Number Generators

Naïve persons tend to make up numbers that do not conform to Benford’s Law.

This includes naïve fraudsters --- this is the power of Benford’s Law.

They tend to believe that 1st digits are distributed approximately uniformly (equal # of 1s, 2s, 3s, ….., 9s).

Real World Look at Naïve Number Generators

Naïve Number Generators

I’ve had Fraud Exam students create fraudulent sales numbers: Pretend that you are a budding fraudster --- just for a moment, not permanently. You need to “boost” sales for the year to meet analysts’ expectations. Your real sales are $13,000,000 and you need to bump them up by about 1.5% ($200,000) to get to the expectation of about $13,200,000.

Student-Generated Fraud Numbers

Naïve Number Generators

You have 2,031 legitimate sales transactions that range from $1,000 to $10,000 for the year. If you were to create 30 fictitious sales amounts to boost your revenue (about $200,000) what amounts would you create (list the 30 fictitious sales amounts you would record in the books and records to perpetrate the fraud)?

Student-Generated Fraud Numbers

Student First Digit vs. Benford's Law

Class % 5.1282051282051294E-2 5.3846153846153898E-2 7.1794871794871831E-2 0.1051282051282052 0.11794871794871803 0.1807692307692311 0.14487179487179491 0.15128205128205141 0.12307692307692321 Benford 0.30102999566398153 0.17609125905568124 0.1249387366083 9.6910013008056448E-2 7.9181246047624901E-2 6.694678963061329E-2 5.7991946977686802E-2 5.1152522447381339E-2 4.5757490560675185E-2

First Digit

Percent

Benford’s Law Applied to Real World Phenomena

Illinois Cities Populations (Source: U.S. Census Bureau, Census 2010) – Actual Application of Benford’s Law

2 = 1.989 (equal distributions)

M.A.D. = 0.000322; close conformity per Nigrini

Illinois Cities 1st Digit Proportions versus Benford 1st Digit Proportions

Illinois Cities 0.30200308166409862 0.17873651771956856 0.12249614791987673 9.4761171032357469E-2 8.3204930662557783E-2 6.7796610169491525E-2 6.2403697996918334E-2 4.8536209553158703E-2 4.0061633281972264E-2 Benford 0.3010299956639812 0.17609125905568124 0.12493873660829993 9.691001300805642E-2 7.9181246047624818E-2 6.6946789630613221E-2 5.7991946977686733E-2 5.1152522447381291E-2 4.5757490560675143E-2

First Digit

Proportion

Intuitive Explanation of Benford’s Law

Use a census data example:

Consider a city with a population of 10,000.

Assume the population is growing at 10% per year.

At 10,000 the first digit 1 will persist until the population doubles to 20,000 (7.27254 years) and the 1st digit goes to 2.

19,999  to  20,000

At 20,000 the first digit 2 will persist until the population goes up 50% to 30,000 (4.254164 years) and the 1st digit goes to 3.

29,999  to  30,000

Intuitive Explanation of Benford’s Law

Use a census data example:

At 30,000 the first digit 3 will persist until the population increases by 1/3 to 40,000 (3.018377 years) and the 1st digit goes to 4.

At 40,000 the first digit 4 will persist until the population increases by 1/4 to 50,000 (2.34124 years) and the 1st digit goes to 5.

So on and so on --- next slide ……..

Intuitive Explanation of Benford’s Law

Use a census data example:

50,000 to 60,000: 1.91293 years.

60,000 to 70,000: 1.61736 years.

70,000 to 80,000: 1.40102 years.

80,000 to 90,000: 1.23579 years.

90,000 to 100,000: 1.10545 years.

From Steps to Benford Proportion

From-To Steps (Yrs.)

10-20 7.27254

20-30 4.25416

30-40 3.01838

40-50 2.34124

50-60 1.91293

60-70 1.61736

70-80 1.40102

80-90 1.23579

90-100 1.10545

Totals 24.15886

(1,000s)

Going from 1 as the 1st digit to 1 as the 1st digit.

Math Fun. Check 24.15886 years to go from 10,000 to 100,000 at a 10% annual growth rate.

10,000x1.1y = 100,000

1.1y = 10

yLog(1.1) = Log(10)

y = Log(10)/Log(1.1)

y = 1/Log(1.1)

y = 1/0.041392685

y = 24.15886

Bingo!

1+Growth Rate

From Steps to Benford Proportion

From-To Steps (Yrs.) Proportion

10-20 7.27254❶ 0.30103 (❶ 7.27254 ÷ ❷ 24.15886)

20-30 4.25416 0.17609

30-40 3.01838 0.12494

40-50 2.34124 0.09691

50-60 1.91293 0.07918

60-70 1.61736 0.06695

70-80 1.40102 0.05799

80-90 1.23579 0.05115

90-100 1.10545 0.04576

Totals 24.15886❷ 1.00000

Look

Familiar?

Benford Proportions

(1,000s)

Going from 1 as the 1st digit to 1 as the 1st digit.

Steps - Years

How’d I get the number of years that it takes for 10,000 to grow to 20,000 at a 10% annual growth rate?

What do we know?

10,000  1.1y = 20,000; y=number of yrs.

Solve for y: 1.1y = 20,000/10,000 = 2

Log10(Xy) = yLog10X or yLog10(1.1)=Log10(2)

y = Log10(2)/Log10(1.1) = 0.30103/0.041393 = 7.27254 years.

Growth Rate

Intuitive Explanation of Benford’s Law-Example (10% Population Growth – 101 periods)

If we grew more periods, the proportions would converge to Benford proportions.

Intuitive Example n=101

Population 1 2 3 4 5 6 7 8 9 0.32673267326732675 0.16831683168316833 0.11881188118811881 9.9009900990099015E-2 6.9306930693069313E-2 5.9405940594059403E-2 5.9405940594059403E-2 5.9405940594059403E-2 3.9603960396039604E-2 Benford 1 2 3 4 5 6 7 8 9 0.30103000000000002 0.17609 0.12494 9.6909999999999996E-2 7.918E-2 6.6949999999999996E-2 5.799E-2 5.1150000000000001E-2 4.5760000000000002E-2

First Digit

Proportion

Intuitive Explanation of Benford’s Law

Use a census data example:

Larger numbers progress on to other numbers quicker than smaller numbers.

So, lower first digits progress on to higher digits at a slower rate.

The progression is a geometric sequence.

Every successive value is a fixed percentage increase over the value before it.

Nature seems to favor geometric sequences.

Data that fits Benford’s Law are called Benford Sets.

Look at the population example: It took the 1 7.27 yrs. to progress to 2, but it took the 9 only 1.1 yrs. to progress to 1 --- 6.6X faster.

That’s why there are more of them!

Proof of Stuff-1st Digit

Why are Benford’s proportions simply the Log of the ratio of digit change (New/Old)? {From 10,000 to 20,000; Log10(20,000/10,000) = Log10(2) and Benford’s formula is Log10(1+1/1) = Log10(2)}

Proof next.

Not that important; just for the mathematically curious.

Proof-1st Digit

Digit  Log10(ratio) Benford’s Formula

1: 12: Log10(2/1)Log10 (1+1/1)=Log10 (2)

2: 23: Log10 (3/2)Log10 (1+1/2)=Log10 (1.5)

3: 34: Log10 (4/3)Log10 (1+1/3)=Log10 (1.3333)

4: 45: Log10 (5/4)Log10 (1+1/4)=Log10 (1.25)

     

9: 910: Log10 (10/9)Log10 (1+1/9)=Log10 (1.1111)

Not that important; just for the mathematically curious.

Some Theory

Recent work (e.g., Hill 1996) has shown that Benford Sets are, essentially, the distribution of all distributions.

Technically: If distributions are selected at random, and random samples are taken from each of these distributions, the significant digits of the resulting collection will converge to the logarithmic (Benford) distribution.

Conforming to Benford’s Law

General Mathematical Rule: To conform to Benford’s Law, the data, when ranked from smallest to largest, should approximate a geometric sequence.

In theory no two numbers should be the same.

But small duplications (like in accounting applications) should not invalidate the comparison to Benford’s Law.

Conforming to Benford’s Law

Auditors generally make a decision about whether data sets are expected to conform to the Law based on the type of data being analyzed.

If the data is expected to conform and it does not conform, then auditors have a reason to question whether the data is valid and free of major recurring errors.

Conforming to Benford’s Law

Practice has show that the following 4 criteria must be met for a data set to conform to Benford’s Law:

The data set should describe the sizes of similar phenomena

Examples: population of towns (recall the real world example), areas of lakes, heights of mountains, market values of companies on the NYSE, daily sales volume of companies on the NYSE, etc.

Conforming to Benford’s Law

There should be no built-in minimum or maximum values in the data set.

Examples: broker with a minimum commission charge of $50 --- a set of commissions would be biased toward numbers with a first digit of 5 and a second digit of 0.

The numbers should not be made up of assigned numbers --- numbers given to things in place of words.

Examples: SS#, bank acct. #, car license plate #, etc.

Conforming to Benford’s Law

The data set should have more small items than big items.

This reflects the fact that in naturally occurring data there are, for example,

more towns than cities,

more small companies than General Electrics,

more small lakes than big lakes, etc.

Conforming to Benford’s Law

Effect of sample size.

Research has shown that the numbers in data sets should have 4 or more digits for a good fit with Benford’s Law.

When numbers have less than 4 digits, there is a slight bias in favor of the lower digits.

Generally, unless working with nothing but single-digit or double-digit numbers, the bias is not severe enough to merit an adjustment to the expected digit frequencies.

A large data set is required for a close fit to a Benford distribution.

Conforming to Benford’s Law

Scale invariance.

If a Benford Set is multiplied by a nonzero constant, the resultant set will also be a Benford Set.

Pinkham 1961 showed that only the frequencies of a Benford Set have this property.

This could be a problem for auditors – you may not find double-counting errors since its just a scaling by 2.

Conforming to Benford’s Law

Scale invariance.

More problems for auditors: When manipulations of accounting data are systematic (a constant percentage increase of decrease), Benford’s Law will not help detect them.

The same with systematic addition or subtraction of a constant.

But this type of manipulation is rare. As we have discussed, frauds are usually perpetrated/concentrated around period-end.

Goodness of Fit Tests

When you use Benford’s law you will want to assess how well your data conforms to Benford’s law.

First, you should graph your digit proportions against a graph of the Benford proportions – a picture is worth a 1,000 words.

Goodness of Fit Tests

Second, you should perform analytical goodness-of-fit tests to determine if your data conforms to Benford’s law. Two goodness-of-fit tests are often used by practitioners of Benford’s law:

the mean absolute deviation (MAD) test and

the chi-square test.

Mean Absolute Deviation (MAD)

Sum the absolute deviations between the Benford proportion and the actual proportion (for each of the 9 digits). This is the Absolute Deviation.

Divide the absolute sum by 9 (to get the Mean Absolute Deviation.

Dr. Frank’s favorite and the easiest to use! And I recommend you start with the MAD.

RECALL: This is for the 1st digit!

Mean Absolute Deviation (MAD)

Mark Nigrini (Digital Analysis Using Benford’s Law, 2000, Global Audit Publications, pages 118-119) suggests the following guidelines for evaluating the 1st digit MAD:

MAD Range Evaluation

0.000 to 0.004 Close conformity

0.004 to 0.008 Acceptable conformity

0.008 to 0.012 Marginally acceptable

> 0.012 Nonconformity

MAD

Strong end of the “marginally acceptable” category

You use the actual digit proportions!

Chi-Square Test

Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis.

The “specific hypothesis” used when using Benford’s law is that the data conform to Benford’s law (i.e., the actual digit frequencies conform to the Benford proportions).

Chi-Square Test

The null hypothesis is that the two distributions are the same (Benford).

In other words, the statistical null hypothesis is that the number of observations in each category (first digit) is equal to that predicted by a theory (Benford’s law), and the alternative hypothesis is that the observed numbers are different from the expected.

Chi-Square Test

The null hypothesis is usually an extrinsic hypothesis, where you knew the expected proportions before doing the experiment (determining the 1st digit proportions).

Chi-Square Test

You then use a mathematical relationship, in this case the chi-square distribution, to estimate the probability of obtaining that value of the test statistic.

If your calculated 2 value is less than the cut-off value, you would accept that your first digit distribution matches Benford’s law (you can’t reject the null).

Chi-Square Test

Chi-Square =  (ACd1 – ECd1)2 / ECd1

d1=1

Where:

ACd1 = Actual Count for digit d1

ECd1 = Expected Count (Benford Proportion  Population count, n) for digit d1

RECALL: This is for the 1st digit!

Chi-Square Test

Get cut-off from Excel: =CHIINV(.05,8)=15.507

CHIINV(significance level, degrees of freedom)

Degrees of freedom* = # digits – 1 (quick calc).

In the Intuitive Example: 2=0.723 < 15.507, we can’t reject the null hypothesis that both distributions are the same (the population growth distribution is a Benford distribution).

*dof is technically (rows-1) x (columns-1). In our example: (9-1)x(2-1)=8x1=8

Chi-Square Test

If computed 2 > chi-square cut-off (from CHIINV(prob., DoF)), then reject the null: There is a significant difference between the data sets that cannot be due to chance alone.

Chi-Square

=CHIINV(prob.=0.05, DoF=8)

Benford Prop. × Population size. For 1: 0.30103 × 101 = 30.40403

Chi-Square

An alternative Excel chi-square test is as follows:

CHISQ.TEST(Actual data range,Expected data range).

CHISQ.TEST returns the probability that a value of the 2 statistic at least as high as the value calculated by the 2 formula could have happened by chance under the assumption of independence.

Actual digit frequencies

Expected digit freq = Benford pop. × Population total

Chi-Square

You can relate CHIINV(prob., DoF) and CHISQ.TEST(Actual data range,Expected data range) as follows:

Use CHISQ.TEST to get a probability.

Now take that probability and put it in CHIINV and it will return a 2 cut-off equal to the computed 2 value.

Chi-Square Test

A good review of chi-square goodness-of-fit can be found at http://www.biostathandbook.com/chigof.html (even though it’s in a biological science context).

Checked hyperlink 10/20/21, it’s ok.

Z-Stat & Confidence Interval

Z-Statistic

|po – pe| - (1/2n)

Z = -----------------------

[(pe  (1 – pe)/n)]1/2

Where:

pe=expected proportion (Benford digit proportion),

po=observed proportion (actual digit proportion),

n=number of observations

Continuity correction factor and is only used when it is smaller than the first term in the numerator.

Z-Stat & Confidence Interval

Z-Statistic 95% Confidence Interval

Upper Bound =

Pe + 1.96 [(pe  (1 – pe)/n)]1/2 + 1/(2n)

Lower Bound =

Pe – 1.96 [(pe  (1 – pe)/n)]1/2 – 1/(2n)

Intuitive Explanation of Benford’s Law-Example

Benford Case – XYZ Corp.

You can find the case instructions and data on Compass2g:

Module IIIC  Benford’s Law Material: Readings + Lecture  Benford Case

Benford CASE – XYZ Corp.

7,320 sales amounts

Total sales $29,590,385

Use Benford’s Law to check the 1st digit.

How do things look? Goodness-of-fit?

Fraud?

No fraud?

Deliverable 

Sales data in an Excel file in Compass2g.

Benford CASE – XYZ Corp.

Memo on your findings:

Pictures (graph of XYZ vs. Benford).

Goodness of fit stats (at least chi-square and M.A.D.)

Does the data fit?

What might your findings mean? What now?

KISS: Keep It Short & Simple.

About 1 page of verbiage.

Benford CASE

Case is due BEFORE class on Mon., 11/15/21.

You don’t need to use any software other than Excel to do the case.

Excel Hints

Use left : =left(target cell,# left side digits)

b1=left(a1,1)  returns the first digit of the number in cell a1.

Use countif : =countif(date range,”=criteria”).

=countif(b1:b7300,”=1”)  counts the 1s in cells b1 thru b7300 (and for =2, =3, . . . , =9).

Assume your actual data is in column A (a1:a7300).

Intuituve Example

Data Summary

DigitBenfordDataAbs. Dev.

10.301030.326730.02570

20.176090.168320.00777

30.124940.118810.00613

40.096910.099010.00210

50.079180.069310.00987

60.066950.059410.00754

70.057990.059410.00142

80.051150.059410.00826

90.045760.039600.00616

Total0.07495

MAD0.00833

Chart1

1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9

Population

Benford

First Digit

Proportion

Intuitive Example n=100

0.3267326733

0.30103

0.1683168317

0.17609

0.1188118812

0.12494

0.099009901

0.09691

0.0693069307

0.07918

0.0594059406

0.06695

0.0594059406

0.05799

0.0594059406

0.05115

0.0396039604

0.04576

Chart2

1	1	1	1
2	2	2	2
3	3	3	3
4	4	4	4
5	5	5	5
6	6	6	6
7	7	7	7
8	8	8	8
9	9	9	9

Population

Benford

First Digit

Proportion

Intuitve Example n=100

0.3954406285

0.2066193715

0.3267326733

0.30103

0.25532575

0.09685425

0.1683168317

0.17609

0.1943764375

0.0555035625

0.1188118812

0.12494

0.1595564277

0.0342635723

0.099009901

0.09691

0.1367916652

0.0215683348

0.0693069307

0.07918

0.120644716

0.013255284

0.0594059406

0.06695

0.1085231582

0.0074568418

0.0594059406

0.05799

0.0990657096

0.0032342904

0.0594059406

0.05115

0.0914642098

0.0000557902

0.0396039604

0.04576

Chart3

0		10000
1		11000
2		12100
3		13310
4		14641
5		16105.1
6		17715.61
7		19487.171
8		21435.8881
9		23579.47691
10		25937.424601
11		28531.1670611
12		31384.28376721
13		34522.712143931
14		37974.9833583241
15		41772.4816941566
16		45949.7298635722
17		50544.7028499295
18		55599.1731349224
19		61159.0904484147
20		67274.9994932561
21		74002.4994425817
22		81402.7493868399
23		89543.0243255239
24		98497.3267580763
25		108347.059433884
26		119181.765377272
27		131099.941915
28		144209.9361065
29		158630.92971715
30		174494.022688865
31		191943.424957751
32		211137.767453526
33		232251.544198879
34		255476.698618767
35		281024.368480643
36		309126.805328708
37		340039.485861578
38		374043.434447736
39		411447.77789251
40		452592.555681761
41		497851.811249937
42		547636.992374931
43		602400.691612424
44		662640.760773667
45		728904.836851033
46		801795.320536137
47		881974.85258975
48		970172.337848725
49		1067189.5716336
50		1173908.52879696
51		1291299.38167665
52		1420429.31984432
53		1562472.25182875
54		1718719.47701163
55		1890591.42471279
56		2079650.56718407
57		2287615.62390248
58		2516377.18629272
59		2768014.904922
60		3044816.3954142
61		3349298.03495562
62		3684227.83845118
63		4052650.6222963
64		4457915.68452593
65		4903707.25297852
66		5394077.97827637
67		5933485.77610401
68		6526834.35371441
69		7179517.78908585
70		7897469.56799444
71		8687216.52479388
72		9555938.17727327
73		10511531.9950006
74		11562685.1945007
75		12718953.7139507
76		13990849.0853458
77		15389933.9938804
78		16928927.3932684
79		18621820.1325953
80		20484002.1458548
81		22532402.3604403
82		24785642.5964843
83		27264206.8561327
84		29990627.541746
85		32989690.2959206
86		36288659.3255127
87		39917525.2580639
88		43909277.7838703
89		48300205.5622574
90		53130226.1184831
91		58443248.7303314
92		64287573.6033646
93		70716330.9637011
94		77787964.0600712
95		85566760.4660783
96		94123436.5126861
97		103535780.163955
98		113889358.18035
99		125278293.998385
100		137806123.398224

Time

Population

Time

Value

Population Growth

Data


Population growth rate			0.1
initial population			10000
Time		Population
0		10,000	1
1		11,000	1
2		12,100	1
3		13,310	1
4		14,641	1
5		16,105	1
6		17,716	1
7		19,487	1
8		21,436		1
9		23,579		1
10		25,937		1
11		28,531		1
12		31,384			1
13		34,523			1
14		37,975			1
15		41,772				1
16		45,950				1
17		50,545					1
18		55,599					1
19		61,159						1
20		67,275						1
21		74,002							1
22		81,403								1
23		89,543								1
24		98,497									1
25		108,347	1
26		119,182	1
27		131,100	1
28		144,210	1
29		158,631	1
30		174,494	1
31		191,943	1
32		211,138		1
33		232,252		1
34		255,477		1
35		281,024		1
36		309,127			1
37		340,039			1
38		374,043			1
39		411,448				1
40		452,593				1
41		497,852				1
42		547,637					1
43		602,401						1
44		662,641						1
45		728,905							1
46		801,795								1
47		881,975								1
48		970,172									1
49		1,067,190	1
50		1,173,909	1
51		1,291,299	1
52		1,420,429	1
53		1,562,472	1
54		1,718,719	1
55		1,890,591	1
56		2,079,651		1
57		2,287,616		1
58		2,516,377		1
59		2,768,015		1
60		3,044,816			1
61		3,349,298			1
62		3,684,228			1
63		4,052,651				1
64		4,457,916				1
65		4,903,707				1
66		5,394,078					1
67		5,933,486					1
68		6,526,834						1
69		7,179,518							1
70		7,897,470							1
71		8,687,217								1
72		9,555,938									1
73		10,511,532	1
74		11,562,685	1
75		12,718,954	1
76		13,990,849	1
77		15,389,934	1
78		16,928,927	1
79		18,621,820	1
80		20,484,002		1
81		22,532,402		1
82		24,785,643		1
83		27,264,207		1
84		29,990,628		1
85		32,989,690			1
86		36,288,659			1
87		39,917,525			1
88		43,909,278				1
89		48,300,206				1
90		53,130,226					1
91		58,443,249					1
92		64,287,574						1
93		70,716,331							1
94		77,787,964							1
95		85,566,760								1
96		94,123,437									1
97		103,535,780	1
98		113,889,358	1
99		125,278,294	1
100		137,806,123	1
		101	33	17	12	10	7	6	6	6	4
	Proportion		0.3267326733	0.1683168317	0.1188118812	0.099009901	0.0693069307	0.0594059406	0.0594059406	0.0594059406	0.0396039604
	Benford		0.30103	0.17609	0.12494	0.09691	0.07918	0.06695	0.05799	0.05115	0.04576
	Test Benford		0.30103	0.17609	0.12494	0.09691	0.07918	0.06695	0.05799	0.05115	0.04576
			1	2	3	4	5	6	7	8	9
					Expected		Chi Square Comp.		Z-CI Computations
			Population	Benford	Count	Count	(A-E)^2	divide E	((1-pe/n)	(pe*colJ)^.5	1/(2n)		UB	LB	Population	Benford
		1	0.3267326733	0.30103	30.40403	33	6.7390602409	0.2216502299	0.006920495	0.0456429252	0.004950495	1	0.3954406285	0.2066193715	0.3267326733	0.30103
		2	0.1683168317	0.17609	17.78509	17	0.6163663081	0.0346563502	0.0081575248	0.0379006403	0.004950495	2	0.25532575	0.09685425	0.1683168317	0.17609
		3	0.1188118812	0.12494	12.61894	12	0.3830867236	0.0303580747	0.0086639604	0.032900991	0.004950495	3	0.1943764375	0.0555035625	0.1188118812	0.12494
		4	0.099009901	0.09691	9.78791	10	0.0449821681	0.0045956867	0.0089414851	0.0294367003	0.004950495	4	0.1595564277	0.0342635723	0.099009901	0.09691
		5	0.0693069307	0.07918	7.99718	7	0.9943679524	0.1243398238	0.0091170297	0.0268679439	0.004950495	5	0.1367916652	0.0215683348	0.0693069307	0.07918
		6	0.0594059406	0.06695	6.76195	6	0.5805678025	0.0858580443	0.0092381188	0.0248695005	0.004950495	6	0.120644716	0.013255284	0.0594059406	0.06695
		7	0.0594059406	0.05799	5.85699	6	0.0204518601	0.0034918721	0.0093268317	0.0232564608	0.004950495	7	0.1085231582	0.0074568418	0.0594059406	0.05799
		8	0.0594059406	0.05115	5.16615	6	0.6953058225	0.1345887794	0.0093945545	0.0219210278	0.004950495	8	0.0990657096	0.0032342904	0.0594059406	0.05115
		9	0.0396039604	0.04576	4.62176	4	0.3865854976	0.08364465	0.0094479208	0.0207927116	0.004950495	9	0.0914642098	0.0000557902	0.0396039604	0.04576
					Computed Chi Sq.			0.7231835113
					Chi Sq. Cut-off			15.507312493
Intuituve Example
Data Summary
Digit	Benford	Data	Abs. Dev.
1	0.30103	0.32673	0.02570
2	0.17609	0.16832	0.00777
3	0.12494	0.11881	0.00613
4	0.09691	0.09901	0.00210
5	0.07918	0.06931	0.00987
6	0.06695	0.05941	0.00754
7	0.05799	0.05941	0.00142
8	0.05115	0.05941	0.00826
9	0.04576	0.03960	0.00616
		Total	0.07495
		MAD	0.00833

First Digit

INTUITIVE BENFORD EXAMPLE
Growth Rate	0.1
Number of Steps From:							D/B
	1	To	2	=	7.27254	0.30103	2
	2	To	3	=	4.25416	0.17609	1.5
	3	To	4	=	3.01838	0.12494	1.3333333333
	4	To	5	=	2.34124	0.09691	1.25
	5	To	6	=	1.91293	0.07918	1.2
	6	To	7	=	1.61736	0.06695	1.1666666667
	7	To	8	=	1.40102	0.05799	1.1428571429
	8	To	9	=	1.23579	0.05115	1.125
	9	To	10		1.10545	0.04576	1.1111111111
	Total				24.15886	1
							10

Second Digit


2nd Digit
Log (GR)	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	2nd Digit	Steps	Proportion
0.0413926852	10	11	0.0413926852	1	20	21	0.0211892991	0.5119092658	30	31	0.0142404391	0.3440327454	40	41	0.0107238654	0.2590763404	50	51	0.0086001718	0.2077703278	60	61	0.0071785846	0.1734264061	70	71	0.0061603087	0.1488260228	80	81	0.0053950319	0.1303378089	90	91	0.0047988829	0.1159355298	0	2.891314447	0.11968
	11	12	0.0377885609	0.9129284738	21	22	0.0202033861	0.4880907342	31	32	0.0137882845	0.333109206	41	42	0.0104654337	0.2528329254	51	52	0.0084331675	0.2037356964	61	62	0.0070618545	0.1706063393	71	72	0.0060741477	0.1467444716	81	82	0.0053288335	0.1287385316	91	92	0.004746435	0.1146684494	1	2.7514548276	0.11389
	12	13	0.0347621063	0.8398127864	22	23	0.0193051552	0.4663905016	32	33	0.0133639616	0.3228580486	42	43	0.0102191652	0.2468833598	52	53	0.008272526	0.1998547795	62	63	0.00694886	0.1678765204	72	73	0.0059903637	0.1447203453	82	83	0.00526424	0.1271780261	92	93	0.0046951212	0.1134287662	2	2.629003134	0.10882
	13	14	0.0321846834	0.777545193	23	24	0.0184834057	0.4465379722	33	34	0.0129649772	0.3132190414	43	44	0.0099842209	0.2412073744	53	54	0.0081178902	0.1961189566	63	64	0.0068394245	0.1652326855	73	74	0.0059088596	0.1427512998	83	84	0.0052011937	0.1256548993	93	94	0.004644905	0.1122156011	3	2.5204830233	0.10433
	14	15	0.0299632234	0.7238772566	24	25	0.017728767	0.4283067622	34	35	0.0125891273	0.3041389381	44	45	0.0097598373	0.2357865225	54	55	0.0079689297	0.1925202398	64	65	0.0067333827	0.162670835	74	75	0.0058295437	0.1408351171	84	85	0.0051396397	0.1241678242	94	95	0.0045957517	0.1110281121	4	2.4233316076	0.10031
	15	16	0.0280287236	0.6771419514	25	26	0.0170333393	0.4115060242	35	36	0.0122344564	0.2955704944	45	46	0.0095453179	0.2306039792	55	56	0.0078253375	0.1890512172	65	66	0.0066305789	0.1601872136	75	76	0.0057523289	0.1389696964	85	86	0.0050795255	0.1227155356	95	96	0.0045476278	0.1098654927	5	2.3356116046	0.09668
	16	17	0.0263289387	0.63607709	26	27	0.0163904162	0.3959737361	36	37	0.0118992233	0.2874716452	46	47	0.0093400263	0.2256443673	56	57	0.0076868287	0.1857050017	66	67	0.0065308672	0.1577782918	76	77	0.0056771329	0.137153047	86	87	0.0050208014	0.1212968271	96	97	0.0045005012	0.1087269697	6	2.2558269759	0.09337
	17	18	0.0248235837	0.5997094325	27	28	0.0157942672	0.381571457	37	38	0.0115818725	0.2798048135	47	48	0.0091433794	0.2208936049	57	58	0.0075531379	0.1824751852	67	68	0.00643411	0.1554407495	77	78	0.0056038775	0.1353832808	87	88	0.0049634195	0.1199105473	97	98	0.0044543414	0.1076118017	7	2.1828008723	0.09035
	18	19	0.0234810958	0.5672764586	28	29	0.0152399666	0.3681801869	38	39	0.0112810104	0.2725363278	48	49	0.0089548427	0.2163387714	58	59	0.0074240181	0.179355798	68	69	0.006340178	0.1531714603	78	79	0.0055324886	0.1336586061	88	89	0.0049073345	0.1185555969	98	99	0.0044091189	0.1065192772	8	2.1155924832	0.08757
	19	20	0.0222763947	0.538172255	29	30	0.0147232568	0.3556970698	39	40	0.0109953843	0.2656359272	49	50	0.0087739243	0.2119679908	59	60	0.0072992387	0.1763412717	69	70	0.0062489493	0.1509674778	79	80	0.0054628957	0.1319773211	89	90	0.0048525028	0.1172309256	99	100	0.0043648054	0.1054487136	9	2.0534389526	0.08500
			0.3010299957	7.2725408973			0.1760912591	4.2541637099			0.1249387366	3.0183771874			0.096910013	2.3412352361			0.079181246	1.9129284738			0.0669467896	1.6173579794			0.057991947	1.401019208			0.0511525224	1.2357865225			0.0457574906	1.1054487136		24.1588579281	1

PopulationBenfordExpected CountActual Count(A-E)^2divide E

10.32673270.3010330.40403336.739060.22165

20.16831680.1760917.78509170.6163660.034656

30.11881190.1249412.61894120.3830870.030358

40.09900990.096919.78791100.0449820.004596

50.06930690.079187.9971870.9943680.12434

60.05940590.066956.7619560.5805680.085858

70.05940590.057995.8569960.0204520.003492

80.05940590.051155.1661560.6953060.134589

90.0396040.045764.6217640.3865850.083645

Computed Chi Sq.0.723184

Chi Sq. Cut-off, 5%15.50731

Chart1

1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9

Population

Benford

First Digit

Proportion

Intuitive Example n=100

0.3267326733

0.30103

0.1683168317

0.17609

0.1188118812

0.12494

0.099009901

0.09691

0.0693069307

0.07918

0.0594059406

0.06695

0.0594059406

0.05799

0.0594059406

0.05115

0.0396039604

0.04576

Chart2

1	1	1	1
2	2	2	2
3	3	3	3
4	4	4	4
5	5	5	5
6	6	6	6
7	7	7	7
8	8	8	8
9	9	9	9

Population

Benford

First Digit

Proportion

Intuitve Example n=100

0.3954406285

0.2066193715

0.3267326733

0.30103

0.25532575

0.09685425

0.1683168317

0.17609

0.1943764375

0.0555035625

0.1188118812

0.12494

0.1595564277

0.0342635723

0.099009901

0.09691

0.1367916652

0.0215683348

0.0693069307

0.07918

0.120644716

0.013255284

0.0594059406

0.06695

0.1085231582

0.0074568418

0.0594059406

0.05799

0.0990657096

0.0032342904

0.0594059406

0.05115

0.0914642098

0.0000557902

0.0396039604

0.04576

Chart3

0		10000
1		11000
2		12100
3		13310
4		14641
5		16105.1
6		17715.61
7		19487.171
8		21435.8881
9		23579.47691
10		25937.424601
11		28531.1670611
12		31384.28376721
13		34522.712143931
14		37974.9833583241
15		41772.4816941566
16		45949.7298635722
17		50544.7028499295
18		55599.1731349224
19		61159.0904484147
20		67274.9994932561
21		74002.4994425817
22		81402.7493868399
23		89543.0243255239
24		98497.3267580763
25		108347.059433884
26		119181.765377272
27		131099.941915
28		144209.9361065
29		158630.92971715
30		174494.022688865
31		191943.424957751
32		211137.767453526
33		232251.544198879
34		255476.698618767
35		281024.368480643
36		309126.805328708
37		340039.485861578
38		374043.434447736
39		411447.77789251
40		452592.555681761
41		497851.811249937
42		547636.992374931
43		602400.691612424
44		662640.760773667
45		728904.836851033
46		801795.320536137
47		881974.85258975
48		970172.337848725
49		1067189.5716336
50		1173908.52879696
51		1291299.38167665
52		1420429.31984432
53		1562472.25182875
54		1718719.47701163
55		1890591.42471279
56		2079650.56718407
57		2287615.62390248
58		2516377.18629272
59		2768014.904922
60		3044816.3954142
61		3349298.03495562
62		3684227.83845118
63		4052650.6222963
64		4457915.68452593
65		4903707.25297852
66		5394077.97827637
67		5933485.77610401
68		6526834.35371441
69		7179517.78908585
70		7897469.56799444
71		8687216.52479388
72		9555938.17727327
73		10511531.9950006
74		11562685.1945007
75		12718953.7139507
76		13990849.0853458
77		15389933.9938804
78		16928927.3932684
79		18621820.1325953
80		20484002.1458548
81		22532402.3604403
82		24785642.5964843
83		27264206.8561327
84		29990627.541746
85		32989690.2959206
86		36288659.3255127
87		39917525.2580639
88		43909277.7838703
89		48300205.5622574
90		53130226.1184831
91		58443248.7303314
92		64287573.6033646
93		70716330.9637011
94		77787964.0600712
95		85566760.4660783
96		94123436.5126861
97		103535780.163955
98		113889358.18035
99		125278293.998385
100		137806123.398224

Time

Population

Time

Value

Population Growth

Data


Population growth rate			0.1
initial population			10000
Time		Population
0		10,000	1
1		11,000	1
2		12,100	1
3		13,310	1
4		14,641	1
5		16,105	1
6		17,716	1
7		19,487	1
8		21,436		1
9		23,579		1
10		25,937		1
11		28,531		1
12		31,384			1
13		34,523			1
14		37,975			1
15		41,772				1
16		45,950				1
17		50,545					1
18		55,599					1
19		61,159						1
20		67,275						1
21		74,002							1
22		81,403								1
23		89,543								1
24		98,497									1
25		108,347	1
26		119,182	1
27		131,100	1
28		144,210	1
29		158,631	1
30		174,494	1
31		191,943	1
32		211,138		1
33		232,252		1
34		255,477		1
35		281,024		1
36		309,127			1
37		340,039			1
38		374,043			1
39		411,448				1
40		452,593				1
41		497,852				1
42		547,637					1
43		602,401						1
44		662,641						1
45		728,905							1
46		801,795								1
47		881,975								1
48		970,172									1
49		1,067,190	1
50		1,173,909	1
51		1,291,299	1
52		1,420,429	1
53		1,562,472	1
54		1,718,719	1
55		1,890,591	1
56		2,079,651		1
57		2,287,616		1
58		2,516,377		1
59		2,768,015		1
60		3,044,816			1
61		3,349,298			1
62		3,684,228			1
63		4,052,651				1
64		4,457,916				1
65		4,903,707				1
66		5,394,078					1
67		5,933,486					1
68		6,526,834						1
69		7,179,518							1
70		7,897,470							1
71		8,687,217								1
72		9,555,938									1
73		10,511,532	1
74		11,562,685	1
75		12,718,954	1
76		13,990,849	1
77		15,389,934	1
78		16,928,927	1
79		18,621,820	1
80		20,484,002		1
81		22,532,402		1
82		24,785,643		1
83		27,264,207		1
84		29,990,628		1
85		32,989,690			1
86		36,288,659			1
87		39,917,525			1
88		43,909,278				1
89		48,300,206				1
90		53,130,226					1
91		58,443,249					1
92		64,287,574						1
93		70,716,331							1
94		77,787,964							1
95		85,566,760								1
96		94,123,437									1
97		103,535,780	1
98		113,889,358	1
99		125,278,294	1
100		137,806,123	1
		101	33	17	12	10	7	6	6	6	4
	Proportion		0.3267326733	0.1683168317	0.1188118812	0.099009901	0.0693069307	0.0594059406	0.0594059406	0.0594059406	0.0396039604
	Benford		0.30103	0.17609	0.12494	0.09691	0.07918	0.06695	0.05799	0.05115	0.04576
	Test Benford		0.30103	0.17609	0.12494	0.09691	0.07918	0.06695	0.05799	0.05115	0.04576
			1	2	3	4	5	6	7	8	9
					Expected	Actual	Chi Square Comp.		Z-CI Computations
			Population	Benford	Expected Count	Actual Count	(A-E)^2	divide E	((1-pe/n)	(pe*colJ)^.5	1/(2n)		UB	LB	Population	Benford
		1	0.3267326733	0.30103	30.40403	33	6.7390602409	0.2216502299	0.006920495	0.0456429252	0.004950495	1	0.3954406285	0.2066193715	0.3267326733	0.30103
		2	0.1683168317	0.17609	17.78509	17	0.6163663081	0.0346563502	0.0081575248	0.0379006403	0.004950495	2	0.25532575	0.09685425	0.1683168317	0.17609
		3	0.1188118812	0.12494	12.61894	12	0.3830867236	0.0303580747	0.0086639604	0.032900991	0.004950495	3	0.1943764375	0.0555035625	0.1188118812	0.12494
		4	0.099009901	0.09691	9.78791	10	0.0449821681	0.0045956867	0.0089414851	0.0294367003	0.004950495	4	0.1595564277	0.0342635723	0.099009901	0.09691
		5	0.0693069307	0.07918	7.99718	7	0.9943679524	0.1243398238	0.0091170297	0.0268679439	0.004950495	5	0.1367916652	0.0215683348	0.0693069307	0.07918
		6	0.0594059406	0.06695	6.76195	6	0.5805678025	0.0858580443	0.0092381188	0.0248695005	0.004950495	6	0.120644716	0.013255284	0.0594059406	0.06695
		7	0.0594059406	0.05799	5.85699	6	0.0204518601	0.0034918721	0.0093268317	0.0232564608	0.004950495	7	0.1085231582	0.0074568418	0.0594059406	0.05799
		8	0.0594059406	0.05115	5.16615	6	0.6953058225	0.1345887794	0.0093945545	0.0219210278	0.004950495	8	0.0990657096	0.0032342904	0.0594059406	0.05115
		9	0.0396039604	0.04576	4.62176	4	0.3865854976	0.08364465	0.0094479208	0.0207927116	0.004950495	9	0.0914642098	0.0000557902	0.0396039604	0.04576
					Computed Chi Sq.			0.7231835113
					Chi Sq. Cut-off, 5%			15.5073130559
Intuituve Example
Data Summary
Digit	Benford	Data	Abs. Dev.
1	0.30103	0.32673	0.02570
2	0.17609	0.16832	0.00777
3	0.12494	0.11881	0.00613
4	0.09691	0.09901	0.00210
5	0.07918	0.06931	0.00987
6	0.06695	0.05941	0.00754
7	0.05799	0.05941	0.00142
8	0.05115	0.05941	0.00826
9	0.04576	0.03960	0.00616
		Total	0.07495
		MAD	0.00833

First Digit

INTUITIVE BENFORD EXAMPLE
Growth Rate	0.1
Number of Steps From:							D/B
	1	To	2	=	7.27254	0.30103	2
	2	To	3	=	4.25416	0.17609	1.5
	3	To	4	=	3.01838	0.12494	1.3333333333
	4	To	5	=	2.34124	0.09691	1.25
	5	To	6	=	1.91293	0.07918	1.2
	6	To	7	=	1.61736	0.06695	1.1666666667
	7	To	8	=	1.40102	0.05799	1.1428571429
	8	To	9	=	1.23579	0.05115	1.125
	9	To	10		1.10545	0.04576	1.1111111111
	Total				24.15886	1
							10

Second Digit


2nd Digit
Log (GR)	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	From	To	Log (Ratio)	Steps	2nd Digit	Steps	Proportion
0.0413926852	10	11	0.0413926852	1	20	21	0.0211892991	0.5119092658	30	31	0.0142404391	0.3440327454	40	41	0.0107238654	0.2590763404	50	51	0.0086001718	0.2077703278	60	61	0.0071785846	0.1734264061	70	71	0.0061603087	0.1488260228	80	81	0.0053950319	0.1303378089	90	91	0.0047988829	0.1159355298	0	2.891314447	0.11968
	11	12	0.0377885609	0.9129284738	21	22	0.0202033861	0.4880907342	31	32	0.0137882845	0.333109206	41	42	0.0104654337	0.2528329254	51	52	0.0084331675	0.2037356964	61	62	0.0070618545	0.1706063393	71	72	0.0060741477	0.1467444716	81	82	0.0053288335	0.1287385316	91	92	0.004746435	0.1146684494	1	2.7514548276	0.11389
	12	13	0.0347621063	0.8398127864	22	23	0.0193051552	0.4663905016	32	33	0.0133639616	0.3228580486	42	43	0.0102191652	0.2468833598	52	53	0.008272526	0.1998547795	62	63	0.00694886	0.1678765204	72	73	0.0059903637	0.1447203453	82	83	0.00526424	0.1271780261	92	93	0.0046951212	0.1134287662	2	2.629003134	0.10882
	13	14	0.0321846834	0.777545193	23	24	0.0184834057	0.4465379722	33	34	0.0129649772	0.3132190414	43	44	0.0099842209	0.2412073744	53	54	0.0081178902	0.1961189566	63	64	0.0068394245	0.1652326855	73	74	0.0059088596	0.1427512998	83	84	0.0052011937	0.1256548993	93	94	0.004644905	0.1122156011	3	2.5204830233	0.10433
	14	15	0.0299632234	0.7238772566	24	25	0.017728767	0.4283067622	34	35	0.0125891273	0.3041389381	44	45	0.0097598373	0.2357865225	54	55	0.0079689297	0.1925202398	64	65	0.0067333827	0.162670835	74	75	0.0058295437	0.1408351171	84	85	0.0051396397	0.1241678242	94	95	0.0045957517	0.1110281121	4	2.4233316076	0.10031
	15	16	0.0280287236	0.6771419514	25	26	0.0170333393	0.4115060242	35	36	0.0122344564	0.2955704944	45	46	0.0095453179	0.2306039792	55	56	0.0078253375	0.1890512172	65	66	0.0066305789	0.1601872136	75	76	0.0057523289	0.1389696964	85	86	0.0050795255	0.1227155356	95	96	0.0045476278	0.1098654927	5	2.3356116046	0.09668
	16	17	0.0263289387	0.63607709	26	27	0.0163904162	0.3959737361	36	37	0.0118992233	0.2874716452	46	47	0.0093400263	0.2256443673	56	57	0.0076868287	0.1857050017	66	67	0.0065308672	0.1577782918	76	77	0.0056771329	0.137153047	86	87	0.0050208014	0.1212968271	96	97	0.0045005012	0.1087269697	6	2.2558269759	0.09337
	17	18	0.0248235837	0.5997094325	27	28	0.0157942672	0.381571457	37	38	0.0115818725	0.2798048135	47	48	0.0091433794	0.2208936049	57	58	0.0075531379	0.1824751852	67	68	0.00643411	0.1554407495	77	78	0.0056038775	0.1353832808	87	88	0.0049634195	0.1199105473	97	98	0.0044543414	0.1076118017	7	2.1828008723	0.09035
	18	19	0.0234810958	0.5672764586	28	29	0.0152399666	0.3681801869	38	39	0.0112810104	0.2725363278	48	49	0.0089548427	0.2163387714	58	59	0.0074240181	0.179355798	68	69	0.006340178	0.1531714603	78	79	0.0055324886	0.1336586061	88	89	0.0049073345	0.1185555969	98	99	0.0044091189	0.1065192772	8	2.1155924832	0.08757
	19	20	0.0222763947	0.538172255	29	30	0.0147232568	0.3556970698	39	40	0.0109953843	0.2656359272	49	50	0.0087739243	0.2119679908	59	60	0.0072992387	0.1763412717	69	70	0.0062489493	0.1509674778	79	80	0.0054628957	0.1319773211	89	90	0.0048525028	0.1172309256	99	100	0.0043648054	0.1054487136	9	2.0534389526	0.08500
			0.3010299957	7.2725408973			0.1760912591	4.2541637099			0.1249387366	3.0183771874			0.096910013	2.3412352361			0.079181246	1.9129284738			0.0669467896	1.6173579794			0.057991947	1.401019208			0.0511525224	1.2357865225			0.0457574906	1.1054487136		24.1588579281	1

Intuitve Example n=100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

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First Digit

Proportion

Population

Benford