behind the veil of power: state statistics and benford’s law in chosǒn korea

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Journal of Human and Social Science ResearchVol. 2, No. 1 (2013), 9-20

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H uman and S ocial

S cience R esearch HSSR

Vol. 2(1), 9-20

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Keywords:

Chosǒ nKoreaHistoryBenford’s Law

Received: 27 Oct 2013 Accepted: 04 Nov 2013

Milan Hejtmanek

Department of Korean History, Collegeof Humanities, Seoul National

University, Gwanak-gu, Seoul, SouthKorea

Behind The Veil of Power: State Statistics and Benford’s Law in Chos n

Korea

Corresponding Author: Milan Hejtmanek

Department of Korean History, College of Humanities, Seoul National University, Gwanak-gu, Seoul, South Korea

Abstract

Situating the Chosǒn period (1392-1910) in Korea within the larger historiographical

framework has proved no easy challenge. While its economy was basic, relying largely onagriculture, its system of rule comprised a complex bureaucracy superintended by highly

educated officials selected by impartially administered examinations. While generallyconsigned to the catch-all historiographical category of “pre-modern” some scholars such

as Alexander Woodside have pointed to the rationality of the Chosǒn bureaucracy and

deemed it an alternative form of modernity. Central in such attempt to reappraise the

sophistication of rule in Chosǒn are the sophistication and accuracy of its administrative

record keeping. This paper uses the mathematical insights garnered over the past century

concerning “Benford’s Law” or the “First Digits Law” to examine the internal consistencyof a set of Chosǒn-era government statistics. Benford and others discovered that the

distribution of the left-most digits in a wide variety of statistics follow a non-uniform

distribution, one not obvious to those unfamiliar with it. Through a comparison of the

actual numbers with their theoretical distribution under Benford’s Law, it has been possible

to uncover a wide variety of malfeasance and accounting fraud. This paper finds thatgovernment documents from both the 15th and 19 centuries generally comport with the

Benford distribution, adding weight to an understanding of the Chosǒn period as more

competent and sophisticated it is rule than has previously been thought.

1. INTRODUCTION

The Chosǒn period (1392-1910) presents the seeming paradox of

understanding how one of the longest continuous dynastic states in world

history achieved such seeming stability in the midst of not infrequent

invasion, domestic rebellions, fratricidal mayhem and murder in the royal

house, the tensions inherent in the existence of widespread hereditary

slavery, and frequent recourse to brigandage. Scholars have tended to locate

the social anchor in the Chosǒn period in its tenacious commitment to

learning generally, especially as articulated through a turn toward

Neo-Confucian thought and social precepts, as propagated by — and

demonstrated through — a state examination system that yielded hard-earned

degrees for which every adult aristocrat (the yangban class) sought, often

for long decades into their late adulthood.1 Other explanations for the

dynastic longevity must surely involve a careful examination of the

elaborate bureaucratic system that superintended the affairs of state.

1For context on Korean history in the Chosǒn period generally see Lee (1984); for specialized information on Confucianism and

its influence on Chosǒn-era institutions, see Palais (1996) and Deuchler (1995 ).



M. Hejtmanek | 11

© ORIC Publications /2013

set of reference comprehensive reference works, providing extensive details on institutions and locales, vital

for effective governance by the capital bureaucracy. Often referenced but relatively little studied, either in

Korea or elsewhere, these texts reveal a systematic approach to rational governance, based on the patient

assembling of a wide variety of useful and accurate information necessary for the development of thoughtful

and well-designed policies.

2. MATERIALS AND METHODS

The counterintuitive fact that for many (but not all) collections of numbers, the first significant digits

(i.e. the first non-zero digit from the left of number) are not distributed uniformly but instead are highly

skewed toward low numbers was realized only in the late nineteenth century by Newcomb (1881) and was

not explored with some thoroughness until the twentieth century by Benford (1938), a physicist at General

Electric. He discovered a wide variety of categories of numbers, including the areas of rivers, baseball

statistics, street addresses all display a characteristic weakly monotonic decreasing distribution of the first

digits (from 1 to 9), for which he devised a formula treating the numbers logarithmically:

( ) (1)

By this model, now known as Benford’s Law (also termed the First Significant Digits (FSD) phenomenon),

numbers beginning with 1 will occur with a frequency of about 30.1%, while those beginning with 9 only

around 4.6%, as shown in Fig. 1.

Figure 1. Benford's Distribution for 1st Digits

A related implication of Benford’s Law is that the second digits (from 0 to 9) also follow a non-uniform

distribution, given by the equation:



12 | Behind The Veil Of Power: State Statistics And Benford’s Law In Chosǒn Korea

Journal of Human And Social Science Research / Vol. 2, No. 1 (2013), 9-20

∑ 0 −9

= (2)

By this rule, the number 0 should appear as the second digit about 12% of the time, in contrast to 9,

which lags at a frequency of about 8.5%, as shown in Fig. 2.

Figure 2. Benford's Distribution for 2st Digits

Subsequently, over decades scholars such as Pinkham (1961), Raimi (1969), Hill (1995), and Berger et

al. (2011) have probed the mathematical foundations of Benford’s Law, while others have sought to apply it

empirically in a wide variety of disciplines, including genome analysis, scientific fraud detections, analysis

of macro-economic data, and forensic accounting. Due to the counter-intuitive nature of the FSD distribution,

tax fraud and other financial malfeasance can at time be ferreted out through its use (Nigrini, 1996, 2012).

Another burgeoning venue for use of Benford’s Law has been the forensic analysis of elections, seeking

irregularities in the vote through unusual patterns in the count (Berber et al., 2012). Active research in the

field continues, even as Grendar et al. (2007) have refined the underlying approach by introducing a family

of related Benford-like distributions. At present Benford’s Law has been employed to seek out possible

irregularities, but while such analyses produce at times highly suggestive and indicative results, they should

not be understood as fully confirmative of potential malfeasance.

This paper uses Benford's Law to examine at two related issues: first, do the counts found in historical

Korean statistical compendia generally follow the distributions predicted by Benford’s Law? And second, if

they do indeed do so, can we find statistically significant departures that might hint at more broadly based

lapses in record keeping, or even deliberate malfeasance?

3. ANALYSIS OF SEJONG ANNALS GEOGRAPHICAL APPENDIX

Using data derived from Park (1987) and Kang (1998), using Stata 12 (2011) software, I analyzed the

household and population statistics given in the Geographical Appendix for 65 locales in 15th century Korea,



M. Hejtmanek | 13


comprising 112,874 persons, living in 37,749 households. As can be seen from Figure 3, the first digits of

the household data, well tracks the Benford distribution. Using the standard Pearson’s chi-squared test to

compare the observed data with the expected data from a Benford first-digit distribution, the statistic

obtained (3.3, p=0.914), with 8 degrees of freedom is far too low to find evidence to reject the null

hypothesis of concordance with the 1BL distribution. Hence, households in this data sample do track the

first-digit Benford’s Law (1BL) quite well. Using the second digit test of Benford’s Law (2BL, see Figure 4),

there remains no evidence (chi2=3.72, p=0.9287) to reject the null hypothesis. This data on households from

the fifteenth century well follows a Benford distribution and provides no evidence of possible irregularities.

Figure 3. First Digit Distribution of Households in Senjong Annals

Figure 4. Second Digit Distribution of Households in Senjong Annals





Figure 5. First Digit Distribution of Population in Senjong Annals

Figure 6. Second Digit Distribution of Population in Senjong Annals

Data on the population in these locales from the Sejong Annals is presented in Figures 5 and 6. Here

too, for the 1st digit, the numbers line up in a distribution statistically indistinguishable from the that

predicted by 1BL, with a chi-square value of 6.84 for 8 degrees of freedom, or p = 0.554, i.e. and hence far

above a p value of 0.05 than could indicated convincing evidence of a difference possibly indicative of



M. Hejtmanek | 15


slipshod data collection or fraudulent recording. For the second digit analysis, the chi-square value of 14.24

also failed to provide strong evidence of a departure from the 2BL distribution, but at p = 0.1139, displayed

marked divergences in the second digits of 0, 1, 4, and 5 especially. The data might bear closer examination

to see if the departures might be seen as part of a local pattern of recording. One might speculate that the

number four was underrepresented in both the 1BL and 2BL counts given the general taboo in East Asia

against the number four by those recording population since the word for “four” ( sa 四) is a homophone

with the word for “death” (sa 死). It is conceivable than some fours were recorded as fives. However

any such taboo concerning four is not in evidence in other data presented in this study. In the end, the data

remain explicable as random departures from the 1BL and 2BL distributions.

4. ANALYSIS OF THE ROYAL COMPENDIUM

Using data from JS Bae (1995) I first examined the statistics concerning the national system of

strategic royal forests ( pongsan), a category of data likely less charged with the potential for malfeasance

than the others examined for the Royal Compendium, and hence useful as a potential baseline for observing

how well the data in this collection might adhere to a Benford-like distribution. Nationally there were 651strategic forests, found in 74 jurisdictions. The results of the analysis for 1BL are presented in Figure 7.

Because only 21 of the 74 locales had 10 or more forests, it was deemed unreasonable to attempt a 2BL

analysis, which requires counts of 10 or higher to obtain a second digit. The first digits in the data generate a

chi-square of 8.89 or p=0.352, providing insufficient evidence to reject the null hypothesis. These data are,

then, statistically concordant with the predictions of the 1BL distribution. Given the small number of second

digits in this data set (21 out of 74), not analysis of the 2BL is possible.

Figure 7. First Digit Distribution of Strategic Forests in Royal Compendium

Next, using raw data compiled by KT Kim et al. (2010) as part of an on-going project in Chosŏn-era

economic history at Sunggyungwan University and Seoul National University, I examined a series of

potentially more contentious statistics from the early 19th century, beginning with tax-exempt land.





Nationwide there was a recorded 338,293 kyŏl (it is difficult to know precisely the precise acreage, since the

kyŏl measures productive capacity of land rather than absolute acreage, but on the order of roughly a million

acres) of land off the tax rolls for various 199 separate purposes and jurisdictions. The 1BL and 2BL

analyses are presented in Figures 8 and 9. Here, as well, it is impossible to reject the null hypothesis of

statistical concordance with the Benford distributions, given a chi-square of 10.31 (p=0.244) for the 1BL and

9.83 (p=0.364) for the 2BL. There is no evidence, at this level of aggregation, of possible irregularities

discernible via Benford distributions, which the data well follow.

Figure 8. First Digit Distribution of Tax Exempt Land in Royal Compendium

Figure 9. Second Digit Distribution of Tax Exempt Land in Royal Compendium



M. Hejtmanek | 17


Figure 10. First Digit Distribution of Grain Tax in Royal Compendium

Figure 11. Second Digit Distribution of Grain Tax in Royal Compendium

Figures 10 and 11 show the 1BL and 2BL analyses of the national grain tax in rice and beans, paid to

the central government by farmers throughout the country in the early 19th century. Split among 70

categories of place and crop and totaling 107,895 sŏk (1 sŏk equals approximately 40 gallons), the numbers





provided for the grain tax again show no statistical difference from the 1BL distribution (chi-square of 11.68,

p=0.166)) and 2BL distribution (chi-square 6.07 p=0.73)). Of course, this level of high aggregation might

well cover a wide variety of departures from Benford distributions that have canceled each other out, but as

they stand, these numbers diplay no discernible irregularities.

The final category examined was the potentially contentious one of military personnel. The

compendium lists 180,729 men in 79 separate categories. The results of the 1BL and 2BL analysis are shown

in Figures 12 and 13. Unlike other data presented in this paper, these provide statistically sufficient evidence

to reject the null hypothesis, both for first and second digits. The chi-square for 1BL is 21.95 (0.005) and for

2BL it is 38.93 (p=.000). Perhaps some of these high chi-square results stem from low cell counts, but

clearly the very high counts for 1 (1BL) and 0 (2BL) bear closer examination. These data appear to break

decisively from a Benford distribution and hence admit the possibility of either special circumstances that

might generate an excess of 1s, 0s, and 5s or, alternatively, potential abuse of the military rolls. One

plausible possibility would seem that size of military units in created an excess of 1s in the first digit

position and of 0s in the second position.

Figure 12. First Digit Distribution of Military Manpower in Royal Compendium



M. Hejtmanek | 19


Figure 13. Second Digit Distribution of Military Manpower in Royal Compendium

5. CONCLUSION

To a surprising degree, a wide variety of data introduced in this paper from 15th- and 19th- centuries

om Chosŏn Korea statistically match Benford 1BL and 2BL distributions. This consonance is both indicative

of the general underlying power of Benford-like distributions to model data drawn from complex social

arrangements on a variety of scales and its potential use for further, more intense investigations at finer

levels of data aggregation. The 19th century military roster data show departures from the Benford

distributions, both1BL and 2BL, that warrant further study to understand their sources, which may well have

stemmed from the nature of military unit sizes.

While the general absence of non-Benford distributions of the data cannot be used as proof that

malfeasance was not occurring, it is nonetheless reassuring that in cases of household numbers and their

recorded population (15th century), strategic forest preserves, tax-exempt land plots, and grain tax figures

(19th century), there appeared no evidence of gross manipulation or wholesale manufacture of numbers. On

balance, this preliminary examination of two of Chosŏn Korea’s most important statistical compendia has

strengthened our general confidence in their utility and give credence to historical interpretations of the

Chosǒn period as superintended by a complex and sophisticated bureaucracy capable of producing a variety

of statistical information that passes twenty-first century tests of overall lack of gross unreliability.

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behind the veil of power: state statistics and benford’s law in chosǒn korea

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