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Breaking Benford’s law: A statistical analysis of Covid-19 data

using the Euclidean distance statistic

Leonardo Campanelli1∗

1All Saints University School of Medicine, 5145 Steeles Ave., Toronto (ON), Canada

(Dated: October 26, 2022)

Using the Euclidean distance statistical test of Benford’s law, we analyze the Covid-19 weekly

case counts by country. While 62% of the 100 countries and territories considered in the present

study conforms to Benford’s law at a signiﬁcant level α= 0.05 and 17% at a signiﬁcant level

0.01 ≤α < 0.05, the remaining 21% shows a deviation from it (pvalues smaller than 0.01). In

particular, 5% of countries “breaks” Benford’s law with a pvalue smaller than 0.001.

I. INTRODUCTION

At the end of the 19th century, Newcomb [1] noticed that the ﬁrst-digit distribution of logarithms were not uniform,

as one would expect, but rather followed the rule

PB(d) = log1 + 1

d,(1)

where PB(d) is the probability of the ﬁrst signiﬁcant digit d. About 60 years later, Benford [2] rediscovered Newcomb’s

rule (hereafter Benford’s law), extended the law to arbitrary logarithmic bases and to multiple digits, and successfully

tested the law against 20 very diﬀerent data sets, like physical constants, deaths rates, populations of cities, length of

rivers, etc.

Although it is now known that some distributions satisfy Benford’s law (see [3] and references therein) and that

particular principles lead to the emergence of the Benford phenomenon in data [4], no general criteria has be found

that fully explain when and why Benford’s law holds for a generic set of data. Compliance to Benford’s law has been

recently tested on very disparate data sets, from natural sciences [5] to the general framework of detecting fraud, as

in payment of taxes [6] and campaign ﬁnance [7] (for theoretical insights and general applications of Benford’s law,

see [8]). However, rejection of tests on data whose underlying distribution is not known to follow Benford’s law should

not be used as a tool to uncover error or, more importantly, fraud. This is particularly true for Covid-19 data since

there is no theoretical basis or suﬃcient empirical evidence that these data follow a Benford distribution.

The ﬁrst application of Benford’s law to the study of Covid-19 data, in particular to daily and cumulative case and

death counts, is due to Sambridge and Jackson [9], while the most recent work on the “Benfordness” of Covid-19 data

is by Farhadi [10]. Using diﬀerent statistical tests, the authors of both studies conclude that, in general, Covid-19

data conform to a Benford’s distribution and also indicate “anomalies” in the data of some countries. The results

of these and similar analyses, however, cannot be completely trusted for reasons discussed in Sec. II. Here, we will

describe the statistical approach used to test the compliance of Covid-19 data to Benford’s law and will also present

our results. These are not in complete disagreement with previous results in the literature and clearly show that, in

some countries, Covid-19 “breaks” Benford’s law.

II. METHOD AND RESULTS

It is well known that the compliance of data sets to Benford’s law improves as the range of the data increases.

Daily conﬁrmed cases and daily death cases are then not appropriate when checking for the compliance of Covid-19

ﬁrst-digit distributions to Benford’s law because they typically extend over very few orders of magnitude. Another

possibility would be the use of cumulative data. The disadvantage of using this type of data is that as cumulative

cases numbers begin to ﬂatten (especially after a Covid-19 “wave” has passed), ﬁrst digits tend to become all the

same, thus distorting relative digit frequencies. In order to overcome the above problems for Covid-19 data, we will

only analyze the data on weekly conﬁrmed cases by country: they extend, at least for about 45.0% of countries, over

4 order of magnitudes, and do not ﬂatten.

The most common tests in use for testing whether an observed sample of size Nsatisﬁes Benford’s law are the

Pearson’s χ2, Kolmogorov-Smirnov, and Kuiper tests. However, such tests are based on the null hypothesis of a

continuous distribution, and are generally conservative for testing discrete distributions as the Benford’s one [11].

2

62%

17%

16%

5%

FIG. 1: Percentages of countries in a given range of pvalues of the Euclidean distance statistic for the ﬁrst-digit distribution of

Covid-19 weekly case counts by country: from top and clockwise, p≥0.05 (green), 0.01 ≤p < 0.05 (yellow), 0.001 ≤p < 0.01

(red), and p < 0.001 (purple).

This problem can be overcome if one uses the results by Morrow [3] who has recently found asymptotically valid test

values for these statistics under the speciﬁc null hypothesis that Benford’s law holds.

Other tests have been recently proposed, based on new statistics such as the “max” statistic, m, introduced by

Leemis et al. [12], and the “normalized Euclidean distance” statistic, d∗, introduced by Cho and Gaines [7]. At the

moment of their introduction, however, the properties of the corresponding estimators were not well understood and

no test values were reported. These problems were solved by Morrow [3], who provided asymptotically test values for

those statistics too.

Recently enough [13], we have found, by means of Monte Carlo simulations, the (empirical) cumulative distribution

function (CDF) of the “Euclidean distance” statistic, d∗

N, which is based on the statistic d∗and was introduced by

Morrow. It is deﬁned as [3]

d∗

N=v

u

u

tN

9

X

d=1

[P(d)−PB(d)]2,(2)

where P(d) is the observed ﬁrst-digit frequency distribution. 1

In the following, we will use this statistic to study the ﬁrst-digit distribution of Covid-19 weekly case counts

by country since this is the only statistic, among the ones discussed before and analyzed by Morrow, with known

distribution. In particular, we will use its CDF to evaluate pvalues as p= 1 −CDF(d∗

N).

Data are from the World Health Organization (WHO) [16] and updated to December 20, 2021 (two years from the

start of the pandemic). Of the 222 countries and territories aﬀected by Covid 19, only 100 have Covid-19 weekly case

counts with range spanning 4, or more, orders of magnitude. These countries and territories are shown in Tab. I and

grouped in six diﬀerent regions [16]. Also shown is the range of weekly cases, [Nmin, Nmax ], the number of weeks, N,

the Euclidean distance, d∗

N, and the corresponding pvalue. Notice that the CDF of d∗

N, and then the pvalues, are

reliable up to the second decimal place if 0.28 < d∗

N<1.85 and up to the third decimal place otherwise [13]. In the ﬁrst

case, the uncertainty on pis ±0.001, while in second case is ±0.0001. In Tab. I, the last digits in parentheses refer to

these errors. For example, p= 0.27(4) stands for p= 0.274 ±0.001, while p= 0.000(2) stands for p= 0.0002 ±0.0001.

As shown in Fig. 1, while the great majority of countries (79%) conform to Benford’s law (p≤0.01), 5% of them

show a large deviation from it, having pvalues smaller than 0.001. 2

In Fig. 2, we show the observed ﬁrst-digit frequency distributions of weekly case counts for 15 selected countries

superimposed to Benford’s law. Represented countries are China (where the pandemic started), the United States of

America (with the largest total number of cases), India (with the largest range of weekly case counts, Nmax /Nmin),

1The d∗statistic is deﬁned as d∗=qP9

d=1 [P(d)−PB(d)]2/D, where D=qP8

d=1 P2

B(d)+[P(9) −1]2'1.03631 is a normalization

factor that assures that the normalized Euclidean distance is bounded by 0 and 1. A measure of ﬁt to check concordance with Benford’s

law has been proposed by Goodman [14]. His “rule of thumb”, which has been used in the literature (see, e.g., [15]), but whose statistical

validity has been criticized in [13], is that compliance to Benford’s law occurs when d∗≤0.25.

2It is worth observing that the use of the Cho-Gaines’ normalized Euclidean distance d∗together with Goodman’s rule-of-thumb for

compliance to Benford’ law would give a highly questionable compliance to Benford’s law for all countries excepted Honduras, for which

d∗= 0.260, and Tanzania, with d∗= 0.251.

3

Tanzania (with the smallest sample size N), Mauritius (with the smallest total number of cases), Algeria (with the

smallest range of weekly case counts), Vietnam, Thailand, and Poland (the outliers in the ﬁrst box plot of Fig. 4 with

the world largest pvalues), Honduras, Qatar, Belarus, Cuba, and Egypt (with the smallest pvalues, p < 0.001), and

Canada (with the smallest pvalue in the interval 0.001 ≤p < 0.01). It is worth noticing that, although the ﬁrst six

countries in Fig. 1 have very disparate statistical properties (such as sample size, total number of cases, and range of

weekly cases), they all conform to Benford’s law at a signiﬁcant level of 0.01 (excluding Mauritius and Algeria, the

other four countries conform to Benford’s law at a signiﬁcant level of 0.05).

In Fig. 3, we show the percentages of countries in a given range of pvalues, as in Fig. 1, this time grouped in six

diﬀerent regions of the world [16]: Africa, Americas, Eastern Mediterranean, Europe, South-Est Asia, and Western

Paciﬁc. As it is clear from the pie charts, Africa conforms very well to Benford’s law, all countries in this region having

pvalues larger than 0.01. Also, South-Est Asian and Western Paciﬁc countries conform well to Benford’s law, the

only countries with a pvalue less than 0.01 being Maldives (for South-Est Asia), and Philippines and Australia (for

Western Paciﬁc). Countries in Americas (Eastern Mediterranean), instead, show the largest deviation from Benford’s

law: only about 41% (53%) of them have pvalues bigger than 0.05, while about 12% (13%) have pvalues below 0.001.

In Fig. 4, we present box-and-whisker plots for the pvalues of all 100 countries and territories analyzed in this

study and countries in the six diﬀerent regions of the world. All distributions are positively skewed, with medians well

below 0.5. This indicates that the ﬁrst-digit distribution of Covid-19 weekly case counts by country deviates somehow

from Benford’s law on a “global” scale. 3Such a deviation is, however, to be expected for the reasons explained

in [13]. Indeed, Benford’s law does not represent a true law of numbers: some distributions can be “close” to but not

exactly Benford’s, and this regardless of data quality; also, Benford’s law emerges in the limit of inﬁnite range of the

underlying distribution, condition which is never realized in practice.

Our conclusion is twofold: since conformity to Benford’s law cannot be rejected at a signiﬁcant level of 0.01 for most

of the countries (79%), the ﬁrst-digit distribution of Covid-19 weekly case counts by country follows Benford’s law

and then can be used to detect possible “anomalies” in Covid-19 count data. In our case, data from Canada, Jordan,

Puerto Rico, Greece, Philippines, Belgium, Tunisia, Latvia, Paraguay, Sweden, Guatemala, Pakistan, Kazakhstan,

Maldives, Australia, and Russia show a possible anomalous behaviour (0.001 ≤p < 0.01), while anomalies are

certainly present in the data of Honduras, Qatar, Belarus, Cuba, and Egypt (p < 0.001).

III. CONCLUSIONS

We have analyzed the Covid-19 weekly case counts by country, as provided by the World Health Organization,

updated to December 20, 2021. We worked under the null hypothesis that the ﬁrst-digit distribution of those counts

follows a Benford’s distribution. The choice of weekly conﬁrmed cases instead of daily ones came from the requirement

of having counts that extended over many order of magnitudes so to improve the compliance of the data sets to

Benford’s law. For the same reason we did not consider daily and weekly death counts. Also, cumulative cases were

not considered as their numbers ﬂatten (especially at the end of a “wave”), thus distorting relative digit frequencies.

Out of the 222 countries aﬀected by Covid 19, we considered only those ones with weekly counts spanning at least 4

orders of magnitude. This choice reduced the study to the analysis of the data from 100 countries and territories. In

order to test the null hypothesis, we used the Euclidean distance test introduced in [3] and developed in [13], which

avoids the speciﬁc problems introduced by other statistical tests.

Our analysis shows that the majority of countries (62%) conforms to Benford’s law at a signiﬁcant level of 0.05.

However, 5% of countries (Honduras, Qatar, Belarus, Cuba, and Egypt) “break” Benford’s law with pvalues smaller

than 0.001.

∗Electronic address: leonardo.s.campanelli@gmail.com

[1] S. Newcomb, “Note on the frequency of use of diﬀerent digits in natural numbers,“ Am. J. Math. 4, 39 (1881).

3Such a deviation can be quantiﬁed by a Kolmogorov-Smirnov (KS) statistical test for the distribution of pvalues, whose CDF is

CDF(p) = p. The values (degrees of freedom) of the KS statistic for all countries and the ones in the six regions are 0.4295 (100),

0.4487 (13), 0.6165 (17), 0.6567 (15), 0.3379 (38), 0.5208 (8), and 0.3639 (9), respectively. Accordingly, conformance to Benford’s law is

rejected at a signiﬁcant level of 0.001 [17] in the case of all countries, and countries in Americas, Eastern Mediterranean, and Europe.

It is rejected at a signiﬁcant level of 0.01 for African countries. It is not rejected at a signiﬁcant level of 0.01 for South-East Asian

countries, and it is not rejected at a signiﬁcant level larger than 0.20 for the case of Western Paciﬁc countries.

4

[2] F. Benford, “The Law of Anomalous Numbers,” Proceedings of the American Physical Society 78, 551 (1938).

[3] J. Morrow, “Benford’s Law, Families of Distributions and a Test Basis” (Centre for Economic Performance, London, 2014).

[4] T. P. Hill, “The signiﬁcant-digit phenomenon,” Am. Math. Mon. 102, 322 (1995); “Base-invariance implies Benford’s law,”

Proc. Am. Math. Soc. 123, 887 (1995); “A statistical derivation of the signiﬁcant-digit law”, Stat. Sci. 10, 354 (1995).

[5] M. Sambridge, H. Tkalˇci´c, and A. Jackson, “Benford’s law in the natural sciences,” Geophys ˙

Res. Lett. 37 L22301 (2010).

[6] M. Nigrini, “A taxpayer compliance application of Benford’s law,” Journal of the American Taxation Association 18, 72

(1996).

[7] W. K. T. Cho and B. J. Gaines, “Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance,” Am.

Stat. 61, 218 (2007).

[8] S. J. Miller (ed.), “Benford’s Law: Theory and Applications,” (Princeton University Press, Princeton, 2015).

[9] M. Sambridge and A. Jackson, “National COVID numbers - Benford’s law looks for errors,” Nature 581, 384 (2020).

[10] N. Farhadi, “Can we rely on COVID-19 data? An assessment od data from over 200 countries worldwide,” Sci. Prog. 104,

1 (2021).

[11] G. E. Noether, “Note on the Kolmogorov statistic in the discrete case,” Metrika 7, 115 (1963)

[12] L. M. Leemis, B. W. Schmeiser, and D. L. Evans, (2000), “Survival Distributions Satisfying Benford’s Law,” Am. Stat.

54, 236 (2000).

[13] L. Campanelli, “On the Euclidean Distance Statistic of Benford’s Law,” submitted to Communications in Statistics -

Theory and Methods.

[14] W. Goodman, “The promises and pitfalls of Benford’s law,” Signiﬁcance 13, 38 (2016).

[15] A. Wei and A. E. Vellwock, “Is COVID-19 data reliable? A statistical analysis with Benford’s law.”

[16] www.covid19.who.int

[17] S. Facchinetti, “A procedure to ﬁnd exact critical values of Kolmogorov-Smirnov test,” Ital. J. Appl. Stat. 21, 337 (2009).

5

TABLE I: The Euclidean distance d∗

Nin Eq. (2) and its corresponding pvalue for the ﬁrst-digit distribution of Covid-19 weekly

case counts for 100 countries. Also indicated is the range of cases, [Nmin , Nmax ], and the number of weeks, N. Counts are from

WHO [16] and are updated to December 20, 2021. (Digits in parentheses indicate a statistical error on those digits of ±1).

Country Range N d∗

Np

Africa

Algeria [5,10524] 96 1.4079 0.02(9)

Botswana [1,15884] 87 1.0374 0.24(3)

Ethiopia [3,19940] 94 0.8937 0.43(8)

Kenya [3,19023] 94 1.2771 0.06(7)

Mauritius [1,10258] 80 1.4535 0.02(1)

Mozambique [2,13268] 92 1.1051 0.17(5)

Namibia [1,12944] 89 1.1731 0.12(2)

Nigeria [1,12531] 95 0.6236 0.84(9)

South Africa [7,162987] 95 1.5317 0.01(2)

Tanzania [4,24307] 23 1.2457 0.08(0)

Uganda [1,22511] 90 0.7271 0.70(8)

Zambia [1,19058] 93 1.3057 0.05(7)

Zimbabwe [1,26671] 93 0.9221 0.39(5)

Americas

Argentina [16,219910] 95 1.5532 0.01(1)

Brazil [6,533024] 96 1.2301 0.08(9)

Bolivia [7,19834] 94 1.0026 0.28(4)

Canada [2,60784] 100 1.8364 0.00(1)

Colombia [5,204556] 95 1.3949 0.03(2)

Costa Rica [9,17469] 95 1.3167 0.05(3)

Cuba [8,64196] 94 2.0674 0.000(1)

Dominican Republic [4,11168] 95 1.3509 0.04(3)

Ecuador [5,14597] 95 1.3332 0.04(8)

Guatemala [5,26678] 94 1.6470 0.00(5)

Honduras [6,10595] 94 2.6172 0.000(0)

Mexico [5,128779] 96 1.1353 0.15(0)

Paraguay [5,20955] 95 1.6844 0.00(4)

Peru [9,60739] 95 1.0690 0.20(9)

Puerto Rico [7,32162] 93 1.7721 0.00(2)

Uruguay [6,26378] 94 0.8801 0.46(0)

U.S.A. [12,1745361] 101 0.7242 0.71(2)

Eastern Mediterranean

Afghanistan [3,12314] 96 1.2214 0.09(3)

Egypt [5,10778] 96 1.9710 0.000(4)

Iran [47,269975] 97 1.3850 0.03(4)

Iraq [2,83098] 96 1.2867 0.06(4)

Jordan [5,57666] 95 1.7989 0.00(1)

Lebanon [5,33605] 97 1.1526 0.13(7)

Libya [1,19510] 92 1.4154 0.02(8)

Morocco [6,64784] 95 1.1256 0.15(8)

Oman [6,17783] 96 1.0093 0.27(6)

Pakistan [2,40287] 95 1.6157 0.00(6)

Palestine [8,17509] 96 1.0512 0.22(8)

Qatar [7,13049] 96 2.4137 0.000(0)

Saudi Arabia [5,30925] 95 1.2266 0.09(1)

Tunisia [5,52076] 95 1.7322 0.00(2)

U.A.E [2,26285] 100 0.9135 0.40(8)

Europe

Armenia [1,14417] 95 0.7368 0.69(3)

Austria [8,96094] 96 0.8381 0.52(8)

Azerbaijan [2,29155] 96 0.6744 0.78(5)

Belarus [1,14213] 96 2.2927 0.000(0)

Country Range N d∗

Np

Belgium [1,125246] 96 1.7387 0.00(2)

Bosnia and Herzegovina [2,11122] 95 0.6642 0.79(9)

Bulgaria [2,32962] 95 1.4023 0.03(0)

Croatia [1,37433] 96 0.6771 0.78(1)

Czechia [27,127489] 95 0.9291 0.38(5)

Denmark [3,78981] 96 1.4868 0.01(7)

Estonia [1,11930] 96 1.4258 0.02(6)

Finland [1,16510] 98 0.7175 0.72(3)

France [1,504469] 100 0.8111 0.57(2)

Georgia [3,33665] 96 0.9460 0.36(0)

Germany [2,406754] 99 0.8982 0.43(1)

Greece [7,47411] 96 1.7715 0.00(2)

Hungary [7,70400] 95 1.1028 0.17(7)

Ireland [1,53846] 96 1.0170 0.26(7)

Israel [1,65917] 97 0.7419 0.68(5)

Italy [3,257579] 98 1.3064 0.05(6)

Kazakhstan [6,56120] 94 1.5923 0.00(8)

Latvia [3,16957] 95 1.6877 0.00(4)

Lithuania [1,20730] 96 0.7973 0.59(5)

Moldova [1,11680] 95 1.4101 0.02(9)

Netherlands [2,156007] 96 1.0607 0.21(8)

Norway [1,33281] 97 1.0084 0.27(7)

Poland [6,192441] 95 0.4811 0.96(3)

Portugal [2,86549] 95 1.4460 0.02(3)

Romania [3,104668] 96 1.1152 0.16(6)

Russia [5,281305] 95 1.5750 0.00(9)

Serbia [1,49995] 95 0.9512 0.35(3)

Slovakia [1,61514] 95 1.2145 0.09(7)

Slovenia [2,22657] 95 1.0019 0.28(5)

Spain [1,245818] 99 0.6382 0.83(2)

Sweden [1,46511] 97 1.6545 0.00(5)

Turkey [6,414312] 94 1.4798 0.01(8)

U.K. [1,683874] 100 1.0711 0.20(7)

Ukraine [1,153131] 95 1.4855 0.01(7)

South-East Asia

Bangladesh [7,99693] 94 1.3715 0.03(7)

India [1,2738957] 97 1.2935 0.06(1)

Indonesia [10,350273] 95 1.2031 0.10(4)

Maldives [1,11401] 94 1.5900 0.00(8)

Myanmar [4,40004] 92 0.8451 0.51(6)

Nepal [4,61814] 91 0.9980 0.29(0)

Sri Lanka [5,41519] 95 1.2816 0.06(6)

Thailand [1,150652] 102 0.4247 0.98(2)

Western Paciﬁc

Australia [3,45560] 100 1.5845 0.00(8)

China [1,31333] 104 0.6523 0.81(4)

Japan [1,156931] 101 1.1777 0.11(9)

Malaysia [3,150933] 100 0.8115 0.57(2)

Mongolia [1,36698] 91 1.0876 0.19(1)

Philippines [1,144991 97 1.7711 0.00(2)

Singapore [4,25950] 101 0.8253 0.54(9)

South Korea [3,47825] 101 1.2551 0.07(7)

Vietnam [1,125955 97 0.4202 0.98(4)

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7

8

9

1

2

3

4

5

6

7

8

9

d

0.1

0.2

0.3

0.4

0.5

fHdL

Belarus

æ

æ

æ

æ

æ

æ

æ

æ

æ

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

d

0.1

0.2

0.3

0.4

0.5

fHdL

Cuba

æ

æ

æ

æ

æ

æ

æ

æ

æ

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

d

0.1

0.2

0.3

0.4

0.5

fHdL

Egypt

æ

æ

æ

æ

æ

æ

æ

æ

æ

1

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

d

0.1

0.2

0.3

0.4

0.5

fHdL

Canada

FIG. 2: Observed ﬁrst-digit frequencies of the Covid-19 weekly case counts for 15 selected countries: China (with the largest

sample size N), USA (with the largest total number of cases), India (with the largest range of weekly case counts), Tanzania

(with the smallest sample size N), Mauritius (with the smallest total number of cases), Algeria (with the smallest range of

weekly case counts), Vietnam, Thailand, and Poland (the outliers in the ﬁrst box plot of Fig. 4 with the world largest pvalues),

Honduras, Qatar, Belarus, Cuba, and Egypt (with the smallest pvalues, p < 0.001), and Canada (with the smallest pvalue in

the interval 0.001 ≤p < 0.01). The (blue) continuous lines represent Benford’s law.

7

Africa

76.9%

23.1%

Americas

41.2%

23.5%

23.5%

11.8%

Eastern Meditteranean

53.3%

13.4%

20.0%

13.3%

Europe

63.2%

18.4%

15.8%

2.6%

South-East Asia

75.0%

12.5%

12.5%

Western Pacific

77.8%

22.2%

FIG. 3: Percentages of countries in diﬀerent regions of the world in a given range of pvalues of the Euclidean distance statistic

for the ﬁrst-digit distribution of Covid-19 weekly case counts by country. Ranges of pvalues in each pie chart are as follows:

from top and clockwise, p≥0.05 (green), 0.01 ≤p < 0.05 (yellow), 0.001 ≤p < 0.01 (red), and p < 0.001 (purple).

ë

ë

ë

ë

ë

ëë

ëë

World

Africa

Americas

E. Mediterranean

Europe

S.-E. Asia

W. Pacific

0.0

0.2

0.4

0.6

0.8

1.0

p

FIG. 4: Box-and-whisker plots for the pvalues of the Euclidean distance statistic for the ﬁrst-digit distribution of Covid-19

weekly case counts of all countries and countries in diﬀerent regions of the world.