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Abstract

To fight COVID-19, global access to reliable data is vital. Given the rapid acceleration of new cases and the common sense of global urgency, COVID-19 is subject to thorough measurement on a country-by-country basis. The world is witnessing an increasing demand for reliable data and impactful information on the novel disease. Can we trust the data on the COVID-19 spread worldwide? This study aims to assess the reliability of COVID-19 global data as disclosed by local authorities in 202 countries. It is commonly accepted that the frequency distribution of leading digits of COVID-19 data shall comply with Benford’s law. In this context, the author collected and statistically assessed 106,274 records of daily infections, deaths, and tests around the world. The analysis of worldwide data suggests good agreement between theory and reported incidents. Approximately 69% of countries worldwide show some deviations from Benford’s law. The author found that records of daily infections, deaths, and tests from 28% of countries adhered well to the anticipated frequency of first digits. By contrast, six countries disclosed pandemic data that do not comply with the first-digit law. With over 82 million citizens, Germany publishes the most reliable records on the COVID-19 spread. In contrast, the Islamic Republic of Iran provides by far the most non-compliant data. The author concludes that inconsistencies with Benford’s law might be a strong indicator of artificially fabricated data on the spread of SARS-CoV-2 by local authorities. Partially consistent with prior research, the United States, Germany, France, Australia, Japan, and China reveal data that satisfies Benford’s law. Unification of reporting procedures and policies globally could improve the quality of data and thus the fight against the deadly virus.

Keywords

COVID-19 data analysis Benford’s law public health data manipulation

Background

The novel virus—also known as Coronavirus Disease 2019, COVID-19, or SARS-CoV-2—is a respiratory illness. The widely accepted understanding is that the virus typically spreads via airborne droplets from an infected patient’s coughs or sneezes.¹ The unpleasant news of SARS-CoV-2 emerged with the World Health Organization (WHO) announcement on January 10, 2020. The WHO distributed a brief report on new pneumonia cases of unknown cause detected in Wuhan City, in the Chinese province Hubei, on December 31, 2019. After several months, the world now faces millions of Coronavirus cases and deaths while COVID-19 has been rapidly propagating.

The severe impact of the fast-paced increase in Coronavirus cases and deaths has intensified in line with the logistic law.² Since the rapid surge of Coronavirus in early 2020, incidents have continued to escalate with unprecedented speed across all countries worldwide. Since the beginning of the pandemic, a tidal wave of data on Coronavirus has emerged, often without scientific proof. The commonly reported measures are “new cases”—individuals testing positive for the virus—and “new deaths”—patients who have died due to infection with COVID-19.

Data on Coronavirus is typically generated and governed by local authorities, governments, national, and subnational agencies around the globe. These statistics, mainly on the known cases, may not tell the entire truth about the pandemic. The unknown instances or asymptomatic infectious individuals—those who carry the virus in their bodies but show no symptoms—add additional complexity in this context, as these are entirely neglected in global measurement.

Balsari et al.³ address their concerns on the quality of information on COVID-19. They propose transparency, thoughtfulness, and steadfast expertise as the foremost principles for the dependability and overall quality of any data, publications, or reports on Coronavirus spread and related crises.

The Johns Hopkins University (JHU) of the United States has been collecting, aggregating, and publishing updates on confirmed cases and deaths for all countries since January 22, 2020. Millions of people track the development of the pandemic by using the John Hopkins data. To effectively fight the pandemic, it is essential to have reliable data that is freely accessible to the global research community. This way, researchers worldwide can create and share new insights. Local authorities can make better policy choices to fight the rapid circulation of the virus. Public confidence in pandemic data can create a sense of urgency for radical but effective socio-economic policies, such as nationwide lockdowns, as witnessed in many territories, for example, in China, France, the United Kingdom, Italy, and Germany.

In its comprehensive assessment and benchmarking of health security and related capabilities across 195 countries—the 2019 Global Health Security Index (GHSI)—the JHU investigated national infrastructures, systems, and policies to respond to epidemics of potential international concern. Only 19% of countries demonstrated sufficient abilities in early detection and reporting.⁴ The inaugural report gave the topmost scores to the United States (ranked 1), Australia (ranked 4), Canada (ranked 5), and other advanced European countries, such as the United Kingdom (ranked 2), France (ranked 11), and Germany (ranked 14). The 2019 GHSI took laboratory systems, real-time surveillance and reporting, epidemiology workforce, and data integration between the human, animal, and environmental health sectors into consideration.

Initial doubts about the credibility of Coronavirus data, especially in the case of China, led to a slow response to the emerging pandemic across the world.⁴ In recent studies on the evolution and reliability of new cases and deaths from countries with the highest COVID-19 incidence, researchers operationalized forensic techniques, such as Benford’s law. Koch and Okamura⁵ confirmed the reliability of the official data coming from COVID-19 flashpoints, such as China, the United States, and Italy. University professors Sambridge and Jackson⁶ tested pandemic data reported by 51 countries from January 16 to April 9, 2020. They found statistical evidence for the authenticity of the Coronavirus data coming from those countries. Wei and Vellwock⁷ analyzed the Benfordness of pandemic data from China and 20 COVID-19 hot spots as of September 1, 2020. By making use of one goodness-of-fit test, Wei and Vellwock found full compliance with the law for the United States, Brazil, India, Peru, South Africa, Colombia, Mexico, Spain, Argentina, Chile, the United Kingdom, France, Saudi Arabia, China, the Philippines, Belgium, Pakistan, and Italy. In this study, significant irregularities were found for Russia combined with a small divergence in case of Iran.⁷ An additional study accompanied by Lee et al. proposed an epidemic growth model and assessed conformation to Benford’s law. Lee et al.⁸ discovered that all countries except for Japan satisfied Benfordness.

All of these studies face three major challenges: first, lack of consistency of data stemming from independent states; second, limitations with the sample size—application of forensic techniques require sufficient levels of data; and third, different approaches to the measurement and statistical techniques used in their own workings. This article overcomes these boundaries by extending the scope of the study to 202 countries worldwide, over a longer timeframe, increasing the number of quantifiable variables for measurement, and operationalizing three proven statistical techniques to advance the goodness-of-fit tests.

Method

Benford’s law and goodness-of-fit tests

Benford’s law is a commonly applied technique for the detection of data manipulation and fraud. Its core idea relates to the frequency of leading digits in naturally generated datasets, known as Benford’s law. In a dataset consisting of arbitrary collected integers, the first digits of the numbers should be distributed across nine orders of magnitude. According to Benford’s law, the leading digits of numbers from randomly generated real-life data are skewed toward Benford’s first digit distribution and follow a very specific logarithmic pattern: 30.1% for one, 17.6% for two, 12.5% for three, 9.7% for four, 7.9% for five, 6.7% for six, 5.8% for seven, 5.1% for eight, and 4.6% for nine.⁹ Benford¹⁰ asserted that, “Benford’s law reflects a profound harmonic truth of nature.”

In an artificially generated dataset distribution, the numbers would not appear in the same frequency as Benford’s law. If data show a geometrical tendency characterized by the non-existence of minima and maxima, such as population or pandemic distributions, one can observe Benford’s law. The idea relies on Newcomb’s¹⁰ findings of 1881, which addressed the probabilities of first-digit numbers as being given by the following equation:

lo g_{10} (\frac{1}{1 + d}), d = 1, 2, 3, \dots, 9

Benford’s law is common practice in social sciences and has been applied in various disciplines, such as finance and accounting,^11–13 politics,¹⁴ and pandemics.^5–8 For example, the groundwork theory was used by Roukema¹⁴ to investigate the reliability of election data in Iran.

The body of knowledge accepts different goodness-of-fit tests to assess the reality to the expected distribution. Three commonly used techniques are known and frequently used: Kolmogorov-Smirnov statistic, Chi-square, and Euclidean distance.

The Kolmogorov-Smirnov statistic (or K-S) is a non-parametric test for discrete data and quantifies the distance between the empirical distribution of samples of observations and the cumulative distribution of Benford’s⁹ first digit probability. The Kolmogorov-Smirnov statistic was applied to detect potential anomalies and incompliance of data in prior research.^12,13 Manipulation is evident if the Kolmogorov-Smirnov statistic is greater than the square root of the total number of the leading digits observed in a probability sample (hereafter referred to as cut-off). Once the K-S statistic is identified, the null hypothesis can be accepted, if: $\sqrt{N} D_{n} > K_{n}$ . The $D_{n}$ is calculated as $D_{n} = Ma x_{x} | F_{n} (x) - F (x) |$ , where $F_{n}$ is the cumulative observed distribution, and F is Benford’s cumulative distribution. For a 5% significance, K is set to 1.36; for a 1%, K is set to 1.63.^15,16 The K-S statistic is the most precise technique to detect anomalies in distributions. It was recently used in a study on COVID-19 by Alberti and Faranda.¹⁷ The non-parametric K-S test is distribution-free and more powerful when the sample size is small.^12,16 This technical characteristic is of particular importance in the case of smaller samples of epidemics.

Another popular technique is the Pearson chi-square ( $χ^{2})$ goodness-of-fit test with a confirmatory null hypothesis that the first digit’s distribution must conform to Benford’s frequency curve.^11,12 The chi-square test is sensitive to the sample size and is not recommended for making inferences when the dataset exceeds 5000 observations.¹² The chi-square statistic makes use of the expected number of observations. If the sample size is too big the null hypothesis will be rejected even if there is no significant difference between the actual and expected subsets. The chi-square test is computed as follows:

χ^{2} = \sum_{i = 1}^{9} \frac{{({\tilde{p}}_{i} - p_{i)})}^{2}}{p_{i}}

For small sample sizes, the chi-square test encounters difficulties too.²⁰ As a less sensitive technique to the sample size, the d-factor (d*) is calculated as follows:

d^{*} = \frac{\sqrt{\sum_{i = 1}^{9} ({\tilde{p}}_{i} - {p_{i)}}^{2}}}{1.03606}

${\tilde{p}}_{i}$ and $p_{i}$ are the observed and expected (Benford) frequencies.¹³ The d*-factor ultimately measures the Euclidian distance. The statistic quantifies the distance between the empirical distribution function of the sample and the cumulative distribution function of the reference dataset after normalization by 1.03606, the maximum possible distance.^7,17 A d^* equal to 0.0 suggests full conformation with the law, while the highest Euclidian distance, d^* = 1.0, would indicate non-conformation to Benford’s distribution. Goodman suggests that a Euclidean distance larger than 0.25 may be an indicator of “non-conforming” to Benford’s expectations.¹⁹

For each jurisdiction in the scope of this study, we postulate and draw the following null hypothesis, H₀: COVID-19 data from J_i adhere to Benford’s law, where J_i stands for individual jurisdiction in the scope of our analysis. This study operationalizes three goodness-of-fit tests stated above based on a 0.05 significance level.

COVID-19 data sampling

A sample consisting of 1000 observations is considered an acceptable size, although the minimum threshold is not specified in the body of knowledge. The measurement of Benfordness only makes sense if the COVID-19 sample size is not too small. Small datasets may not indicate nonconformities to the expected distribution.

It is logical to assume that conformity to Benford’s law rises if the size and range of the underpinning dataset grows. In contrast to prior research, this study employs the theory based on a larger dataset from 202 countries worldwide and a longer time frame.

Datasets used in this research stem from the COVID-19 Data Repository by the Centre for Systems Science and Engineering at Johns Hopkins University in the US; they include daily submissions of 202 national authorities on daily new cases and new deaths. The John Hopkins Institute provides one of the first glimpses of a global view of how the COVID-19 virus is spreading; the related online services were made available to the public from January 22, 2020. All numbers are taken into consideration to evaluate the compliance of the first digit distribution appearing in the 106,274 cumulative integers reported by 202 nations worldwide, from December 31, 2019, to September 24, 2020.

Our observations are built using three variables quantifiable on a daily basis: (1) new cases, (2) new deaths, and (3) new tests. This study uses a new variable, “new tests,” as an additional measure, which was technically not available in prior research, or was purposefully neglected by researchers. New tests indicate the number of individuals identified for being contaminated with the novel Coronavirus. As of writing this report, the author documents 106,274 integers in total (N, or sample size), including 45,031 on new cases and 16,212 on new tests from all countries in the scope. The data contained either no information, or incomplete information, on new tests for some countries, such as China.

Most of the countries affected by COVID-19 have reported sharp growth in new cases and new deaths, and with some delay, new tests. Some countries report data with no substantial growth rates. The largest sample size of the data on the pandemic stems from the United States. On the contrary, the smallest sample sizes, less than 100, belong to Anguilla, the Falkland Islands, Montserrat, the Vatican, the Western Sahara, Greenland, Dominica, Saint Kitts, and Nevis and can be excluded from the analysis. A list of 155 countries with a sample size larger than n = 100 was extracted, resulting in 384 observations on average.

Results

For each country disclosing big data on the COVID-19 spread in their territories, we evaluated the consistency of the leading digits, mainly by applying Benford’s law and Kolmogorov-Smirnov, chi-square, and d*-factor test statistics. Evaluation of the leading digits reported by 202 jurisdictions since the beginning of the pandemic revealed surprising anomalies. Inconsistencies of the COVID-19 data worldwide are illustrated in Figure 1. Countries that fully comply with the distribution of the leading digits are highlighted green. All countries that do not comply with Benford’s law based on the three goodness-of-fit tests are flagged red. Territories highlighted yellow do not obey the law in at least one of the tests conducted in this study. Gray jurisdictions are those areas for which no sufficient sample size was available, and therefore, a statistical assessment was not applicable.

Figure 1.

Worldwide reliability of data on Coronavirus at a glance.

When aggregating all reports from 202 countries, the global data fully complies with Benford’s law (see Figure 2).

Figure 2.

Distribution of the worldwide data on Coronavirus—green: complaint countries; yellow: evidence found for incompliance; red: incompliant countries.

Overall, the author operationalized three variables, new cases, new deaths, and new tests, from 202 countries. For 49 countries or 24% of the observations, the sample size was smaller than 100 (n < 100), which did not allow a statistically significant assessment of Benfordness for these countries. Twenty-seven percent of the countries in scope showed full compliance with the law, followed by 69% (42 territories) that complied based on two measures of goodness-of-fit test. Most notably, 4% of the countries worldwide did not conform to Benford’s Law (see Figure 3). Thus, the null hypothesis can be rejected for these jurisdictions.

Figure 3.

Compliance of countries reviewed at a glance.

Table 1 summarizes the results of the statistical analyses of the COVID-19 data worldwide. Table 1 portrays the frequency of the leading digit distributions for 154 countries. It is evident that the K-S and chi-square tests reveal similar patterns, while Euclidean distance statistics do not always flag anomalies in the distribution of first digits.

Table 1.

Results of the Benford’s leading digit distribution analysis.

Location	N	GHSI Score	GHSI Rank	d*	χ²	p-value	K-S Statistic	Cut-off	Difference	LD1	LD2	LD3	LD4	LD5	LD6	LD7	LD8	LD9
Afghanistan	324	32.3	130	0.05892	3.82103	0.14800	0.03557	0.07556	0.47075	0.26543	0.17901	0.16049	0.09259	0.09877	0.06481	0.07407	0.03395	0.03086
Albania	324	52.9	39	0.11825	16.44921	0.00027	0.10332	0.07556	1.36748	0.40432	0.15432	0.07716	0.10494	0.06790	0.08642	0.03086	0.03704	0.03704
Algeria	393	23.6	173	0.12858	42.30935	0.00000	0.09607	0.06860	1.40032	0.36387	0.09160	0.06616	0.08142	0.07888	0.07634	0.07125	0.08651	0.08397
Andorra	140	30.5	143	0.18386	20.75460	0.00003	0.18014	0.11494	1.56726	0.47143	0.18571	0.11429	0.07857	0.02143	0.01429	0.04286	0.04286	0.02857
Angola	200	25.2	170	0.08443	6.21099	0.04480	0.07300	0.09617	0.75910	0.31500	0.23500	0.12000	0.06500	0.11500	0.04500	0.05000	0.04000	0.01500
Argentina	580	58.6	25	0.09043	20.11116	0.00004	0.07314	0.05647	1.29514	0.37414	0.16379	0.08448	0.07586	0.06552	0.05172	0.08448	0.04655	0.05345
Armenia	363	50.2	44	0.05166	2.94749	0.22907	0.04741	0.07138	0.66422	0.31680	0.15427	0.14325	0.10468	0.09642	0.06887	0.06612	0.03857	0.01102
Aruba	107	-	-	0.12508	6.58341	0.03719	0.09152	0.13148	0.69612	0.35514	0.16822	0.06542	0.01869	0.10280	0.04673	0.08411	0.06542	0.09346
Australia	497	75.5	4	0.02922	3.07255	0.21518	0.03085	0.06100	0.50575	0.30181	0.17304	0.12676	0.11871	0.08853	0.06439	0.04427	0.04829	0.03421
Austria	533	58.5	26	0.06698	5.26173	0.07202	0.05923	0.05891	1.00538	0.29268	0.14822	0.13321	0.06567	0.09756	0.11069	0.07129	0.05629	0.02439
Azerbaijan	352	34.2	117	0.06277	1.23754	0.53861	0.03382	0.07249	0.46653	0.28977	0.15341	0.16761	0.07386	0.11080	0.06534	0.06250	0.03977	0.03693
Bahamas	156	30.6	142	0.09218	8.61333	0.01348	0.10228	0.10889	0.93934	0.36538	0.16026	0.14744	0.12821	0.07051	0.05769	0.02564	0.00641	0.03846
Bahrain	448	39.4	88	0.11430	22.30074	0.00001	0.10234	0.06425	1.59273	0.19866	0.18080	0.15179	0.10938	0.10491	0.07143	0.09375	0.02902	0.06027
Bangladesh	561	35	113	0.10775	43.06962	0.00000	0.13344	0.05742	2.32400	0.34046	0.18538	0.20856	0.09804	0.06061	0.02852	0.03030	0.02139	0.02674
Belarus	413	35.3	108	0.17037	25.50544	0.00000	0.22185	0.06692	3.31516	0.22276	0.07264	0.08475	0.13801	0.16465	0.10169	0.05327	0.07022	0.09201
Belgium	608	61	19	0.07271	10.08304	0.00646	0.08221	0.05516	1.49053	0.35855	0.20066	0.10033	0.06908	0.07895	0.05757	0.06086	0.03783	0.03618
Benin	101	28.8	150	0.05215	1.69873	0.42769	0.06137	0.13533	0.45347	0.32673	0.18812	0.14851	0.08911	0.07921	0.06931	0.02970	0.04950	0.01980
Bolivia	551	35.8	102	0.04306	2.05381	0.35811	0.03480	0.05794	0.60059	0.32849	0.18330	0.11252	0.08348	0.07441	0.07623	0.05263	0.06534	0.02359
Bosnia and Herzegovina	347	42.8	79	0.10367	18.81294	0.00008	0.09160	0.07301	1.25464	0.26225	0.24784	0.15850	0.09510	0.08646	0.08646	0.01153	0.02017	0.03170
Brazil	395	59.7	22	0.06836	1.89846	0.38704	0.03065	0.06843	0.44784	0.33165	0.12658	0.12405	0.12658	0.08608	0.07848	0.03544	0.04304	0.04810
Bulgaria	462	45.6	61	0.04123	1.80008	0.40655	0.04952	0.06327	0.78256	0.32251	0.18831	0.14069	0.08874	0.06061	0.06277	0.03896	0.06061	0.03680
Burkina Faso	195	30.1	145	0.13811	16.93120	0.00021	0.13133	0.09739	1.34851	0.43077	0.16923	0.13333	0.06154	0.05641	0.04103	0.05128	0.02051	0.03590
Cameroon	173	34.4	115	0.08423	7.04689	0.02950	0.09869	0.10340	0.95444	0.36416	0.19075	0.13873	0.10405	0.05780	0.06936	0.01156	0.03468	0.02890
Canada	580	75.3	5	0.07156	4.25977	0.11885	0.04810	0.05647	0.85183	0.30345	0.13966	0.15345	0.14483	0.07069	0.08103	0.04138	0.03103	0.03448
Cape Verde	205	-	-	0.08646	5.42766	0.06628	0.07461	0.09499	0.78548	0.37561	0.17561	0.10732	0.09268	0.04878	0.07805	0.06829	0.03415	0.01951
Central African Republic	135	27.3	159	0.09701	4.88803	0.08681	0.08419	0.11705	0.71922	0.38519	0.17037	0.08148	0.08889	0.05926	0.06667	0.05926	0.02963	0.05926
Chad	159	28.8	150	0.13540	13.67373	0.00107	0.16451	0.10786	1.52528	0.40881	0.23270	0.11950	0.08176	0.03774	0.04403	0.01887	0.03145	0.02516
Chile	571	58.3	27	0.06224	15.94647	0.00034	0.07858	0.05691	1.38073	0.34851	0.19440	0.13135	0.10333	0.06655	0.05604	0.03678	0.02452	0.03853
China	392	48.2	51	0.05438	3.86447	0.14482	0.06382	0.06869	0.92904	0.33163	0.20663	0.12755	0.09439	0.05612	0.05102	0.04337	0.03571	0.05357
Colombia	581	44.2	65	0.08465	1.30055	0.52190	0.08991	0.05642	1.59353	0.29776	0.23580	0.15835	0.06713	0.03614	0.05680	0.06540	0.04475	0.03787
Congo	527	27.2	160	0.06129	7.29080	0.02611	0.07330	0.05924	1.23722	0.34345	0.17837	0.14991	0.10057	0.04744	0.05313	0.05123	0.03226	0.04364
Costa Rica	481	45.1	62	0.11157	26.56976	0.00000	0.10233	0.06201	1.65014	0.40333	0.16008	0.08316	0.07277	0.06237	0.06237	0.05821	0.05198	0.04574
Cote d’Ivoire	410	45.1	62	0.04285	2.67676	0.26227	0.02932	0.06717	0.43649	0.32927	0.16098	0.13415	0.08780	0.09512	0.04634	0.05854	0.04146	0.04634
Croatia	498	53.3	38	0.07123	9.76902	0.00756	0.05111	0.06094	0.83869	0.34739	0.18072	0.10643	0.04819	0.07229	0.07631	0.04819	0.05823	0.06225
Cuba	440	35.2	110	0.07064	20.08709	0.00004	0.07827	0.06484	1.20725	0.28409	0.22045	0.15909	0.11364	0.07727	0.04091	0.04773	0.03636	0.02045
Cyprus	189	43	77	0.08111	2.66605	0.26368	0.10699	0.09893	1.08157	0.34392	0.22222	0.14286	0.05820	0.05291	0.05820	0.04762	0.04762	0.02646
Czech Republic	577	52	42	0.06569	7.46975	0.02388	0.06647	0.05662	1.17404	0.25130	0.17158	0.11265	0.11438	0.09879	0.06759	0.09012	0.06066	0.03293
Democratic Republic of Congo	413	52	42	0.05671	1.93230	0.38054	0.05417	0.06692	0.80952	0.31477	0.16949	0.17191	0.09685	0.05327	0.06053	0.05327	0.03390	0.04600
Denmark	520	70.4	8	0.14067	44.12875	0.00000	0.13362	0.05964	2.24037	0.43462	0.13462	0.11538	0.07692	0.05769	0.04231	0.05192	0.04231	0.04423
Djibouti	171	23.2	175	0.08462	2.21787	0.32991	0.09391	0.10400	0.90293	0.35088	0.20468	0.14035	0.05848	0.05263	0.03509	0.04094	0.07602	0.04094
Dominican Republic	546	38.3	91	0.03805	1.28909	0.52490	0.03726	0.05820	0.64024	0.30037	0.19231	0.14652	0.09707	0.06044	0.05128	0.04945	0.06227	0.04029
Ecuador	557	50.1	45	0.04613	4.05822	0.13145	0.02990	0.05763	0.51896	0.27110	0.19749	0.11490	0.12208	0.08438	0.06643	0.05386	0.03950	0.05027
Egypt	384	39.9	87	0.16732	46.55207	0.00000	0.15733	0.06940	2.26698	0.45833	0.13281	0.09635	0.08073	0.03646	0.04948	0.04427	0.05729	0.04427
El Salvador	455	44.2	65	0.23408	17.14973	0.00019	0.15816	0.06376	2.48071	0.23956	0.39560	0.07033	0.07033	0.03736	0.03077	0.05714	0.04615	0.05275
Estonia	428	57	29	0.07941	10.79072	0.00454	0.07283	0.06574	1.10791	0.37383	0.17290	0.11916	0.06542	0.07009	0.06542	0.05841	0.04673	0.02804
Ethiopia	465	40.6	84	0.08580	14.90725	0.00058	0.07534	0.06307	1.19464	0.37634	0.14624	0.09677	0.07957	0.07527	0.05591	0.06452	0.05161	0.05376
Finland	483	68.7	10	0.06274	6.92929	0.03128	0.06958	0.06188	1.12446	0.35197	0.19462	0.09938	0.08489	0.08075	0.05176	0.04348	0.04555	0.04762
France	519	68.2	11	0.04589	0.55742	0.75676	0.03962	0.05970	0.66369	0.28709	0.15029	0.12524	0.12524	0.09056	0.05588	0.07129	0.04046	0.05395
Gabon	142	20	186	0.18313	19.69751	0.00005	0.17083	0.11413	1.49683	0.47183	0.16197	0.09859	0.05634	0.04225	0.03521	0.02817	0.07746	0.02817
Gambia	118	34.2	117	0.16104	14.77914	0.00062	0.16071	0.12520	1.28366	0.44915	0.18644	0.12712	0.08475	0.03390	0.05085	0.03390	0.00000	0.03390
Georgia	208	52	42	0.07379	2.74890	0.25298	0.04515	0.09430	0.47884	0.34615	0.16827	0.10096	0.06250	0.09135	0.07212	0.02885	0.08173	0.04808
Germany	408	66	14	0.03210	0.25456	0.88049	0.02592	0.06733	0.38499	0.31127	0.16912	0.12990	0.09559	0.07108	0.08333	0.06863	0.02941	0.04167
Ghana	346	35.5	105	0.04689	1.20564	0.54727	0.02643	0.07311	0.36154	0.28902	0.16763	0.16474	0.10405	0.06936	0.04913	0.06647	0.04913	0.04046
Greece	485	53.8	37	0.05685	9.01946	0.01100	0.05779	0.06175	0.93587	0.29485	0.21443	0.15052	0.08247	0.08660	0.06598	0.04742	0.02268	0.03505
Guam	150	-	-	0.08399	6.44431	0.03987	0.10300	0.11104	0.92756	0.36000	0.22000	0.12000	0.08667	0.07333	0.04000	0.02667	0.04000	0.03333
Guatemala	339	32.7	125	0.06138	2.62983	0.26850	0.06762	0.07387	0.91541	0.29794	0.21829	0.15339	0.09440	0.04720	0.05605	0.04130	0.05015	0.04130
Guinea	412	32.7	125	0.07391	7.95633	0.01872	0.06308	0.06700	0.94142	0.36408	0.14563	0.10194	0.10437	0.08981	0.06311	0.05583	0.03883	0.03641
Guyana	184	31.7	137	0.12139	12.89060	0.00159	0.12626	0.10026	1.25933	0.41304	0.19022	0.11957	0.08152	0.04891	0.03804	0.05435	0.02174	0.03261
Haiti	243	31.5	138	0.08882	10.31992	0.00574	0.11993	0.08724	1.37465	0.35802	0.21399	0.14403	0.10288	0.04115	0.05761	0.04115	0.02058	0.02058
Honduras	353	27.6	156	0.07894	2.64277	0.26677	0.09208	0.07239	1.27215	0.30312	0.11898	0.08782	0.12465	0.11048	0.07932	0.06516	0.05666	0.05382
Hungary	526	54	35	0.04971	3.20087	0.20181	0.06530	0.05930	1.10121	0.31939	0.19202	0.15589	0.08365	0.05894	0.04943	0.04373	0.05323	0.04373
Iceland	366	46.3	58	0.06730	3.55047	0.16944	0.06785	0.07109	0.95448	0.30328	0.12842	0.11749	0.08197	0.10929	0.09563	0.06831	0.06557	0.03005
India	573	46.5	57	0.08859	43.89474	0.00000	0.10435	0.05681	1.83671	0.31065	0.11344	0.10995	0.07853	0.06108	0.08551	0.06806	0.07853	0.09424
Indonesia	541	56.6	30	0.05831	6.10700	0.04719	0.04835	0.05847	0.82696	0.34935	0.14972	0.13678	0.09612	0.06654	0.06285	0.05360	0.05545	0.02957
Iran	548	37.7	97	0.27089	44.30276	0.00000	0.31862	0.05810	5.48435	0.39416	0.40146	0.03467	0.02190	0.03285	0.02555	0.03285	0.02190	0.03467
Iraq	513	25.8	167	0.05289	4.13433	0.12654	0.04188	0.06005	0.69749	0.28460	0.20078	0.11501	0.09357	0.04678	0.06238	0.08577	0.06433	0.04678
Ireland	515	59	23	0.04818	2.36925	0.30586	0.04593	0.05993	0.76644	0.26990	0.16117	0.15146	0.11456	0.07184	0.06796	0.04854	0.06019	0.05437
Israel	597	47.3	54	0.08123	17.97054	0.00013	0.08749	0.05566	1.57182	0.37688	0.18760	0.11390	0.07873	0.06533	0.05360	0.05025	0.03518	0.03853
Italy	587	56.2	31	0.08763	25.49936	0.00000	0.06420	0.05613	1.14376	0.23680	0.21976	0.15332	0.11584	0.09199	0.07496	0.04600	0.02896	0.03237
Jamaica	208	29	147	0.08741	2.00040	0.36781	0.08550	0.09430	0.90669	0.34135	0.22115	0.09615	0.07212	0.07692	0.10096	0.03365	0.04808	0.00962
Japan	423	59.8	21	0.05300	3.51027	0.17288	0.03706	0.06613	0.56047	0.33806	0.14894	0.11820	0.11584	0.08511	0.06619	0.04965	0.05201	0.02600
Jordan	211	42.1	80	0.06803	1.73080	0.42088	0.05855	0.09363	0.62531	0.34123	0.19431	0.08531	0.10900	0.05213	0.08531	0.05687	0.03791	0.03791
Kazakhstan	459	40.7	83	0.13934	36.60016	0.00000	0.15917	0.06348	2.50736	0.42048	0.21569	0.11111	0.07843	0.03268	0.03050	0.03268	0.03268	0.04575
Kenya	469	47.1	55	0.05529	9.39866	0.00910	0.05052	0.06280	0.80451	0.27292	0.19403	0.14925	0.12154	0.08102	0.07676	0.03838	0.03838	0.02772
Kosovo	299	-	-	0.11847	22.40348	0.00001	0.10034	0.07865	1.27574	0.40134	0.14716	0.08696	0.07023	0.04013	0.06355	0.07358	0.05686	0.06020
Kuwait	500	46.1	59	0.19464	68.90437	0.00000	0.18300	0.06082	3.00883	0.13200	0.16200	0.17400	0.18200	0.09200	0.10800	0.06800	0.05400	0.02800
Kyrgyzstan	263	-	-	0.05160	1.66759	0.43440	0.05765	0.08386	0.68749	0.30418	0.19772	0.13688	0.11787	0.05323	0.04183	0.07224	0.03422	0.04183
Latvia	416	62.9	17	0.20395	76.72280	0.00000	0.19660	0.06668	2.94838	0.49760	0.16827	0.08654	0.06490	0.03365	0.04327	0.04087	0.03606	0.02885
Lebanon	290	43.1	73	0.06173	2.24802	0.32497	0.03693	0.07986	0.46244	0.33793	0.15862	0.08621	0.09655	0.09655	0.08621	0.06207	0.04138	0.03448
Liberia	192	35.1	111	0.14714	24.03366	0.00001	0.15321	0.09815	1.56097	0.43750	0.18229	0.13542	0.07292	0.06771	0.03646	0.01563	0.03125	0.02083
Libya	396	25.7	168	0.04879	8.06399	0.01774	0.05073	0.06834	0.74225	0.31313	0.14141	0.12626	0.08081	0.06566	0.08586	0.07576	0.05808	0.05303
Lithuania	366	55	33	0.10286	15.45418	0.00044	0.08629	0.07109	1.21384	0.24863	0.14208	0.17213	0.13661	0.12842	0.06011	0.06284	0.03005	0.01913
Luxembourg	444	43.8	67	0.07480	16.00578	0.00033	0.04827	0.06454	0.74788	0.34685	0.13514	0.09685	0.07883	0.07207	0.07432	0.07658	0.07658	0.04279
Madagascar	253	40.1	86	0.08334	0.82851	0.66083	0.03618	0.08550	0.42312	0.27668	0.20158	0.10277	0.15415	0.04348	0.09091	0.04348	0.05929	0.02767
Malawi	345	28	154	0.07303	0.77668	0.67818	0.05707	0.07322	0.77947	0.29565	0.13043	0.11884	0.12754	0.12174	0.04638	0.07246	0.03768	0.04928
Malaysia	458	62.2	18	0.10324	27.05785	0.00000	0.06363	0.06355	1.00126	0.36463	0.10699	0.08952	0.07642	0.08297	0.08734	0.07205	0.06769	0.05240
Maldives	364	33.8	121	0.15960	54.00684	0.00000	0.12482	0.07128	1.75110	0.42582	0.11264	0.06044	0.07143	0.04121	0.07967	0.06868	0.06044	0.07967
Mali	239	29	147	0.13518	18.14028	0.00012	0.17154	0.08797	1.94991	0.38912	0.25941	0.11715	0.05858	0.05858	0.02929	0.03766	0.02929	0.02092
Malta	390	37.3	98	0.13200	29.48816	0.00000	0.11438	0.06887	1.66097	0.41538	0.14872	0.07436	0.06154	0.06923	0.05641	0.05385	0.04615	0.07436
Mauritania	130	27.5	157	0.10658	8.38475	0.01511	0.09900	0.11928	0.82998	0.40000	0.16154	0.11538	0.10000	0.08462	0.03846	0.04615	0.03846	0.01538
Mexico	647	57.6	28	0.09477	5.42082	0.06651	0.09659	0.05347	1.80654	0.26275	0.12519	0.11747	0.14065	0.10510	0.11283	0.07264	0.04019	0.02318
Moldova	373	42.9	78	0.04943	2.08433	0.35269	0.03144	0.07042	0.44647	0.32172	0.14477	0.12064	0.08043	0.09383	0.08311	0.07507	0.04558	0.03485
Montenegro	215	43.7	68	0.06039	3.89547	0.14260	0.07593	0.09275	0.81864	0.33488	0.18605	0.12558	0.09302	0.09302	0.08837	0.02791	0.03256	0.01860
Morocco	546	43.7	68	0.14029	29.96671	0.00000	0.17685	0.05820	3.03846	0.39377	0.26007	0.09524	0.06044	0.04396	0.03480	0.04762	0.02198	0.04212
Mozambique	370	28.1	153	0.09656	14.70393	0.00064	0.06657	0.07070	0.94151	0.36757	0.17027	0.05676	0.07568	0.07568	0.08108	0.05676	0.06486	0.05135
Namibia	282	35.6	104	0.15868	31.69942	0.00000	0.14935	0.08099	1.84418	0.45035	0.13475	0.09574	0.06738	0.04610	0.06028	0.05674	0.03901	0.04965
Nepal	448	35.1	111	0.08207	8.41155	0.01491	0.06284	0.06425	0.97798	0.36384	0.12723	0.12054	0.12054	0.06473	0.06696	0.05357	0.04464	0.03795
Netherlands	373	75.6	3	0.08115	7.16578	0.02780	0.05557	0.07042	0.78912	0.35657	0.14477	0.08043	0.08311	0.10456	0.07775	0.05362	0.05630	0.04290
New Zealand	354	54	35	0.06827	3.01711	0.22123	0.04959	0.07228	0.68602	0.27966	0.23446	0.13277	0.10169	0.05367	0.05367	0.06497	0.04520	0.03390
Niger	163	32.2	132	0.11946	8.09486	0.01747	0.13036	0.10652	1.22379	0.39877	0.20859	0.09816	0.06135	0.06748	0.08589	0.01840	0.03067	0.03067
Nigeria	457	37.8	96	0.05216	12.16552	0.00228	0.06528	0.06362	1.02619	0.30416	0.17724	0.15755	0.10722	0.09628	0.06783	0.03282	0.03282	0.02407
Norway	449	64.6	16	0.06981	14.90613	0.00058	0.08274	0.06418	1.28910	0.34076	0.16036	0.16258	0.11804	0.07127	0.04454	0.03786	0.04009	0.02450
Oman	360	43.1	73	0.04838	2.81323	0.24497	0.03789	0.07168	0.52860	0.33889	0.16389	0.13056	0.08889	0.05556	0.06944	0.04444	0.05278	0.05556
Pakistan	562	35.5	105	0.09054	8.23524	0.01628	0.05147	0.05737	0.89718	0.26868	0.25979	0.11032	0.09253	0.07473	0.07295	0.05160	0.03381	0.03559
Palestine	259	-	-	0.13227	11.48345	0.00321	0.14109	0.08451	1.66961	0.20463	0.13127	0.15830	0.16602	0.09266	0.09266	0.07722	0.03089	0.04633
Panama	576	43.7	68	0.02917	3.52989	0.17120	0.02903	0.05667	0.51225	0.30729	0.17882	0.11285	0.07813	0.07292	0.06597	0.07292	0.05903	0.05208
Paraguay	470	35.7	103	0.09253	17.16408	0.00019	0.10598	0.06273	1.68939	0.38085	0.20213	0.11489	0.08511	0.05106	0.04894	0.03191	0.04894	0.03617
Peru	562	49.2	49	0.05480	0.36721	0.83227	0.03216	0.05737	0.56059	0.30783	0.13701	0.16014	0.11032	0.07295	0.07473	0.04804	0.04982	0.03915
Philippines	558	47.6	53	0.07195	7.54699	0.02297	0.06108	0.05757	1.06095	0.27599	0.21326	0.17384	0.08961	0.05735	0.04839	0.04301	0.05735	0.04122
Poland	538	55.4	32	0.09395	17.92717	0.00013	0.11919	0.05863	2.03278	0.35502	0.23792	0.12825	0.06506	0.06320	0.05019	0.04461	0.02230	0.03346
Portugal	593	60.3	20	0.09057	16.79380	0.00023	0.09783	0.05585	1.75173	0.37774	0.18044	0.14165	0.06239	0.04722	0.06745	0.04047	0.04384	0.03879
Qatar	494	41.2	82	0.13908	92.68630	0.00000	0.14508	0.06119	2.37095	0.24696	0.23887	0.14777	0.18219	0.10729	0.03036	0.01619	0.01619	0.01417
Romania	569	45.8	60	0.09861	24.02083	0.00001	0.12350	0.05701	2.16605	0.36731	0.21968	0.13708	0.09842	0.03339	0.04394	0.02460	0.04394	0.03163
Russia	570	44.3	63	0.04844	2.42940	0.29680	0.05339	0.05696	0.93719	0.29123	0.17544	0.11228	0.06667	0.10526	0.08596	0.04737	0.06140	0.05439
Rwanda	375	34.2	117	0.05565	7.19213	0.02743	0.05667	0.07023	0.80687	0.33333	0.15733	0.14400	0.11467	0.08533	0.04800	0.06133	0.03200	0.02400
San Marino	112	31.1	139	0.19726	17.81227	0.00014	0.20157	0.12851	1.56855	0.48214	0.19643	0.07143	0.04464	0.06250	0.07143	0.03571	0.00893	0.02679
Sao Tome and Principe	101	31.1	139	0.21072	27.79948	0.00000	0.20988	0.13533	1.55094	0.49505	0.18812	0.12871	0.06931	0.07921	0.00000	0.00990	0.01980	0.00990
Saudi Arabia	583	49.3	47	0.10518	23.16571	0.00001	0.08935	0.05633	1.58632	0.24871	0.13894	0.18868	0.13551	0.11321	0.07204	0.03431	0.03087	0.03774
Senegal	522	37.9	95	0.16443	62.42164	0.00000	0.13195	0.05953	2.21670	0.43295	0.10728	0.08238	0.07663	0.04598	0.04598	0.04406	0.06705	0.09770
Sierra Leone	205	38.2	92	0.16177	20.76978	0.00003	0.17666	0.09499	1.85983	0.44390	0.20976	0.11220	0.03415	0.04878	0.03902	0.04390	0.02927	0.03902
Singapore	259	58.7	24	0.05282	3.24704	0.19720	0.03459	0.08451	0.40933	0.26641	0.17761	0.15444	0.12355	0.07722	0.06950	0.05405	0.04247	0.03475
Slovakia	385	47.9	52	0.09150	9.75680	0.00761	0.10222	0.06931	1.47479	0.32727	0.25195	0.10130	0.10909	0.05455	0.04156	0.04675	0.03896	0.02857
Slovenia	443	67.2	12	0.10697	20.48284	0.00004	0.09855	0.06462	1.52515	0.39955	0.14673	0.12190	0.06546	0.07223	0.04289	0.04966	0.05192	0.04966
Somalia	155	16.6	194	0.05211	1.73398	0.42022	0.03648	0.10924	0.33399	0.26452	0.20000	0.14194	0.09032	0.09677	0.07742	0.04516	0.04516	0.03871
South Africa	564	54.8	34	0.05868	4.11495	0.12778	0.06821	0.05727	1.19115	0.33156	0.20035	0.13830	0.09574	0.04255	0.04610	0.04610	0.05851	0.04078
South Korea	577	70.2	9	0.09200	18.38158	0.00010	0.07855	0.05662	1.38737	0.37955	0.13172	0.10572	0.07972	0.08666	0.07452	0.04506	0.05199	0.04506
South Sudan	128	21.7	180	0.08677	3.81662	0.14833	0.06619	0.12021	0.55061	0.36719	0.15625	0.14063	0.06250	0.10938	0.04688	0.03906	0.03906	0.03906
Spain	353	65.9	15	0.08566	9.33411	0.00940	0.07437	0.07239	1.02743	0.22663	0.18414	0.14731	0.10765	0.09348	0.05666	0.06516	0.08499	0.03399
Sri Lanka	410	33.9	120	0.19927	67.27079	0.00000	0.21080	0.06717	3.13858	0.48537	0.20244	0.06829	0.05122	0.04878	0.05122	0.02683	0.03415	0.03171
Sudan	340	26.2	163	0.06730	6.96931	0.03066	0.06371	0.07376	0.86373	0.36471	0.16765	0.12647	0.07647	0.07647	0.05882	0.05294	0.04118	0.03529
Swaziland	216	36.5	100	0.07923	4.85263	0.08836	0.05100	0.09254	0.55114	0.25000	0.20833	0.17130	0.08796	0.08333	0.08333	0.05093	0.03241	0.03241
Sweden	474	72.1	7	0.08977	26.72636	0.00000	0.09657	0.06247	1.54587	0.24262	0.16245	0.13924	0.09072	0.04641	0.07595	0.10970	0.08228	0.05063
Switzerland	570	67	13	0.04726	0.83152	0.65984	0.04937	0.05696	0.86666	0.28421	0.15263	0.11579	0.11053	0.10526	0.08596	0.06491	0.03684	0.04386
Syria	187	19.9	188	0.10200	2.30271	0.31621	0.09244	0.09945	0.92952	0.32620	0.17647	0.18182	0.10695	0.03209	0.11230	0.03743	0.02139	0.00535
Taiwan	364	-	-	0.23204	85.19144	0.00000	0.22630	0.07128	3.17461	0.52198	0.18132	0.07967	0.04670	0.03571	0.03571	0.02198	0.04121	0.03571
Tajikistan	181	32.3	130	0.23082	18.23662	0.00011	0.19523	0.10109	1.93131	0.20442	0.07735	0.19337	0.27072	0.09392	0.07735	0.02210	0.05525	0.00552
Thailand	383	73.2	6	0.09132	12.67668	0.00177	0.08991	0.06949	1.29375	0.37598	0.16449	0.15144	0.06005	0.08094	0.06527	0.03655	0.02611	0.03916
Togo	401	32.5	129	0.03355	0.46532	0.79242	0.02345	0.06792	0.34523	0.28928	0.16708	0.12219	0.11721	0.08728	0.04988	0.06484	0.04489	0.05736
Trinidad and Tobago	134	36.6	99	0.09850	4.05525	0.13165	0.10509	0.11749	0.89448	0.38060	0.20149	0.07463	0.08209	0.05970	0.05224	0.05970	0.05224	0.03731
Tunisia	415	33.7	122	0.04698	3.53984	0.17035	0.03876	0.06676	0.58057	0.33976	0.16627	0.12048	0.09157	0.08675	0.07470	0.05783	0.03133	0.03133
Turkey	540	52.4	40	0.13472	28.06820	0.00000	0.10456	0.05853	1.78651	0.40556	0.13333	0.11111	0.10741	0.05556	0.03889	0.02778	0.01852	0.10185
Uganda	259	44.3	63	0.09803	9.13089	0.01041	0.13112	0.08451	1.55155	0.35135	0.20463	0.16988	0.10425	0.02317	0.05019	0.02317	0.03861	0.03475
Ukraine	503	38	94	0.07918	9.95250	0.00690	0.06574	0.06064	1.08417	0.35388	0.18887	0.07356	0.08350	0.06362	0.06561	0.04374	0.06163	0.06561
United Arab Emirates	560	46.7	56	0.14686	53.10640	0.00000	0.13136	0.05747	2.28565	0.16964	0.20357	0.14464	0.15893	0.10357	0.07321	0.06250	0.05357	0.03036
United Kingdom	593	77.9	2	0.07561	23.80012	0.00001	0.05119	0.05585	0.91654	0.34907	0.13322	0.09612	0.08600	0.06239	0.07420	0.07589	0.06914	0.05396
United States	742	83.5	1	0.03319	5.88059	0.05285	0.03676	0.04993	0.73627	0.29111	0.16712	0.10916	0.09973	0.07412	0.08221	0.05391	0.05526	0.06739
United States Virgin Islands	106	-	-	0.18157	14.68265	0.00065	0.17070	0.13209	1.29224	0.47170	0.12264	0.10377	0.07547	0.05660	0.07547	0.02830	0.01887	0.04717
Uruguay	362	41.3	81	0.14698	34.68235	0.00000	0.12994	0.07148	1.81784	0.43094	0.14088	0.07182	0.06077	0.06630	0.07459	0.06354	0.06630	0.02486
Uzbekistan	295	34.3	116	0.19473	25.56781	0.00000	0.19183	0.07918	2.42264	0.18305	0.10847	0.11864	0.12203	0.15254	0.17966	0.09492	0.00678	0.03390
Venezuela	275	23	176	0.08817	15.88615	0.00036	0.09591	0.08201	1.16946	0.30909	0.12727	0.09455	0.10545	0.05091	0.06182	0.07636	0.10909	0.06545
Vietnam	168	49.1	50	0.08063	11.54043	0.00312	0.11052	0.10493	1.05335	0.34524	0.20833	0.14286	0.11310	0.07738	0.02976	0.04762	0.02381	0.01190
Yemen	264	18.5	190	0.08292	9.55645	0.00841	0.10403	0.08370	1.24286	0.31818	0.18182	0.14015	0.16288	0.05303	0.04167	0.05303	0.02652	0.02273
Zambia	348	28.7	152	0.04778	2.05270	0.35831	0.03521	0.07290	0.48292	0.33621	0.14655	0.11782	0.10920	0.08046	0.06322	0.06322	0.04598	0.03736
Zimbabwe	336	38.2	92	0.09713	10.37822	0.00558	0.07995	0.07419	1.07761	0.38095	0.12798	0.09524	0.09524	0.09524	0.06548	0.06250	0.03571	0.04167

As stated earlier, the tests quantify the distance between the reported and referred distributions. According to the K-S statistics, the most substantial evidence for abnormalities with the law was found with Iran’s data, with a cut-off $D_{n}$ = 0.06, $KS$ = 0.32, and a distance between the observed and reference samples of 548% at a 5% confidence level, followed by Belarus with 332% and Taiwan with 318%. Based on the chi-square goodness-of-fit test, it can be indicated that there was a significant difference in the proportion of first digits reported for COVID-19 new cases, new deaths, and new tests by the current sample of the jurisdictions stated—particularly for Iran (n: 548; χ²: 44.3; p < 0.000).

Consequently, Iran also fails to meet the d^*-factor/Euclidean distance requirements with a substantial result of 0.271. As suggested by Goodman, as well as Wie and Vellwock, a Euclidean distance with 0.25 or above indicates irregularities in the reported distribution. Notably, Iran shows by far the largest distance from the Benford distribution according to all statistical assessments, followed by Taiwan, Latvia, El Salvador, Tajikistan, and Sao Tome and Principe. These countries fail to confirm the null hypothesis based on all three goodness-of-fit tests. Some countries meet Benford’s expectations in all three tests, such as the United States, Germany, France, New Zealand, China, and Japan—followed by 35 other countries (see Figure 4).

Figure 4.

Distribution of leading digits of Benford’s violators.

The Euclidean distance suggests compliance with Benford’s law for all nations, except for Iran, El Salvador, Latvia, Sao Tome and Principe, Taiwan, and Tajikistan. The chi-square and K-S statistics indicate that some countries do not conform to the first-digit distribution law, such as Denmark, Norway, and Sweden. Figure 2 illustrates the observed distributions of the top violators of Benford’s law.

To better visualize the results of the most compliant countries, see Figure 5, illustrating the countries that fully meet the requirements of the statistical tests.

Figure 5.

Distribution of the first leading digits of compliant countries.

Conclusion

The author extended the sample size and related goodness of fit tests, as stated earlier. This approach has improved the quality of the analysis and, thus, the reliability of the findings in this study. According to the K-S statistic, Germany seems to disclose the most compliant data worldwide. Other countries obeying Benford’s law, such as China, the United States, and France, pass all goodness-of-fit tests applied in this study. These outcomes are, to some extent, in agreement with prior research.^5–8

The records of cumulative infections and deaths from the United States, Japan, Indonesia, and most European nations adhere well to the law. Koch and Okamura came to a similar conclusion and also confirmed China’s compliance with the law. Furthermore, consistent with prior research, all European countries (with the exception of Latvia), demonstrate Benfordness by satisfying at least one of the goodness-of-fit tests. This is also the case with North and Latin American as well as Asia Pacific countries.

In contrast to Germany, the most significant irregularities concerning data compliance with the law occurred in the Islamic Republic of Iran, El Salvador, Latvia, Sao Tome and Principe, Taiwan, and Tajikistan. In other words, these countries do not provide reliable data on the COVID-19 epidemics since they do not pass all goodness-of-fit tests.

Notably, and in particular, Iran merits our attention since the Islamic Republic shows by far the largest distance to Benford’s distribution of leading digits—five times larger than expected. This is consistent with the WHO assessment in March 2020, suggesting that the number of cases reported by the Iranian authorities could represent only about a fifth of the real numbers in the early stages of the epidemic.²² Our analysis, moreover, suggests that Iran reveals a persistent pattern for reporting questionable data within a large period. Iran was also crystallized out as an outlier in earlier studies.⁷ In addition, the Islamic Republic was blatantly stated as demonstrating statistical inconsistencies during the growing outbreak in early 2020. The Washington Post published satellite pictures of mass graves for Coronavirus victims in Qom, the primary hub of Iran’s outbreak in 2020.²¹ By looking at all records, one can recognize that Iran decided to report equally distributed leading integers starting with all first digits (see Figure 2).

Last but not least, the 2019 GHSI seems to partially explain why countries demonstrating overwhelming compliance with the law belong to the same groups in this study, such as Russia and Mali, flagged yellow or the United States and Germany, flagged green. Iran ranked 103 and Tajikistan 144 for early detection and reporting. Latvia, however, was placed second in the same benchmark after the United States and Australia and before Germany and France.⁴ In a further attempt, Pearson product-moment correlation analysis was used to explore the relationship between the GHSI scores for early detection and reporting and the goodness of fit tests for all countries. Preliminary analyses were conducted to ensure no violation of assumptions of normality, linearity, and homoscedasticity. The author identified a moderate, negative partial correlation between the GHSI score and K-S statistic for all countries (r = −0.264, n = 154, p < 0.0005). This suggests that, at a moderate level, countries with a high GHS Index score are prone to report reliable COVID-19 data.

Contradictions with the law can be explained through artificially fabricated data.²⁰ They may pertain to varying national policies and limited abilities to detect and collect data on the spread of submicroscopic infectious organisms. For example, some nations may not have access to sufficient testing kits, as already reported by the WHO.²²

Future research

One cannot claim that lack of compliance with Benford’s law is always strong evidence for fraud. Our analysis did not investigate the formative indicators of incompliance with the natural distribution of leading digits. The most important question that emerges in our study is why countries report data with considerable inconsistencies with Benford’s law. To address this question, one shall further examine the social-economic conditions of those countries. Local authorities may have additional reasons for providing incompliant data on COVID-19, such as fearing a rise in opposition or a decline in inbound tourism.

Based on the key results of this paper, the author recommends establishing a global governance model and unifying measurement standards to examine better and review distributed statistics on the spread of any disease in the future. For the sake of reliability and accuracy in such extraordinary situations, having access to compliant data is vital in the fight against disastrous epidemics of potential international concern. To effectively protect people from pandemics and save lives globally, political will and global leadership appear to be essential for success.

Limitation

The author acknowledges a fundamental limitation in this study related to the statistical regime of Benfordness. The underpinning data collected from divergent regions and countries might not be comparable. They are generated based upon different public health systems and policy decisions, locally controlled and enforced by unaligned authorities. Small countries, mainly, may not possess the required capabilities and capacity to fully detect COVID-19 cases and precisely report the number of incidents.

Footnotes

Acknowledgements

The author sincerely thanks the editorial team and peer reviewers of this paper for critical review.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Noah Farhadi

Author biography

Noah Farhadi is Professor of Strategy at the IU International University of Applied Science of Germany. He founded the Berlin-based innovation hub Mammuttree Analytics, focusing on strategy and data science. Dr. Farhadi completed his doctoral studies on Corporate Strategy at Henley Business School in the UK.

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Can we rely on COVID-19 data? An assessment of data from over 200 countries worldwide