Note on Pre-Taxation Data Reported by UK FTSE-Listed Companies: Search for Compatibility with Benford’s Laws

Abstract

Pre-taxation analysis plays a crucial role in ensuring the fairness of public revenue collection. It can also serve as a tool to reduce the risk of tax avoidance, one of the UK government’s concerns. Our report utilises pre-tax income ($PI$) and total assets ($TA$) data from 567 companies listed on the FTSE All-Share index, gathered from the Refinitiv EIKON database, covering 14 years, i.e., the period from 2009 to 2022. We also derive the $PI/TA$ ratio, and distinguish between positive and negative $PI$ cases. We test the conformity of such data to Benford’s Laws, specifically studying the first significant digit ($Fd$), the second significant digit ($Sd$), and the first and second significant digits ($FSd$). We use and justify two pertinent tests, the ${\chi }^2$ and the Mean Absolute Deviation (MAD). We find that both tests do not lead to conclusions in complete agreement with each other—in particular, the MAD test entirely rejects the Benford’s Laws conformity of the reported financial data. From the mere accounting point of view, we conclude that the findings not only cast some doubt on the reported financial data, but also suggest that many more investigations should be considered on closely related matters. On the other hand, the study of a ratio, like $PI/TA$, of variables that are (or are not) Benford’s Laws-compliant adds to the literature concerning whether such indirect variables should (or should not) be Benford’s Laws-compliant.

1. Introduction

Effective pre-audit accounting data analysis is crucial in detecting possible earnings management and tax planning, thereby enhancing the expected integrity of financial reporting. In UK, the tax gap, i.e., the difference between the amount of tax that should theoretically be paid to HMRC, and what is actually paid, was estimated at GBP 39.8 billion for the 2022 to 2023 tax year, accounting for 4.8% of the known total liabilities (https://www.gov.uk/government/statistics/measuring-tax-gaps/1-tax-gaps-summary; accessed on 29 January 2025).

During our investigation of tax optimization by listed FTSE companies, we had to use data on their pre-tax income ($PI$) and data on their Total Assets ($TA$) [or “$SIZE$” = $ln$ ($TA$)] in a given year over a large time interval. In fact, the ratio $PI/TA$ weighted through some irrelevant factor, at this stage, is a correction term to the deferred tax expense ($TXDI$) needed in order to calculate the (finally relevant) studied tax avoidance, estimated from the measure of total book–tax differences ($TBTD$) [1].

In so doing, we came across a huge dataset for $TA$ and $PI$, i.e., 6811 and 6768 data points, respectively. One pertinent question delves into the reliability of such pre-taxation data. Its analysis plays a crucial role in ensuring the fairness of public revenue collection. Econometrics, mixing statistical analysis and economic considerations, should increase the confidence of the population, and even more so that of taxation officers. The findings of this study, if the data are found to be markedly bizarre, can subsequently serve as a disciplinary tool toward reducing the risk of tax avoidance, and even tax evasion, as well as manipulations of profit shifts, which (among other things) are great concerns of the UK government.

Benford’s Laws, mathematically written in Section 3, state that the leading digits of (naturally occurring) numbers follow specific $log$ distributions [2,3]. A deviation from this empirical distribution is often considered as a warning suggesting a more detailed examination. Therefore, it has seemed to us of pertinent interest to observe whether such $PI$ and $TA$ data fulfil the expectations of BLs. This is the main aim of this report—i.e., the research questions and the distributions of the first, second, and first–second digits in such data are tested according to the empirical BLs, called BL1, BL2, and BL12, respectively.

In other words, through the null hypothesis (agreement between the empirical data and expected laws), we aim to assess the discrepancy between secondary (financial) data and their possible theoretical Benford distributions, through two statistical tests based on rather different concepts: the ${\chi }^2$ test and the Mean Absolute Deviation (MAD), both outlined in Section 4.

The BLs are of interest for identifying irregularities in financial reports [4,5,6,7,8,9,10], but the literature is too huge to be summarized here. Let us mention that BLs have been considered in many fields, not just finance, but also academia, elections [11], engineering, medicine, psychology, physics, religion, scientometrics, sports, and likely many others; some of these are discussed in [1], with which the following subsection sometimes overlaps.

Moreover, because of the $PI$ and $TA$ data type and size, it is possible to distinguish between negative and positive PIs and to study them in the BLs framework. Notice that this sign distinction has rarely been considered; indeed most authors, except a few to our knowledge, have searched for BL obedience in absolute values of data [10].

A short and focused literature review is found in Section 2. The literature is vast. In order not to add to useless (and, necessarily or often, incomplete) literature reviews, we reduce the present literature review section to the essential papers pertinent to our aim; i.e., we include a mention of (i) the pioneering papers, (ii) the next most often quoted ones, which are likely of general interest, and (iii) the most recent ones, essentially to pin point the “state of the art”. We focus on papers at the intersection of three sets: (i) high order BLs, in particular BL2 and BL12, (ii) financial data relevant to pre-taxation, and (iii) statistical tests, in particular papers considering the Mean Absolute Deviation (MAD).

A warning: irregularities may not always be revealed through the use of Benford’s Law [12,13,14,15]. Moreover, financial data might not necessarily strictly conform to BLs: they may depend on the data ranges [16,17].

Notice that within a statistical framework, it is sometimes debated whether BL compliance can be extended to derived, correlated, or combined quantities. Therefore, we also calculated $PI/TA$ when possible, i.e., when both $PI$ and $TA$ data were reported for a company in a given year.

To conclude these remarks, a fundamental limitation of studies reported so far should be pointed out: this limitation is their common focus on BL1 tests, except for a few authors who have approached income items and BL2 from a behavioural perspective [18,19,20,21]. Consequently, our research aims to significantly advance the literature by conducting an in-depth statistical analysis of (accounting and tax) variables in financial statements, exploring potential conformity to BL1, BL2, and BL12.

We fully present the data acquisition and study methodology in Section 3. We run two null hypothesis significance testing methods to investigate conformity to BLs. More precisely, as measures to assess the discrepancy between the empirical and the theoretical Benford distributions, we use the ${\chi }^2$ test and the MAD test as described in detail in Section 4. We report our findings through Tables and Figures. Interestingly, we find that the MAD test entirely rejects the conformity of the reported financial data with Benford’s Laws, i.e., the relevant null hypothesis.

We conclude in Section 5 both with remarks on the statistical tests and their disagreement, and on the implications for practical accounting research. Indeed, we find that both ${\chi }^2$ tests and MAD tests do not lead to conclusions in complete agreement with each other. Therefore, we reject the null hypothesis. Therefore, this finding demands further studies on the validity ranges of statistical test applications. Notice that the study of a ratio, like $PI/TA$, with variables that are (or not) Benford’s Laws-compliant (as is differently found a posteriori), adds to the literature concerning whether indirectly measured variables should be (or not) Benford’s Laws-compliant.

Furthermore, from the mere accounting point of view, we conclude that the findings not only cast some doubt on the reported financial data, but also suggest that many more empirical and thorough investigations may be needed regarding closely related financial data reported by listed companies.

2. Literature Review

Benford’s Laws have been like a “sleeping beauty” sleeping in the dirty pages of logarithmic tables [22]. But they have been revived in the strategic management literature [23] and in accounting [10,24], both of which are fields of interest here. The range of applications is huge, i.e., as long as large natural datasets are available. For conciseness, we focus on papers at the intersection of three sets: (i) high-order BLs, (ii) tax-releated financial data, and (iii) specific statistical tests, in particular papers considering the Mean Absolute Deviation (MAD). We also comment on pertinent papers at the intersection of pairs of such sets.

In brief, one should start recognizing the pioneer work of Nigrini [4,5,10,25]: he introduced and developed statistical research based on BLs to estimate the compliance of taxpayers and of companies engaging in tax planning strategies. In [10], Nigrini provides a review of the literature on audit sampling and perspectives.

One of the most relevant papers at the above-mentioned triple set intersection is that of Alali and Romero from 2013 [26]. They discovered significant differences in various accounting figures within US financial statements of companies, either audited by Big Four firms or by non-Big Four firms. Thereafter, Druica et al. studied Romanian banks along BL2 [12]. Also, Prachyl and Fischer evaluated the conformity of municipality financial data with BL2 [27]. Cheuk et al. assessed the “financial reporting quality of Company Limited by Guarantee charities in Malaysia” [28].

More oriented toward comparing tests, Kössler et al. looked at share prices [29] essentially through BL2. Along the same lines, da Silva Azevedo [30] as well as Cerqueti and Lupi [31] considered BL12.

In a more comprehensive way, Sadaf [32], Patel et al. [33], and Sylwestrzak [34] studied both BL2 and BL12 with respect to possible data manipulation by managers.

Concerning the pertinent literature at the intersection of pairs of sets, let us recognize the keystone and pioneering observation of Carslaw in 1988 [35]: he observed while working on BL2 that New Zealand companies’ income statements showed a markedly higher occurrence of 0s and a lower occurrence of 9s in the second digit position than should be expected, implying voluntary roundings. Similar findings were obtained by Niskanen and Keloharju in 2000 on Finnish public companies [36]. Using both BL2 and BL12, Ausloos et al. also found that “Benford’s laws tests on S&P500 daily closing values and the corresponding daily log-returns both pointed to huge non-conformity” [16]. Similar studies on two joint BLs by Das et al. [37] and by Jordan and Clark [38] can be mentioned.

For completeness, let us also mention that authors have used various values of financial results and tests to detect potential data manipulation. Of interest with respect to our report are, e.g., Van Caneghem [39] who investigated with the aid of BL2 tests a sample of 1256 UK companies that reported pre-tax income for the accounting year 1998. He found results similar to those of Carslaw [35]. Other recent works containing BL2 studies are [40,41,42,43,44,45]. Last but not least, Günnel and Tödter considered BL12 [46], as did Le and Mantelaers, who even discussed the state of the art up to BL123 [47]; Sardar and Sharma also studied BL3 and BL4 in respect to the financial reports of several listed companies of the Adani Group [48].

Of course, many other works discuss statistical tests and financial reports sometimes using ${\chi }^2$ or MAD tests; many other statistical tests are available and have been considered. However, because such papers restrict their consideration to BL1, for the sake of the brevity of this literature review, we reiterate that we limit ourselves to the above works for the sake of framing our field of interest.

3. Data Acquisition and Study Methodology

3.1. Data Acquisition

We utilise data from Refinitiv EIKON, focusing on pre-tax income ($PI$) and the total assets ($TA$) as pre-taxation indicators from 2014 to 2022, both years being included in the time interval for the study. The financial data are recorded in GBP, ensuring uniformity in currency representation across all datasets. The FTSE All-Share list is updated annually.

We should emphasise several steps throughout the data retrieval from the Refinitiv EIKON database. Firstly, we utilise the list of companies indexed in the FTSE All-Share, with the (final) year being 2022, which comprises 567 companies—2022 marks the commencement of our investigation into tax avoidance [1]. Secondly, we encountered a lot of missing data. To maintain the authenticity of the dataset, we leave these data points blank in our “Master Data Bank”. Lastly, to facilitate our calculations, we employ the term “Absolute Fiscal Year (FY)” within Refinitiv EIKON, which defines the financial period as a continuous 12-month interval commencing on January 1 and concluding on December 31. This warning is emphasized since the “real” UK fiscal year usually goes from April in a given year till March the next year.

The data extraction process from the Refinitiv Eikon database was based on a connection to a server at the University of Leicester. We first downloaded and installed the file using the link https://eikon.refinitiv.com to install the Eikon add-in for Excel. It is important to note that this add-in is exclusively compatible with non-Mac computers. Upon completion of the download, we launched Microsoft Excel. Within the “File” menu, we selected “Options” to enable the Refinitiv Eikon add-in. After making the required adjustments, it is advisable to restart Excel by closing and reopening the application. Following this action, the Refinitiv Eikon should be successfully integrated with Microsoft Excel.

After entering the instrument, we searched for “total assets” and “pre-tax income”. We used “total assets reported,” representing the company’s total asset; we used the net income before taxes to describe pre-tax income representing the sum of operating income. Many other financial data are available within the Refinitiv Eikon database, but are not presently discussed here.

In so doing, one obtains 6768 and 6811 data points for $PI$ and $TA$, respectively. Due to the large size of the dataset, one can split the $PI$ data into $P{I}^{(-)}$ and $P{I}^{(+)}$ and analyze them independently, in order to observe if any compliance holds for both types of pre-tax incomes.

We wish to warn the reader that a few data points are markedly (i.e., after displaying the pertinent histograms, logically deduced to be) incorrect, but this can be assumed to be a result of transcription errors relating to the decimal point. B As should be clear, this typo is irrelevant for a BLs study. However, a small number (≃3) of data points reported by Refinitiv EIKON are ”anomalously extreme outliers” with quite incomprehensible values. We do not have access to the sources used by Refinitiv EIKON and cannot pursue further the “correction” of these data points. Rather than, as is often carried out, winsorising (modifying the first few digits of) these numbers, we preferred to conserve them. This voluntary step, amounting to not disregarding three bizarre data points, should only have a very weak influence on the BLs analysis due to the large number of data points, called N in

Table 1. Summary of the (rounded) main statistical characteristics of the downloaded $PI$, further distinguishing between $P{I}^{(-)}$ and $P{I}^{(+)}$, $TA$, and the subsequently deduced $PI/TA$. Financial data are originally in GBP.

	N	Min.	Max.	Mean	St. Dev.	Skew.	Kurt.	CV
$PI$	6768	−19,722,854,000	53,579,321,000	316,593,080	1,752,653,700	11.26074	227.834	5.53598
$TA$	6811	15,580	2,438,029,900,000	20,513,747,000	130,724,180,000	10.51739	128.530	6.37252
$P{I}^{(-)}$	1160	−9.9748	−1.0000	−3.7738	2.4381	−0.79236	−0.50234	−0.6461
$P{I}^{(+)}$	5608	1.0000	9.9996	3.8908	2.4777	0.78672	−0.52936	0.6368
$PI/TA$	6765	−1.81579	3.11173	0.07390	0.16680	4.14856	82.5632	2.25731
$P{I}^{(-)}/TA$	1158	−1.81579	−0.00004	−0.10559	0.16054	−4.33229	27.73320	−1.52035
$P{I}^{(+)}/TA$	5607	0.00002	3.11173	0.11096	0.14221	9.97602	161.7935	1.28154

.

A “detail” of financial interest appears when observing whether a relationship might exist between $TA$ and $PI$. It is displayed on Figure 1. “Interestingly”, it seems that there are two clusters. Whether this implies different pre-tax income strategies by companies seems to be a theoretical question in itself and demands some thought before further modelling. Nevertheless, this matter is far beyond the scope of the present paper and is left for further investigation.

As justified in the introduction, we calculated $PI/TA$ for common (company-year) cases before further analysis; one obtains 6765 data points, despite absences of data for some companies in some years. Notice also the three “anomalous outliers”.

The statistical characteristics are given in Table 1.

3.2. Study Methodology

The study of pre-income tax (and total assets) reported data by FTSE-listed companies as being BL-compatible starts with the following formulae for the first digit (BL1), which is used for calculating the probability that a number will have a non-zero first digit [16]:

$$P(d_1=i)=lo{g}_{10}(1+{\displaystyle \frac{1}{i}}),$$

(1)

for i = 1, 2, 3, 4, 5, … 9. So, $P(d_1=i)$ represents the likelihood that any number starts with a digit (i), between 1 and 9, according to a $log$ law. In other words, according to the first digit Benford Law (BL1), a number from the selected dataset should start with “1” approximately 30.1% of the time; “9”, the least frequently occurring first digit, should be the first digit approximately 4.6% of the time.

This formula BL1 can be extended to the first two digits, resulting in BL12 [16]:

$$P_{12}(d_1{d}_2=ij)=lo{g}_{10}(1+{\displaystyle \frac{1}{d_1{d}_2}}).$$

(2)

In this context, $d_1$ indicates the first digit (i), ∈ [1, 9], while $d_2$ signifies the second digit (j) of a number, which may be a 0. Thus, $d_1{d}_2\in$ [10; 99].

Furthermore, a Benford ($log$) Law can also be applied to test the empirical occurrence of the second digit in the dataset numbers, using the following formula, called BL2 [16]:

$$P(d_2=j)=\sum _{k=1}^{9}lo{g}_{10}(1+{\displaystyle \frac{1}{10k+j}}),$$

(3)

for j = 0, 1, 2, 3, 4, … 9 and k = 1, 2, 3, … 9. So, $P(d_2=j)$ represents the likelihood that the second digit (j) of a number is any digit between 0 and 9. The calculation methodology is relatively straightforward: k ranges from 1 to 9; consequently, 10 k serves as a multiple of these integers, resulting in the series of numbers 10, 20, …, and 90.

For the reader, we report the expected frequency values of the pertinent digits in BL1, BL2, and a few for BL12 in

Table 2. Expected proportion values for BL1, BL2, and BL12.

d	BL1	BL2	BL12	${\mathit{d}}_1{\mathit{d}}_2$
0		0.1197
1	0.3010	0.1139	0.0414	10
2	0.1761	0.1088	0.0378	11
3	0.1250	0.1043	0.0348	12
4	0.0969	0.1003	0.0322	13
5	0.0792	0.0967	0.0230	14
6	0.0669	0.0934	0.0280	15
7	0.0580	0.0904	0.0264	16
8	0.0512	0.0876	0.0248	17
9	0.0460	0.0850	0.0235	18
			…	…
			0.0086	50
			…	…
			0.0044	99

.

One may wonder about the origin and significance of such BLs—a “mysterious law of nature” [49,50]. This matter is still an open question. Roughly speaking, a $y=log\left(x\right)$ law can be contrasted to a $y=exp\left(x\right)$ law. In the latter case, it results from the “growth equation” $dy/y=dx$, leading to Keynes infinite growth, e.g., in the economy. In contrast, the former arises from $dy=dx/x$. This means that an elementary increase $dy$ has a reducing effect (by $1/x$) due to the accumulation of previous xs. In the economy, this corresponds to one of the Marxist principles. The BL1, as the basic example, derives from

$${\displaystyle \frac{dy}{dx}}=-{\displaystyle \frac{1}{x(x+1)}}$$

(4)

meaning a reduction in previous values, resulting in subsequent $log$ decay through a complex factor $-1/\left[x\right(x+1\left)\right]$. Notice that such a factor occurs in calculating the geometric means (of size x quantities) [51].

4. Findings

We report the results in the following Tables:

• Table 3. Summary of the (rounded) first digit ($Fd$) of the downloaded $PI$, further distinguishing between $P{I}^{(-)}$ and $P{I}^{(+)}$, and the pertinent frequencies to be compared to the BL1.

  ${\mathit{d}}_1$	$\mathit{Fd}$  $\mathit{PI}$	$\mathit{Fd}$  ${\mathit{PI}}^{(-)}$	$\mathit{Fd}$  ${\mathit{PI}}^{(+)}$	BL1 $\mathit{PI}$	BL1 ${\mathit{PI}}^{(-)}$	BL1 ${\mathit{PI}}^{(+)}$	BL1
  1	2035	374	1661	0.30103	0.30068	0.32241	0.29618
  2	1262	214	1048	0.17609	0.18647	0.18448	0.18688
  3	785	120	665	0.12494	0.11599	0.10345	0.11858
  4	701	111	590	0.09691	0.10358	0.09569	0.10521
  5	508	96	412	0.07918	0.07506	0.08276	0.07347
  6	436	84	352	0.06695	0.06442	0.07241	0.06277
  7	416	67	349	0.05799	0.06147	0.05776	0.06223
  8	332	51	281	0.05115	0.04905	0.04397	0.05011
  9	293	43	250	0.04576	0.04329	0.03707	0.04458
  ${\chi }^2$	16.5706	10.3233	15.7448
  $MAD$				0.04103	0.07764	0.04664
  	${\chi }_c^2$ = 15.507, at 5%, for D = 8.			close conformity if MAD ≤ 0.006

  contains the (rounded) first digit ($Fd$) of the downloaded $PI$, as well as $P{I}^{(-)}$ and $P{I}^{(+)}$, and their pertinent frequencies to be compared to the BL1. From the results of the two tests, we can notice that the $PI$ variable shows non-conformity (both in the “gain” and “loss” domains) considering the MAD test. The ${\chi }^2$ analysis shows mixed results, pointing to some violation for the $PI$ variable and its “gain” domain. Notably, we observe “under-occurrence” of the last three digits (7, 8, 9) as the first digit concerning BL1 distribution, potentially pointing to corporate “rounding up” practices.
• Table 4. Summary of the (rounded) first digit ($Fd$) counted occurrences of the downloaded $TA$ and $PI/TA$ values and their pertinent frequencies to be compared to the empirical BL1.

  ${\mathit{d}}_1$	$\mathit{Fd}$  $\mathit{TA}$	$\mathit{Fd}$  $\mathit{PI}/\mathit{TA}$	BL1 $\mathit{TA}$	BL1 $\mathit{PI}/\mathit{TA}$	BL1
  1	2084	2223	0.30598	0.32860	0.30103
  2	1186	1100	0.17413	0.16260	0.17609
  3	829	655	0.12171	0.09682	0.12494
  4	761	564	0.11173	0.08337	0.09691
  5	494	523	0.07253	0.07731	0.07918
  6	400	508	0.05873	0.07509	0.06695
  7	399	446	0.05858	0.06593	0.05799
  8	340	386	0.04992	0.05706	0.05115
  9	318	360	0.04669	0.05322	0.04576
  ${\chi }^2$	27.757	117.287
  $MAD$			0.04258	0.11404
  	${\chi }_c^2$ = 15.507, at 5%, for D = 8.		close conformity if MAD ≤ 0.006

  contains the (rounded) first digit ($Fd$) occurrences of the downloaded $TA$ and the deduced $PI/TA$ values and their pertinent frequencies to be compared to the empirical BL1. For the total assets and the $PI/TA$ ratio, we again observe complete non-conformity based on the MAD test and non-conformity of the ratio variable considering the ${\chi }^2$ analysis. We can observe for the $TA$ variable an “over-occurrence” of the last digit $\left(9\right)$, which also leads to some interesting behavior for the occurrence of $\left(1\right)$ and $\left(9\right)$ as in the first digit of the $PI/TA$ ratio. The findings can be of interest for suggesting a deeper investigation of different ratios as part of financial statement analyses.
• Table 5. Summary of the (rounded) second digit ($Sd$) of the downloaded $PI$, further distinguishing between $P{I}^{(-)}$ and $P{I}^{(+)}$, and the pertinent frequencies to be compared to the BL2.

  ${\mathit{d}}_2$	$\mathit{Sd}$  $\mathit{PI}$	$\mathit{Sd}$  ${\mathit{PI}}^{(-)}$	$\mathit{Sd}$  ${\mathit{PI}}^{(+)}$	BL2 $\mathit{PI}$	BL2 ${\mathit{PI}}^{(-)}$	BL2 ${\mathit{PI}}^{(+)}$	BL2
  0	875	166	709	0.12928	0.14310	0.12643	0.11968
  1	753	121	632	0.11126	0.10431	0.11270	0.11389
  2	745	129	616	0.11008	0.11121	0.10984	0.10882
  3	707	126	581	0.10446	0.10862	0.10360	0.10433
  4	675	116	559	0.09973	0.10000	0.09968	0.10031
  5	626	118	508	0.09249	0.10172	0.09058	0.09668
  6	648	94	554	0.09574	0.08103	0.09879	0.09337
  7	606	115	491	0.08954	0.09914	0.08755	0.09035
  8	595	78	517	0.08791	0.06724	0.09219	0.08757
  9	538	97	441	0.07949	0.08362	0.07864	0.08500
  ${\chi }^2$	9.855	15.208	10.742
  $MAD$				0.02741	0.08787	0.03560
  	${\chi }_c^2$ = 16.919, at 5%, for D = 9.			close conformity if MAD ≤ 0.008

  reports the (rounded) second digit ($Sd$) occurrence of $PI$ also distinguishing between $P{I}^{(-)}$ and $P{I}^{(+)}$, and the pertinent frequencies for comparison to BL2. We can observe that the results of the BL2 analysis are more mixed, with an absolute non-conformity for both “loss” and “gain” domains of pre-tax income with the MAD test, and a conformity of all second digits, considering the ${\chi }^2$ analysis.
• Table 6. Summary of the (rounded) second digit ($Sd$) of the downloaded $TA$ and $PI/TA$ further distinguishing and their pertinent frequencies to be compared to the BL2.

  ${\mathit{d}}_2$	$\mathit{Sd}$  $\mathit{TA}$	$\mathit{Sd}$  $\mathit{PI}/\mathit{TA}$	BL2 $\mathit{TA}$	BL2 $\mathit{PI}/\mathit{TA}$	BL2
  0	821	853	0.12054	0.12609	0.11968
  1	764	854	0.11217	0.12624	0.11389
  2	758	696	0.11129	0.10288	0.10882
  3	717	681	0.10527	0.10067	0.10433
  4	686	666	0.10072	0.09845	0.10031
  5	635	650	0.09323	0.09608	0.09668
  6	621	594	0.09118	0.08780	0.09337
  7	634	605	0.09308	0.08943	0.09035
  8	616	574	0.09044	0.08485	0.08757
  9	559	592	0.08207	0.08751	0.08500
  ${\chi }^2$	3.6678	18.063
  $MAD$			0.02057	0.04253
  	${\chi }_c^2$ = 16.919, at 5%, for D = 9		close conformity if MAD ≤ 0.008

  summarizes the (rounded) second digit ($Sd$) occurrence of $TA$ and $PI/TA$ and their pertinent frequencies to be compared to the BL2. For the $TA$ variable and the derived ratio, we observe the same picture as for the $PI$ analysis: non-conformity with MAD for all variables and also non-conformity considering the behavior of the $PI/TA$ ratio with the ${\chi }^2$ test. For the ratio variable, we can observe an “over-occurrence” of $\left(0\right)$ and $\left(1\right)$ as well as $\left(9\right)$ as the second digit. These observations point to possible irregularities in $PI$ reporting and in the valuation of companies’ assets.
• Table 7. Summary of the (rounded) first and second digit ($FSd$) of the downloaded $PI$, further distinguishing between $P{I}^{(-)}$ and $P{I}^{(+)}$, and the pertinent frequencies to be compared to the BL12.

  ${\mathit{d}}_{12}$	$\mathit{FSd}$  $\mathit{PI}$	$\mathit{FSd}$  ${\mathit{PI}}^{(-)}$	$\mathit{FSd}$  ${\mathit{PI}}^{(+)}$	BL12 $\mathit{PI}$	BL12 ${\mathit{PI}}^{(-)}$	BL12 ${\mathit{PI}}^{(+)}$	BL12
  10	264	56	208	0.039007	0.048276	0.037090	0.04139
  11	251	43	208	0.037086	0.037069	0.037090	0.03779
  12	246	46	200	0.036348	0.039655	0.035663	0.03476
  …	…	…	…	…	…	…	…
  20	162	31	131	0.023936	0.026724	0.023359	0.02119
  21	148	21	127	0.021868	0.018103	0.022646	0.02020
  …	…	…	…	…	…	…	…
  30	100	18	82	0.014775	0.015517	0.014622	0.01424
  …	…	…	…	…	…	…	…
  50	56	11	45	0.008274	0.009483	0.008024	0.00860
  …	…	…	…	…	…	…	…
  90	39	9	30	0.005762	0.007759	0.005350	0.00480
  …	…	…	…	…	…	…	…
  99	19	4	15	0.002807	0.003448	0.002675	0.00436
  ${\chi }^2$	116.213	93.989	128.654
  $MAD$				0.09418	0.21754	0.11131
  	${\chi }_c^2$ = 113.145, at 5%, for D = 89			close conformity if MAD ≤ 0.0012

  reports the (rounded) first and second digit ($FSd$) of the downloaded $PI$, again further distinguishing between $P{I}^{(-)}$ and $P{I}^{(+)}$, and the pertinent frequencies to be compared to BL12.
• Table 8. Summary of a few values of the (rounded) first and second digit ($FSd$) of the downloaded $TA$ and $PI/TA$ data and the pertinent frequencies to be compared to the BL12.

  ${\mathit{d}}_{12}$	$\mathit{FSd}$  $\mathit{TA}$	$\mathit{FSd}$  $\mathit{PI}/\mathit{TA}$	BL12 $\mathit{TA}$	BL12 $\mathit{TA}/\mathit{PI}$	BL12
  10	308	343	0.04522	0.05070	0.04139
  11	259	318	0.03803	0.04701	0.03779
  12	233	249	0.03421	0.03681	0.03476
  …	…	…	…	…	…
  20	144	154	0.02114	0.02276	0.02119
  21	135	150	0.01982	0.02217	0.02020
  …	…	…	…	…	…
  30	95	74	0.01395	0.01094	0.01424
  …	…	,,,	…	…	…
  50	63	44	0.00925	0.00650	0.00860
  …	…	…	…	…	…
  90	32	30	0.00470	0.00443	0.00480
  …	…	…	…	…	…
  99	19	38	0.00279	0.00562	0.00436
  ${\chi }^2$	26.5234	196.574
  $MAD$			0.04658	0.13625
  	${\chi }_c^2$ = 113.145, at 5%, for D = 89		close conformity if MAD ≤ 0.0012

  contains a summary of the (rounded) first and second digit ($FSd$) of $TA$ and $PI/TA$ values and their pertinent frequencies to be compared to the BL12.
  Based on the results in Table 7 and Table 8, the BL12 conformity leads to a relatively complex analysis, especially considering the evaluation of derived variables (such as the $PI/TA$ ratio). The results from Table 7 and Table 8 are again characterized by non-conformity when considering the MAD test. But the ${\chi }^2$ statistics point to the conformity of the $PI$ values in the “gain” domain. Potentially, as a result, this leads to non-conformity of the $PI/TA$ ratio.

The analysis of BL1 and BL2 is also illustrated in Figure 2 and Figure 3, respectively. BL12 is illustrated on Figure 4 without the $TA$ data, otherwise the figure would be somewhat unreadable. Let it be observed that when supplementing the data reported in Table 7 and Table 8 with the data in Figure 4, some salient irregularities appear in BL12, in particular in the last higher two digits’ occurrence (such as $60,\dots ,99$). The worst is for the $P{I}^{(-)}$ data, which seem to have much (unexpected) deviation from the empirical BL12.

The results from the statistical tests, in order to assess the discrepancy between the observed frequency distributions and the theoretical BL distributions, are reported in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8. Recall that the ${\chi }^2$ test and the MAD tests concern two different concepts of distances:

• The ${\chi }^2$ test, as usual defined through the number of observations (O) and the number of expected (E) ones, distributed in a number of bins equal to $D+1$, where D is the number of degrees of freedom,

  $${\chi }^2=\sum _{i=1}^{D+1}{\displaystyle \frac{{(O_i-E_i)}^2}{E_i}},$$

  (5)

  is classically used, but is said to tend toward rejecting the Benford compliance of observations even when the deviations from the theoretical BL (E) are negligible, mainly in large samples.
• The MAD test, defined through the observed ($f_o$) and the expected ($f_e$) frequencies, in the pertinent number K of bins, as

  $$MAD=\sum ^K|f_o-f_e|,$$

  (6)

  is thought to be the most reliable test for checking the validity of the BL [31], but not always [12]. In brief, for BL1, a value below 0.006 allows for deducing close conformity, while an MAD between 0.006 and 0.012 refers to only acceptable conformity; marginally acceptable conformity occurs for values between 0.012 and 0.015; nonconformity is the conclusion otherwise. For BL2, close conformity occurs for MAD ≤ 0.008, while close conformity occurs for MAD ≤ 0.0012 for BL12 [4].

Of course, many different statistical tests have been used for assessing conformity (of financial data, for example) to BLs: a short list may include, without any hierarchical order, the Kolmogorov–Smirnov test, the Chebyshev distance, the Kullback–Leibler divergence, the Freedman–Watson ($U^2$) test, the Joenssen JP-square test, z-statistics, the Financial Statement Divergence Score, the Kuiper test, the Binomial Probability test, and even the Euclidean distance or regression approaches and other ”smooth tests”.

Recent discussions and works of interest are covered with pertinent recommendations by Cerqueti and Maggi [14], Lesperance et al. [52], Ducharme et al. [53], Henselmann et al. [54], Cerqueti and Lupi [31,55], and Barabesi et al. [56].

5. Conclusions

In brief, recall that we aim to assess the discrepancy between secondary (financial) data and their possible theoretical Benford distributions through two statistical tests based on different concepts: the ${\chi }^2$ test and the Mean Absolute Deviation (MAD), both outlined in Section 4. We have used a rather large set of financial data, i.e., the pre-tax income ($PI$) and the total assets ($TA$) of 567 companies listed on the FTSE All-Share index, gathered from the Refinitiv EIKON database, covering 14 years from the period from 2009 to 2022, as described in Section 3. Since the validity and applicability of Benford’s Laws (BLs) are still debated, in particular on indirect measures, we also derived the $PI/TA$ dataset. Furthermore, due to the data size (a little bit less than 7000 data points), we examined cases of either positive or negative $PI$.

Indeed, Benford’s Law, which describes the frequency distribution of digits in many real-world datasets, has been extensively applied in accounting to detect anomalies such as rounding up or other irregularities. While much research focuses on the first digit, some studies (including the present study) have also examined the distribution of second digits and the first and second digits’ occurrence on income and total assets data integrity. Benford’s Law provides a method to detect such anomalies by comparing the expected distribution of digits to the observed distribution in the dataset. The present study has examined such deviations for the first two digits for the $PI$ variable in the profits and losses domain, identifying significant deviations for ${PI}^{(+)}$ and ${PI}^{(-)}$ from the theoretical distribution.

Thus, whether the actual $TA$, $PI$, and $PI/TA$ proportions do not statistically differ from the proportion expected from Benford’s Laws (BL2 and BL12), i.e., the null hypothesis, can be now verified according to the ${\chi }^2$ and MAD statistical tests.

From the tables summarizing the MAD values, it seems obvious that all data appear to be non-conforming to BLs. This null hypothesis is rejected.

In contrast, conclusions from the ${\chi }^2$ tests are more ambiguous. Indeed, the smallest ${\chi }^2$ with a value below the critical one for the pertinent number of degrees of freedom occurs for several cases:

firstly, for $Fd$ $P{I}^{(+)}$,

secondly, for $Sd$ $PI$, $Sd$ $P{I}^{(-)}$, $SDd$ $P{I}^{(+)}$, and $Sd$ $TA$,

and finally, for $FSd$ $P{I}^{(-)}$ and $FSd$ $TA$.

A few cases are close to the ${\chi }^2$ critical value like

$Fd$ $PI$ and $Fd$ $P{I}^{(+)}$,

$Sd$ $TA$,

$FSd$ $PI$, and $FSd$ $P{I}^{(+)}$.

It is remarkable that no $PI/TA$ ($Fd$, $Sd$, and $FSd$) ratio obeys the 5% ${\chi }^2$-test criterion. Although there is no proof of the following deduction, one may assume that this finding might be due to the bizarre $TA/PI$ distribution, which is illustrated in Figure 1.

In fact, this observation allows us to point to one still-open question on the applicability, and further validity, of Benford’s Laws for derived measures, if it is found that initial, raw measures obey the empirical BLs. Our results through the ${\chi }^2$ test values seem to indicate that the universally extended validity is dubious. As a suggestion for further work, the $U^2$ test could be seriously considered [52,53].

In conclusion, one might again be amazed that two different statistical tests do not lead to similar deductions regarding the conformity of large, a priori, unmanipulated datasets. On the other hand, one may further question the lack of conformity with respect to the MAD test. Indeed, this implies that the data might be manipulated, likely in order to maintain a balance between the reported income, thus lowering the tax amount, and the positive show of benefits for shareholders. Nevertheless, our findings add to the observations of Alali and Romero [26] on the one hand and on those of Henselmann et al. [54] on the other. Although we have used quite different datasets, and different tests, we confirm that much remains to be understood and explained.

To sum up, this note enriches the application of Benford’s Law as a robust tool for detecting anomalies and potential fraud in accounting data, utilizing an extensive dataset derived from FTSE-listed companies. We also stress the need for more elaborate statistical analyses before drawing conclusions.

Thus, future research directions may be highlighted here, in particular studies on other financial data through BLs—if one is interested in financial data manipulation, but also other statistical tests, we recommend the recent work of Lesperance et al. [52], Ducharme et al. [53], Henselmann et al. [54], Cerqueti and Lupi [31,55], and Barabesi et al. [56].