Benford’s Law as a Tool for Audit Planning and Control: An Analysis of Municipal Expenditures

Author: Georgios Stefanou, M.Sc., Ph.D. Head of Department, Hellenic Court of Audit – Commissioner’s Office in the Prefecture of Messinia

Abstract

This study presents a practical application of Benford’s Law to the expenditure data of two Greek municipalities, Messini and Trifylia, aiming to evaluate the extent to which these financial transactions conform to the expected digit distributions as defined by Benford’s Law. The primary goal is to assess the utility of Benford’s Law as a tool in public sector auditing processes and determine whether it can raise red flags to prompt further investigative procedures.

We applied Z-tests and Chi-square tests to expenditure contracts from 2021 to measure compliance with Benford’s distribution for both the first and second digits. Statistically significant deviations from the expected digit frequencies were treated as indicators of potential irregularities, thereby warranting further audit review.

The analysis revealed that Benford’s Law can be a valuable preliminary assessment tool in the detection of anomalies. Contracts deviating from expected distributions can inform audit prioritization. Results show that the Municipality of Trifylia demonstrated greater compliance with Benford’s Law, whereas the Municipality of Messini showed more frequent and pronounced deviations.

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1. Introduction

Advancements in information technology have significantly increased the volume and complexity of financial data. In modern audit practices, analytical techniques are increasingly utilized to streamline audit processes and enhance the detection of anomalies and irregularities in public spending. As noted by Durtschi, Hillison, and Pacini (2004), these techniques simplify and enhance audit procedures.

This study explores the use of Benford’s Law as a statistical method for identifying irregularities in public expenditures, specifically within the financial data of two municipalities. The research assesses whether these expenditures align with the expected digit distribution under Benford’s Law and investigates the implications of any observed deviations.

Benford’s Law has been previously applied in numerous studies for fraud detection in public finance, including notable works by Carslaw (1988), Nigrini (2005), and others. The importance of this study lies in its practical application of predictive irregularity models for transparency assessment in public administration.

2. Benford’s Law and Its Audit Relevance

Benford’s Law predicts the frequency distribution of digits in naturally occurring datasets. Introduced by Simon Newcomb (1881) and later formalized by Frank Benford (1938), the law suggests that lower digits occur more frequently as leading digits. The law has been applied across diverse datasets such as financial transactions, river lengths, and population figures.

In audit settings, researchers (Durtschi et al., 2004; Arkan, 2010; Ardiansah & Sudarto, 2017) have demonstrated that Benford’s Law can identify data irregularities that merit further investigation. It is now integrated into audit software tools such as ACL and CaseWare.

Applications range from IT data integrity (Debreceny & Gray, 2010), fraud detection in national statistics (Holz, 2014), environmental reporting (Stoerk, 2016), to electoral fraud analysis (Leeman & Bochsler, 2014). KPMG and other global firms have recommended its use in forensic auditing (Pavlovic, 2019).

3. Research Methodology

3.1 Data Collection

The study analyzed 129 contracts from the Municipality of Messini and 230 from the Municipality of Trifylia, all retrieved from Greece’s electronic procurement platform (Promitheas). These contracts reflect 2021 expenditures totaling approximately €3.73 million and €10.22 million, respectively.

3.2 Statistical Model

We employed Z-tests and Chi-square tests to measure deviations from Benford’s expected distribution for both the first and second digits. The tests evaluate the null hypothesis (H0: no irregularities) versus the alternative (H1: potential irregularities). A significance level of α = 0.05 was adopted.

Benford’s probability distributions for digit frequencies were used as reference, and critical values were based on standard statistical tables (Z-critical = 1.96; Chi-squared critical values = 15.507 for the first digit, 16.919 for the second digit).

4. Results and Discussion

4.1 Municipality of Messini

Significant deviations were observed in the digit frequencies. For the first digit (see Table 1), the values 1, 2, and 9 exceeded the Z-critical threshold. The overall Chi-square value (22.250) also surpassed the critical limit, indicating a rejection of the null hypothesis. Similar deviations were noted for second-digit values 3 and 8 (see Table 2).

Table 1. Distribution of First Digits in Messini Municipality Expenditures (2021).

1st digit	number of observations	Po	Pe	Po-Pe	Fo	Fe	Fo-Fe	Z	x^2	chi-square critical
1	61	0.473	0.301	0.172	61	39	22	3.319	12.659
2	13	0.101	0.176	-0.075	13	23	-10	2.812	4.148
3	11	0.085	0.125	-0.040	11	16	-5	1.491	1.629
4	14	0.109	0.097	0.012	14	13	1	0.278	0.177
5	9	0.070	0.079	-0.009	9	10	-1	0.240	0.139
6	6	0.047	0.067	-0.020	6	9	-3	0.906	0.808
7	7	0.054	0.058	-0.004	7	7	0	-0.007	0.031
8	6	0.047	0.051	-0.004	6	7	-1	0.033	0.051
9	2	0.016	0.046	-0.030	2	6	-4	2.486	2.608
total	129	1	1	0	129	129	0	–	22.250	15.507

Table 2. Distribution of Second Digits in Messini Municipality Expenditures (2021).

2nd digit	number of observations	Po	Pe	Po-Pe	Fo	Fe	Fo-Fe	Z	x^2	chi-square critical
0	25	0.194	0.120	0.074	25	15	10	1.923	5.855
1	10	0.078	0.114	-0.036	10	15	-5	1.413	1.506
2	17	0.132	0.109	0.023	17	14	3	0.627	0.614
3	7	0.054	0.104	-0.050	7	13	-6	2.362	3.068
4	12	0.093	0.100	-0.007	12	13	-1	0.122	0.063
5	14	0.109	0.097	0.012	14	13	1	0.278	0.177
6	8	0.062	0.093	-0.031	8	12	-4	1.298	1.332
7	16	0.124	0.090	0.034	16	12	4	1.019	1.660
8	4	0.031	0.088	-0.057	4	11	-7	3.587	4.761
9	16	0.124	0.085	0.039	16	11	5	1.185	2.312
total	129	1	1	0	129	129	0		21.348	16.919

4.2 Municipality of Trifylia

The data showed fewer deviations. Only the first-digit value 2 (see Table 3) and second-digit values 4 and 9 (see Table 4) exceeded the Z-critical threshold. However, the overall Chi-square result for the first digit (11.657) did not exceed the critical value, implying general conformity with Benford’s Law.

Table 3. Distribution of First Digits in Trifylia Municipality Expenditures (2021).

1st digit	number of observations	Po	Pe	Po-Pe	Fo	Fe	Fo-Fe	Z	x^2	chi-square critical
1	83	0.361	0.301	0.060	83	69	14	1.742	2.739
2	30	0.130	0.176	-0.046	30	40	-10	2.007	2.713
3	21	0.091	0.125	-0.034	21	29	-8	1.691	2.089
4	22	0.096	0.097	-0.001	22	22	0	-0.043	0.004
5	23	0.100	0.079	0.021	23	18	5	0.941	1.284
6	12	0.052	0.067	-0.015	12	15	-3	0.870	0.755
7	17	0.074	0.058	0.016	17	13	4	0.790	1.004
8	14	0.061	0.051	0.010	14	12	2	0.486	0.439
9	8	0.035	0.046	-0.011	8	11	-3	0.753	0.629
total	230	1	1	0	230	230	0		11.657	15.507

Table 4. Distribution of Second Digits in Trifylia Municipality Expenditures (2021).

2nd digit	number of observations	Po	Pe	Po-Pe	Fo	Fe	Fo-Fe	Z	x^2	chi-square critical
0	34	0.148	0.120	0.028	34	28	6	1.079	1.484
1	30	0.130	0.114	0.016	30	26	4	0.636	0.545
2	20	0.087	0.109	-0.022	20	25	-5	1.083	1.025
3	26	0.113	0.104	0.009	26	24	2	0.327	0.181
4	15	0.065	0.100	-0.035	15	23	-8	2.041	2.783
5	15	0.065	0.097	-0.032	15	22	-7	1.850	2.395
6	20	0.087	0.093	-0.006	20	21	-1	0.209	0.090
7	19	0.083	0.090	-0.007	19	21	-2	0.289	0.140
8	18	0.078	0.088	-0.010	18	20	-2	0.429	0.248
9	33	0.143	0.085	0.058	33	20	13	2.357	9.253
total	230	1	1	0	230	230	0		18.144	16.919

4.3 Summary Comparison

The Municipality of Messini exhibited more significant and numerous deviations across both digit positions, suggesting higher risk of irregularities. Approximately 74.10% of Messini’s contract values involved suspect digits, compared to 22.03% for Trifylia (see Table 5). These results support the use of Benford’s Law in prioritizing audit efforts and optimizing resource allocation.

Table 5. Contracts Issued which First Digits Disagree with Benford’s Law

	Non-Conformity Digits		Contracts (n)	Contracts (€)	% Total Contracts
Municipality of Messini	1st digit	1, 2 and 9	76	2,433,183.67 €	65.18
	2nd digit	3 and 8	11	333,030.90 €	8.92
Municipality of Trifylia	1st digit	2	30	782,085.53 €	7.65
	2nd digit	4 and 9	48	1,469,650.74 €	14.38

5. Conclusion

The analysis confirms the value of Benford’s Law as a preliminary audit tool for public expenditures. The Municipality of Messini showed greater deviation and thus may present higher audit risk. With the statistically significant deviations found, which are indicators of potential irregularities, public auditing organizations could identify cases for audit review. Benford’s Law enables data-driven prioritization in audit planning, offering transparency indicators and supporting strategic decisions.

Future research could expand the dataset across more municipalities to establish a transparency index incorporating Benford compliance as a core variable.

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References

1. Benford, F. (1938). The Law of Anomalous Numbers. Proceedings of the American Philosophical Society.
2. Carslaw, C. (1988). Anomalies in Income Numbers: Evidence of Goal-Oriented Behavior. The Accounting Review.
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4. Nigrini, M. (2011). Forensic Analytics: Methods and Techniques for Forensic Accounting Investigations. Wiley.
5. Pavlovic, V. (2019). Benford’s Law Analysis to Determine Audit Priorities. Case Study.
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