Romania’s direct public purchases and Benford’s law: what the first-digit test sees and what it misses

A direct purchase is a public purchase without a tender, allowed when the estimated value, without VAT, is under a cap; the law still requires publicity, through the electronic catalogue or a notice. Until 9 September 2022 the cap for goods and services was 135,060 lei; from 10 September 2022 it is 270,120 (Law 208/2022). Romania’s Authority for Digitalisation publishes quarterly exports from the electronic procurement system on data.gov.ro.

Below, the same purchases seen two ways. Left: how many sit at each amount, between 100,000 and 400,000 lei. Right: which digit their amounts start with. Pick a quarter; the cap moves, the digits stay.

Why the cap does not show in the digits

Two reasons. First, the cap cuts only the top end of the amounts: in Q4 2023, 90% of the awards are under 10,000 lei. The first digit follows the size of the need: a 150-lei toner cartridge, 37,000 lei of firewood, a 2,500-lei service. A register is a mixture of needs at very different scales, and such a mixture can come close to the law (a plausible explanation, not a proof for this register).

The second reason: a small adjustment near the cap often leaves the first digit unchanged. 269,999 and 270,000 both start with 2, just as 135,000 and 134,999 start with 1. The cap does leave its mark in the amounts. In Q1 2022, 448 awards are exactly 135,000 lei, and 211 fall from 135,060 to 270,119 lei. In Q4 2023 there are 3,625 in that interval and 165 at exactly 270,000 lei; in Q2 2024, 4,755 and 268. The first digit moves between quarters by at most 0.5 percentage points. Amounts above the general cap exist too (279 in Q1 2022, 149 in Q4 2023, 263 in Q2 2024); I did not look into why: possible explanations are a different rule for some entities, amounts entered with VAT, or entry errors.

In Q2 2024 the 132,500–135,000 bin holds 780 awards, against 293 and 202 in its neighbours; three repeated amounts (134,397, 134,331, 134,375) account for 129. I did not look into where they come from.

Counting the amounts that sit exactly under a threshold in the electronic catalogue is the subject of One leu below the line. The question here is different: what the first digit can say about them.

Because on a logarithmic ruler the digit 1 has the longest segment. From 1 to 2 is 30.1% of the ruler’s length; from 9 to 10, only 4.6%. A number growing by a fixed percentage spends equal time on each centimetre of ruler, so it lands most often in the long segment. The formula is P(d) = log₁₀(1 + 1/d).

The mechanism can be watched. The number below starts at 1 and grows by 8% at each step, like compound interest; the bars count the digit each step stood on.

Simon Newcomb noticed the pattern in 1881 from the wear on the pages of a logarithm table. Frank Benford checked it in 1938 on 20,229 values from 20 lists (river areas, populations, costs). Theodore Hill showed in 1995 that mixtures of distributions, sampled without favouring any one scale, tend toward the same law.

Distance from the law is measured by the mean absolute deviation (MAD): the average gap between each digit’s share and the law. Mark Nigrini, who wrote the auditors’ handbook, proposes the cut-offs 0.006 (close conformity), 0.012 (acceptable) and 0.015 (marginal). The dashed line on each row shows the MAD a perfect series of that size would average: 0.0159 at 217 values, 0.0003 at 594,448. A MAD near the dashed line is what a perfect series would give too; one far beyond it shows a real gap.

Three lessons from the chart above

The size of the set changes everything. Country populations (217 values) have a MAD of 0.0152, above 0.015, but a perfect series of 217 values would average 0.0159: the gap is within chance (p = 0.50). Nigrini’s cut-offs are empirical conventions, not tests calibrated to the size of the set. At 594,448 rows it is the other way round: even a small gap like the one observed here (MAD 0.0018) gives p below 10⁻⁴⁰, so what counts is the size of the gap, not p.

A set that does not span several scales has no reason to follow the law. The 3,181 communes, towns and cities (88 to 1,716,961 residents at the 2021 census) are 89% between 1,000 and 10,000 residents. The digit 2 opens 25.5% of them, against 17.6% by the law (MAD 0.030). Life expectancy, between 54 and 86 years, starts only with 5, 6, 7 or 8.

Small sets come out further. Subsequent contracts (n = 85,504) have a MAD of 0.0025, framework agreements (n = 27,429) 0.0055, direct purchases of works (n = 13,669) 0.0053: all under 0.006, but further from the dashed line than the direct register. The value of a framework agreement is often an estimate, not a payment; the three contract types are counted separately, never pooled. Rows with the same authority, number, lot and value (usually one framework agreement listed once per supplier) count once; I did not test counting each supplier.

The first digit shows the size of the need. The last digits show how the amount was written. In a natural process the last digit would be close to uniform: 10% each. In procurement it is not.

Round prices are natural: people ask for and offer 2,500 lei, not 2,487. In a register like this they prove nothing about any single contract. They only show where the human hand leaves its print, and that the print is not in the first digit.

The test measures the shape of the whole register. For the MAD verdict to move, many rows must change their first digit. Set below how many awards change their first digit: how many would it take to push the Q4 2023 MAD (0.0018) out of "close conformity"? At least 11,169 of 594,448, or 1.9%, however the moves are chosen. If all of them move from digit 1 to digit 9, as in the chart, it takes 12,737.

A hundred awards of 2xx,xxx lei brought to exactly 270,000, or a thousand, move nothing here: they keep their digit 2. What piles up under a threshold shows in the exact values, not in the digits.

By authority: why a traffic light would raise false alarms

A school that buys only textbooks or a town hall that buys only fuel probably buys on a narrow scale, so it is unlikely to follow the law. A plausible explanation for the deviations is what each authority buys; I did not test it. A "deviating" row is not a finding about an authority. That is why no name appears on this page.

Paste at least 20 positive values (amounts, populations, lengths) or pick a real set. The page shows the first digit with the chance band for your sample size, the MAD, what a perfect series would give, and warns when the law has no reason to apply. Nothing leaves your browser.

What the test cannot show

• It neither proves nor rules out fraud. A series with doctored amounts can follow the law, and an honest authority buying one category may not.
• It cannot see movement inside a digit: 269,999 or 270,000, 121 or 120 lei under the 270,120 cap.
• The MAD classification does not change until at least 1.9% of the rows change their first digit (11,169 awards in a quarter). The χ² test rejects any tiny gap at this many rows anyway.
• Three quarters (Q1 2022, Q4 2023, Q2 2024), fixed before the calculation, not the whole year. Q4 2023 and Q2 2024 differ by at most 0.13 percentage points on any digit.
• Values are as published. I dropped zero or negative amounts (1,512 in Q4 2023, 6,296 in Q2 2024) and malformed rows (27 and 31); the export format differs from one quarter to the next. I did not check individual contracts.

What I predicted before seeing the data

Sources, and how to repeat the calculation

1. Romania’s Authority for Digitalisation, Public procurement 2022, 2023, 2024 (SEAP exports, CC BY 4.0). The files used, with SHA-256, are in the manifest.
2. Law 208/2022, Official Gazette no. 697 of 12 July 2022, applicable from 10 September 2022: text on the ANAP site (in Romanian); the consolidated Law 98/2016, art. 7.
3. World Bank indicators SP.POP.TOTL, AG.SRF.TOTL.K2, NY.GDP.MKTP.CD, SP.DYN.LE00.IN; regional aggregates excluded.
4. INS (Romania’s statistics institute), 2021 Population and Housing Census (recensamantromania.ro): residents per administrative unit, 3,181 units.
5. Mark J. Nigrini, Benford’s Law: Applications for Forensic Accounting, Auditing, and Fraud Detection, Wiley, 2012 (the MAD cut-offs); Simon Newcomb, American Journal of Mathematics, 1881; Frank Benford, Proceedings of the American Philosophical Society, 1938; Theodore P. Hill, Statistical Science, 1995.

The calculation has four steps, each with its own script: downloading the eight exports, reading them row by row, counting the control sets, and computing the numbers on the page. The rules (column, filters, deduplication, statistic) were written in the pre-registration before the first histogram; changes after that are listed there. All files: /reproduce/.

Every set

MAD: the mean of the nine absolute gaps between a digit’s share and Benford’s law. w: the square root of χ²/n, an effect size that does not grow with n. p: the probability that a perfect series would deviate at least this much (χ², 8 degrees of freedom); with hundreds of thousands of rows it comes close to zero even for small gaps.

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