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BENFORD'S LAW AND NUMERICAL STYLIZATION OF MONETARY VALUATIONS IN CLASSICAL LITERATURE
Published online by Cambridge University Press: 29 July 2016
Extract
In an article published in this journal in 1996, I surveyed number stylization in monetary amounts recorded in Roman-era literature up to the Severan period. I argued that certain leading digits such as 1, 3 and 4 were heavily over-represented in the evidence. For the limited samples I used at the time these findings are not in need of revision. However, as I show here, a more inclusive approach to the material produces a substantially different picture. The most significant shortcoming of my study was my failure to take account of the probable distribution of leading digits in a random sample, which may serve as a benchmark for assessing the nature and extent of number preference. While I noted that lower leading digits were inherently more likely to occur than higher ones, I schematically related observed frequencies to an even distribution of leading digits (in which each of them is expected to make up one-ninth of the total). This benchmarking strategy is invalidated by a widely observed phenomenon known as Benford's Law, according to which leading digits frequently conform to a predictable pattern that greatly favours lower over higher numbers. This is true in particular if observations are spread across several orders of magnitude. Ancient monetary valuations satisfy this condition since recorded amounts range from single digits to hundreds of millions. Yet, to the best of my knowledge, Benford's Law has never been applied to the study of these data.
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References
1 Scheidel, W., ‘Finances, figures and fiction’, CQ 46 (1996), 222–38CrossRefGoogle Scholar.
2 See Scheidel (n. 1), 225-9 for the popularity of these digits, and also 235 fig. 3 for the preponderance of 1, 4, 10, 30, 60, 100, 200, 300 and 400 million in thematically specific samples.
3 Introduced by Benford, F., ‘The law of anomalous numbers’, Proceedings of the American Philosophical Society 78 (1938), 551–72Google Scholar. The Wikipedia site https://en.wikipedia.org/wiki/Benford's_law (accessed 29 July 2015) provides a convenient introduction. See now also A. Berger and T.P. Hill, An Introduction to Benford's Law (Princeton, 2015) for a comprehensive survey. A. Berger, T.P. Hill and E. Rogers curate the ‘Benford Online Bibliography’ that refers to hundreds of relevant studies (http://www.benfordonline.net, 2009, accessed 29 July 2015).
4 R.P. Duncan-Jones, ‘Numerical distortions in Roman writers’, in J. Andreau, P. Briant and R. Descat (edd.), Économie antique: prix et formation des prix dans les économies antiques (Saint-Bertrand-de-Comminges, 1997), 147–59, at 149, merely noted, with reference to the distribution of leading digits in Diocletian's Price Edict, that ‘reducing frequencies are to some small extent a result of there being fewer high prices’. This greatly underestimates the quantitative impact of this effect as demonstrated by Benford's Law.
5 Based on Scheidel (n. 1), 235 fig. 4.
6 W. Scheidel, ‘Prices and other monetary valuations in Roman history: ancient literary evidence’, http://www.stanford.edu/~scheidel/NumIntro.htm (accessed 18 December 2014). This collection excludes legal sources.
7 See Duncan-Jones (n. 4), 154-5.
8 Note the absence of any compelling evidence for a strong preference for the leading digit 5 in existing studies of Greek number stylization, most notably Rubincam, C., ‘Casualties in the battle descriptions of Thucydides’, TAPhA 121 (1991), 181–98Google Scholar; id., ‘Numbers in Greek poetry and historiography: qualifying Fehling,’ CQ 53 (2003), 448–63CrossRefGoogle Scholar. But much of Greek literature remains untouched by this kind of inquiry.
9 Compare the insights gained from studies of age-rounding and digit preference in inscriptions and papyri: e.g. R. Duncan-Jones, Structure and Scale in the Roman Economy (Cambridge, 1990), 79–92; W. Scheidel, Measuring Sex, Age and Death in the Roman Empire: Explorations in Ancient Demography (Ann Arbor, 1996), 53–91.