Summary
Share of the original coinage struck by dies already in your sample.
Die estimators
Recommended standard Esty (2011) is the formula most die studies now report, and the one SILVER (KBR) gives for every series alongside Carter (1983). Lead with it for comparability; use the others as cross-checks.
Intervals marked "sim." come from 2,000 simulated resamples under the geometric die-output model. Intervals marked "Lyon" use the published coverage limits (Lyon 1989, formula 5, after Esty 1986).
| Method | Estimate | 95% interval | Basis |
|---|
Bayesian estimate
The probability of every possible die total under five assumptions about how unevenly dies were used, and how well each assumption explains your singletons. Select a row to chart it. A sensitivity check, not a replacement for Esty.
| Die-output model | Most likely | Median | 95% credible interval | Fit to your data |
|---|
"Fit" compares the models with each other; it is not an absolute test of any one of them.
Prior limits and Albarède model settings
Albarède et al. model die output as two populations: dies that fail early (infant mortality) and dies that wear out from metal fatigue. Defaults are the paper's typical case (Fig. 8). Across their 29 issues, the early-failing share averaged 0.63 (sd 0.13), shape β₁ 0.79 (0.16) and β₂ 2.43 (0.46).
Coin output
Dies multiplied by an assumed output per die. SILVER uses 20,000 coins per obverse die as a working average, with 10,000 to 40,000 as the plausible range.
Coins struck uses the working figure per die; the range pairs the low end of the die interval with the low output per die, and the high end with the high output, so it is deliberately wide. Albarède et al. (2021) warn that many singletons are early-failing dies that struck few coins, so high singleton counts push these figures up.
Die frequencies
How many dies are known from 1, 2, 3… specimens. A tall first bar (many singletons, in gold) means many dies are still unrecorded.
About this calculator
The Ancient Die Study Calculator is a free tool for quantitative numismatics. A die study records which dies struck each surviving coin of an issue. From those counts, the calculator estimates the original number of obverse or reverse dies used to strike an ancient coinage, how much of that coinage the known coins cover, and, from an assumed output per die, the size of the coinage in coins and silver.
It leads with the Esty formula (Warren W. Esty, 2011), the recommended standard for estimating coin dies and the figure the SILVER database reports, and checks it against Carter (1983), Lyon (1989), Good's coverage estimator and Bayesian models. It suits Greek, Roman and other ancient coinages, and any die study where the coin and die counts are known.
How do you estimate the original number of dies?
Count the coins in your die study (n) and the different dies they were struck from (d). Esty's formula gives the estimated original number of dies as n·d / (n − d). Results are reliable when there are at least three coins per die, and need care below that.
What is coverage in a die study?
Coverage is the share of the original coinage struck by dies already represented in your sample. It is estimated as 1 − d₁/n, where d₁ is the number of dies known from a single coin (singletons).
How many coins did an ancient die strike?
No figure is certain. A common working average for large silver coins is 20,000 coins per obverse die, with 10,000 to 40,000 as the plausible range. The calculator lets you change these figures and shows the resulting range of coins struck, metal coined and survival rate.
How this is calculated
Esty (2011), geometric model: D = n·d / (n − d), where n is the number of coins and d the number of dies observed. This is the recommended standard and the figure SILVER (KBR) reports. The interval comes from 2,000 simulated resamples under the same model. Coverage is 1 − d₁/n (Good; Esty 1986). W. W. Esty, "The geometric model for estimating the number of dies", in F. de Callataÿ (ed.), Quantifying Monetary Supplies in Greco-Roman Times (Bari 2011), 43–58.
Switch to Full for Carter, Lyon, coverage-based and Bayesian estimates, coin output in metal, and the die-frequency chart.
Formulas and references
Carter (1983): D = n·d / (a·n − b·d), with (a, b) = (1.214, 1.197) when n/d < 2, (1.124, 1.016) when 2 ≤ n/d < 3, and (1.069, 0.843) when n/d ≥ 3. G. F. Carter, ANS Museum Notes 28 (1983), 195–206.
Esty (2011), geometric model: D = n·d / (n − d). W. W. Esty, "The geometric model for estimating the number of dies", in F. de Callataÿ (ed.), Quantifying Monetary Supplies in Greco-Roman Times (Bari 2011), 43–58.
Coverage (Good; Esty 1984, 2006): C = 1 − d₁/n, the share of the original coinage struck by dies already in the sample. The coverage-based die estimate is d / C. W. W. Esty, NC 166 (2006), 359–364.
Lyon (1989): formula (2) D = d + d₁²/(2d₂), the same estimator ecologists know as Chao1; formula (3) D = d + d₁(d₁ + d₂)/(2d₂ + 3d₃), which Lyon prefers because it ignores dies known from many coins and so resists blocks of die-duplicates from hoards. Coverage limits, formula (5): c ± (2/N)·√(d₁ + 2d₂ − d₁²/N); die limits are D × c / cmax and D × c / cmin. Formula (1) is the coverage-based estimate d/(1 − d₁/N) above. S. Lyon, "Die estimation: some experiments with simulated samples of a coinage", British Numismatic Journal 59 (1989), 1–12. The Lyon example reproduces his appendix, Example 1 (sample of 100 coins from a coinage of exactly 100 dies): formula (3) gives 96 dies, with limits 77 to 126.
Albarède et al. (2021): die output follows F(k) = ωF₁(k) + (1−ω)F₂(k), two Weibull distributions for early-failing and fatigue-failing dies. Typical case: ω = 0.54, β₁ = 0.7, β₂ = 2.45, λ₁ = 80, λ₂ = 600 for a 1,000-coin sample. F. Albarède, F. de Callataÿ, P. Debernardi and J. Blichert-Toft, "Model for ancient Greek and Roman coinage production", Journal of Archaeological Science 131 (2021), 105406. Its use in a Bayesian die estimate here is this tool's own construction, not a method proposed in the paper.
Bayesian estimates use a uniform prior between the limits you set, and treat each die's coin count as Poisson given its output, so the probabilities of a die being unseen, a singleton or repeated follow directly from each output model.
Sample-size guidance follows SILVER (KBR): results are reliable when n/d ≥ 3, need care below 3, are risky below 2, and are best avoided below 1.5. The example reproduces SILVER's published values for Naxos staters (Carter 11.49, Esty 13.82, coverage 84.21%).