Weight decomposition Weight loss estimation Example of deconvolution of observed weights Comparison of different methods

Weight decomposition

The observed weight of each coin W_observed can be expressed as the difference between its original weight W_original when it left the mint and the weight loss L due to wear, physico-chemical processes caused by environmental influences over time, and possibly other factors, such as cutting metal off the edge (the so-called clipping).1 Formally expressed:

Wobserved = Woriginal – L.

(1)

The quantities W_observed, W_original and L are random variables. The distribution of W_observed is usually unimodal2 and asymmetrically skewed to the left, that is the part to the left of the mode is longer and with a more gradual slope than the part to the right of the mode. In other words, the distribution of observed weights is negatively-skewed.

The weight standard is the mean value of original weights of coins of a given denomination minted in a given time period. Formally, the weight standard is the expected value of the random variable W_original. Thus, according to equation (1):

weight standard = E(Woriginal) = E(Wobserved) + E(L).

(2)

For each individual coin, we can find its current weight W_observed, but both the W_original and L values are unknown and unobservable. So, de facto, we are dealing with the deconvolution of sum of two random variables and this problem does not have a unique solution without additional information or assumptions.

Weight loss estimation

Possible approaches to estimate the average weight loss L, which is not directly observable, are as follows:

1.

Identify unworn coins with good metal quality, i.e. coins that were lost or hoarded shortly after their mintage and were preserved in conditions that prevented metal degradation. For such coins, we can assume negligible weight loss and, according to equation (2), the weight standard can be expressed as E(Wobserved), i.e., we can estimate it as the average weight of these coins. This is the most accurate estimate, but only if enough such coins can be reliably identified so that the statistical error of the estimate is within the required limits.

2.	Choose suitable parametric models of probability distributions of the random variables Woriginal and L, estimate the parameters of these distributions from the values of Wobserved, and subsequently calculate the weight standard according to equation (2). The problem of this method is the choice of appropriate probability distributions. The random variable Woriginal can probably be assumed to have a normal or truncated normal distribution. However, this is not certain, since we do not know the process by which coin flans were prepared in the given mint.3
	The bigger problem, however, is the choice of the type of probability distribution of the random variable L. It is determined not only by the time of circulation of coins and the degradation of the metal over time, but also by the selection criteria of museums and auction houses, from whose catalogues the data are drawn. Unless they are rare coin types, better preserved specimens are included in museum collections and major auctions. The appropriate choice of the parametric distribution of the random variable L is therefore extremely debatable, and a good approximation of the observed data by convolution of the estimated distributions of Woriginal and L may not at all indicate the correctness of the estimate of the value of Woriginal.
	A numerical illustration of the weight standard estimate for one possible choice of distributions of the random variables Woriginal and L is given below. However, for the reasons stated, the results cannot be considered reliable.

3.	Model the observed distribution of coin weights based on the loss of weight due to wear over time of its cicrulation and on the probability of losing a coin after a certain time. Based on the assumptions that the weight of a coin at a given time after minting has a normal distribution, whose mean and variance are linear functions of time, and that the probability of losing a coin has an exponential distribution, Müller 1977 derived an approximate estimate
	E(L) = (-M3/2)1/3,
	where M3 is the 3rd central moment of the observed coin weight distribution (ibid., p. 194). The weight standard according to equation (2) is then estimated as
	weight standard = Mean(Wobserved) + (-h3/2)1/3,
	where Mean(Wobserved) is the arithmetic mean of the observed coin weights and h3 is the unbiased sample estimate of M3.
	Müller notes (ibid., p. 194) that given the debatable assumptions used, the approximation used in deriving this estimate, and the possibility that some coins may have been hoarded and thus removed from circulation, this is only a rough estimate. However, it is based on a consistent model, uses only data derived directly from the sample, and is easy to apply.

4.	Model the weight loss L also based on other physical properties of the coins, not just their weight.
	4.1.	Hemmy 1938 presented one such possible approach. Hemmy replaces the mean value E(Wobserved) with the mode of Wobserved in equation (2), the reasons for using the mode being given in ibid., p. 66. More importantly, Hemmy assumes that the weight loss L is given by wear and it is proportional to the area of coins neglecting the area of the rim. Based on this assumption, we get (ibid., p. 67, equation ii):
		E(L) = jq2,
		where q is the diameter and j is the measure of wear defined as j = πtρ/2, where t is the average thickness of metal removed and ρ is the density of the metal. Based on an analysis of various sets of ancient coins and their different denominations, Hemmy arrived at estimates of the measure of wear j of 0.033 for gold coinage and 0.07 for silver coinage, where the unit of mass is the gram and the unit of size is the centimeter.4 Equation (2) is thus transformed into the equation:
		weight standard = Mode(Wobserved) + 0.033×q2   for gold coinage,
		weight standard = Mode(Wobserved) + 0.07×q2   for silver coinage,
		where weights are in grams and the average diameter of the coins q is in centimeters.
		Hemmy’s method is also not without problems, see ibid., pp. 69 ff. The use of the mode instead of a suitably chosen robust estimate of the expected value of Wobserved is also debatable. From a practical point of view, there may also be a problem with determining the average diameter q, because many ancient coins do not have a precise circular shape, and therefore for each coin it is necessary to measure its largest and smallest diameters, or two diameters forming a right angle (the latter approach was chosen by Hemmy). However, this information is often missing from coin catalogues.
	4.2.	François Delamare (Delamare 1991 and Delamare 1994) developed a tribological approach taking into account both abrasion and corrosion of coins:
		E(L) = Ka × (Kc×ρ/Hv) × (F×kc×t),
		where Ka is the abrasion coefficient, Kc is the corrosion coefficient, ρ is the specific weight of the monetary alloy, Hv is its hardness, F is the force applied to the coin by some rough surface, kc is the intensity of circulation expressed in meters per year and t is the time of circulation of the coin. The first factor in this equation represents the abrasion of the coin, the second represents the nature of the monetary alloy (sensitivity to corrosion, specific weight and hardness), and the third represents the conditions of circulation (contact force, intensity and length of circulation). The loss of weight of a coin is proportional to the density of the alloy, the sensitivity of the alloy to corrosion, the contact force and the intensity and the length of the circulation, and inversely proportionnal to the hardness of the alloy (Delamare 1991, pp. 353–4; Delamare 1994, pp. 97–9).
		This rigorously physical approach separates the physical parameters relevant to wear of coins from those related to their circulation, and the model fits well with both published data on the wear of coins and laboratory data. If the average weight loss per unit time can be determined from the data,5 then, for example, this model can explain changes in the trend of weight loss over time based on changes in certain parameters, or enable a quantitative comparison of the circulation intensity of different coin issues. However, in the case of analyses of undated coins, for which neither the time of minting nor the time of their circulation can be precisely determined, this approach does not allow for a more precise estimate of the weight standard.

Example of deconvolution of observed weights

Let us give an example of the application of approach 2 above. We will assume that W_original has a normal distribution with parameters μ and σ and L has an exponential distribution with parameter λ, i.e. its probability density is given as

f_L(x) = λe^-λx  for  x≥0.

We will also assume that these random variables are independent. It follows that the random variable W_observed has a mirror-reversed exponentially modified Gaussian distribution with parameters -μ, σ and λ, because W_observed can be expressed as -(-W_original + L).

We will consider three groups of Kelenderis staters, namely Groups 1, 2 and 3A (for the definition of these groups, see the Overview of Kelenderis coins in the Coin Catalogue and the weight analysis of Kelenderis staters). The numbers of coins with known weight in the corpus as of the date of this analysis (16 August 2025) are as follows:

Group 1:	13
Group 2:	237
Group 3A:	191

Estimates obtained by the minimum-distance method using the Anderson-Darling metric are given in Table 1 (the Kolmogorov-Smirnov metric and the maximum likelihood method give similar results). Estimates of the weight standard are equal to the parameter μ based on the choice of the distribution of W_original. As Table 1 shows, the estimated mean weight loss varies from 0.07 g to 0.19 g, but the estimate for Group 1 may be affected by the low number of observations.

	Group 1	Group 2	Group 3A
μ	10.932	10.808	10.797
σ	0.083	0.067	0.085
λ	5.146	12.479	6.028
observed average weight	10.73	10.73	10.62
estimated mean weight loss	0.19	0.08	0.17
	1.78%	0.74%	1.54%
estimated weight standard	10.93	10.81	10.80

Table 1: Minimum-distance estimates

Figure 1 shows relative frequency histograms of observed weights and the estimated distributions of observed weights given by the mirror-reversed exponentially modified Gaussian distribution. Thus, the bars represent the relative frequencies of observations ranging from 8.70 to 11.10 g in increments of 0.10 g and the continuous curves represent the estimated distributions of the random variables W_observed. Figure 2 shows the estimated distributions of observed weights W_observed and of the original weights W_original. Figure 3 shows the P-P and Q-Q plots.

Figure 1: Relative frequency histograms and estimated observed distributions

Figure 2: Estimated observed and original distributions

Figure 3: P-P plots (top row) and Q-Q plots (bottom row)

Comparison of different methods

Table 2 shows a comparison of approach 2 (the deconvolution method applied in the previous section) with approaches 3 (Müller 1977) and 4.1 (Hemmy 1938).6 For the application of Hemmy’s method, average coin diameters of 19 mm are assumed for Group 1 and 20 mm for Groups 2 and 3A. This means that the average weight loss according to Hemmy’s formula is equal to 0.07×1.9² = 0.25 g for Group 1 and 0.07×2² = 0.28 g for Groups 2 and 3A.

	Group 1	Group 2	Group 3A
observed statistics
mean	10.73	10.73	10.62
mode	10.80	10.75	10.75
standard deviation	0.23	0.10	0.25
skewness	-1.75	-0.52	-3.22
3rd central moment	-0.021	-0.001	-0.052
deconvolution
estimated mean weight loss	0.19	0.08	0.17
	1.78%	0.74%	1.54%
estimated weight standard	10.93	10.81	10.80
Müller 1977
estimated mean weight loss	0.22	0.06	0.30
	2.00%	0.60%	2.71%
estimated weight standard	10.95	10.79	10.91
Hemmy 1938
estimated mean weight loss	0.25	0.28	0.28
	2.30%	2.54%	2.57%
estimated weight standard	10.99	11.01	10.90

Table 2: Comparison of different methods

For Groups 1 and 2, the deconvolution approach and Müller’s formula give very similar results. For Group 3A, the estimate of the average weight loss according to Müller’s formula seems unrealistically high. This is due to the highly negative skewness of the weight distribution, which, however, may have other causes than weight loss due to the long circulation period, which would be more likely to be expected for Group 2. In any case, the results of both of these methods should be taken with great caution for the reasons stated above. As for Hemmy’s formula, which gives the highest values of the average weight loss for all groups, its results are significantly influenced by the used estimates of the average coin diameters, which are not based on measurements of a larger number of specimens.