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. Possible approaches to solve this problem 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 weight loss L also based on other physical properties of the coins, not just their weight. Such an approach is presented by Hemmy 1938. 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.

Example of deconvolution of observed weights

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 (18 May 2025) are as follows:

Group 1:	13
Group 3:	236
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.

	Group 1	Group 2	Group 3A
μ	10.932	10.810	10.798
σ	0.083	0.066	0.084
λ	5.146	12.312	5.993
observed average weight	10.73	10.73	10.62
estimated mean weight loss	0.19	0.08	0.17
	1.78%	0.75%	1.55%
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)

As Table 1 shows, the estimated mean weight loss varies from 0.07 g to 0.19 g (note that the estimate for Group 1 may be affected by the low number of observations). If we consider the average diameter of these coins to be 19 mm (a rough estimate), then according to Hemmy’s method we get a weight loss of 0.07×1.9² = 0.25 g. This value is not very far from the results for Groups 1 and 3A, but neither of these estimates can be considered reliable for the reasons stated above.