# Weight of Coins

Founded 9-Jul-2006
Last update 9-Jul-2006

### Weight standards

Value of ancient Greek coins made from precious metals (silver, gold or electron) was based on the weight of the metal. The production of individual coins of the same denomination was therefore closely monitored to ensure their constant weight. Nevertheless, the ancient minting technology had its limitations, so that weights of individual coins slightly vary. We can suppose that, in addition to a minimization of deviations of weights of individual coins, a fixed number of coins per weight unit was produced (e.g. 25 tetradrachms to one mina of silver) to reduce the weight variation and to ensure controllability of the minting process. The weight standard of a given issue is therefore given by the average weight of freshly struck coins.

### Original and observed weights of coins

The weight of a freshly struck coin can be expressed as

original weight = M + D,

where M is the mean value of original weights of coins of the given issue, i.e. the weight standard, and D is the deviation from the weight standard given by technological imperfections of the minting process. The weight standard M is constant in the given time period whereas the deviation D varies from coin to coin and it can be both positive and negative. Thus, D is a random variable with zero mean.

The present weight of a preserved coin can be expressed as

present weight = M + DL,

where L is the loss of weight of the coin during its lifetime. This loss is, roughly speaking, a sum of four factors:

1. wear which the coin suffered during period of its circulation,
2. intentional modifications of the coin during its circulation (test cuts, piercing, intentional cutting off pieces of coins to obtain the precious metal by fraud etc.),
3. physical and chemical changes of the metal which occurred from antiquity to present days (corrosion, crystallization and other changes of the metal),
4. contemporary causes (harsh cleaning, unintentional damages etc.).

Finally, measurements may be affected by measurement errors. Thus, the observed weight can be expressed as

observed weight = M + DL + e,

where e is the error given by imperfections of the weighing. Usually it is reasonable to suppose that this variable has normal distribution with zero mean. Summarize that M is a constant, D is a random variable with zero mean, L is a nonnegative random variable and e is a normally distributed random variable with zero mean. For completeness, note that observed weights are usually available with a precision of one hundredth of a gram.

In fact, the measurement error e is negligible with respect to other parts of the observed weight because non-accuracies of measurements seldom exceed several hundredths of a gram (typos and mistakes of curators, dealers and collectors excepted). Likewise, losses of weight having arisen in modern times can be usually, although not always, taken as negligible. Finally, the intentional modifications of coins during their circulation can be often omitted too because such specimens can be easily recognized and excluded from an analysis. Thus, the above model can be usually simplified in the following way:

observed weight = M + DL1L2,

where L1 is the loss of weight given by wear which the coin suffered during period of its circulation and L2 is the loss of weight given by physical and chemical changes of the metal from antiquity to present days. Provided that we examine well-preserved non-circulated or little-circulated coins only, we can suppose that the observed weight is approximately equal to the original weight, i.e.

observed weight = original weight = M + D.

### Statistical analyses

We observe the total sum of M, D, L and e, but these individual components themselves are generally unobservable. The basic goal is to estimate characteristics of the distribution of observed weights such as mean, median and quartiles, or to identify the distribution itself. These characteristics complement other metrological information about an examined series such as metal composition or die links. The second goal may be a comparison of different series. For example, provided that two samples contain coins in similar states of preservation, we may be able to say that a higher weight standard was used for one series than for the other, although we do not know the precise weight standards themselves. Similarly, we may be able to say that less careful minting process was used for one series than for the other in the sense of greater dispersion of weights. Of course, all such conclusions must take into consideration sizes of samples, their representativity and comparability and other possible factors.

Without additional information or assumptions, it is not possible to precisely estimate the weight standard M itself. If we can be sure that the loss of weight L is negligible (i.e. if we examine well-preserved non-circulated or little-circulated coins) then the weight standard M can be estimated as the mean value of observed weights and we can also estimate characteristics of the distribution of the original weight M + D.

### Representativity of data

An analyzed data set can contain all or nearly all specimens of a studied coin population (for example, if an individual hoard is studied and information about all coins from the hoard is available). More often, we interpret data as a sample from a larger coin population and we want to generalize results of an analysis of our sample to the entire coin population. In such case, two main conditions must be fulfilled. First, the studied coin population must be clearly defined from the numismatic point of view. Second, our data sample should be representative of the studied coin population. This means that the data can be (approximately) considered as a random sample from the defined population in the sense that each coin of the population entered the sample independently with equal likelihood, so that important characteristics correlated (or potentially correlated) with the weights of coins (such as states of preservation, epigraphic styles, control marks etc.) are represented in the sample according to their frequencies in the entire population. In another words, the data sample should not be affected by factors not related to the studied coinage itself such as, for example, preferences of collectors.

This is particularly important as for states of preservation. Individual deviations D from the weight standard M can be taken as independent identically distributed random variables within one series. However, individual losses of weight L can have significantly different distributions within different hoards. For example, data mixed from well-preserved coins and from corroded coins from a shipwreck can be characterized by a bimodal distribution, even if all coins in the data sample come from the same series (weight losses of coins from a shipwreck are usually much higher than weight losses of coins deposited in a less aggressive environment). As for a weight analysis, a data sample constructed in this way has little sense.

### Outlying data

Coins with markedly atypical weight with respect to their state of preservation should be excluded from an analyzed data sample. As we noted, coin production was closely monitored to ensure approximately constant weights of coins within one denomination. Coins highly above the weight standard consumed needlessly much of the precious metal. High upper deviations from the weight standard used for the given series are therefore very extraordinary. A large upper deviation may also indicate a modern forgery. Since such outlying data can distort statistical conclusions, it is reasonable to determine an upper exclusion limit and to exclude each coin from an analysis whose weight is greater than this limit. This limit should be based on known facts about the coinage in the given period and area and also on a preliminary examination of given data. Too high limit fails to meet its purpose of excluding completely non-typical coins; too low limit excludes a part of standard production and therefore distorts statistical conclusions. Usually, there should be at most few coins above the limit.

Coins highly bellow the weight standard were undesirable too because such coins threatened credibility of the given coinage. Unlike the upper deviations, larger lower deviations are more often. In fact, data are often spread out more to the left of the mode than to the right (so called negative skewness) because of the loss of weight L. For this reason, a lower exclusion limit should be chosen cautiously with respect to states of preservations of examined coins.