Methodology for assessing weight standards from a sample of coins

[Andrew McCabe]

I'm placing this post under the conservation and storage thread because it deals with technical matters and I can't see where else to place it. I have a straightforward question on a matter that has probably bothered numismatists for centuries: When you have a large sample of coins, of an assured single type, with listed weights but without photographs, how should one report the weights, and what methodological tricks are there to assess the standard to which the coins may have been originally struck?

In Crawford’s RRC page 590 he says “there are notorious practical difficulties in the way of establishing actual weight standards and in the way of expressing the theoretical weight standards of Rome in modern terms” and page 592: “of the coinage which survives today almost all is worn or corroded, hence the difficulty in establishing the actual weight standard of an issue of Roman coinage. It has been argued that a frequency table may be a more reliable guide to the weight standard of an issue than an arithmetical averaging process” (citing G.F. Hill NC 1923, and T.Hackens RBN 1962). He goes on to say that “there are cases where a frequency table can be seen to be totally unreliable as an indication of weight standard. The best way therefore to discover the weight standard of an issue is to take the mean of the weights of unworn specimens” (citing P.Grierson NC 1963, President’s address).

I’ve no argument with this in theory, but in practice one may just have a long list of coins with weights but without photos, or where there are photos it can be that practically none of the coins is truly unworn (EF or better), or that a separating line between fairly unworn and fairly worn cannot be drawn.

What tricks can be used? I take it that the Mean and Standard Deviation of a sample can be established easily, but I would not expect a Normal distribution in coin weights in a sample – a long lightweight tail being balanced by a shorter tail on the heavy weight side (if there were an extending heavyweight tail one would except the very heaviest coins to have been pulled via Gresham’s Law). I’d imagine that the typical Mean would be to the left (lighter) than the distribution peak due to the long lightweight tail, and furthermore that even the distribution peak will not represent the “standard” but rather will be shifted to the left of the “standard” due to wear. So the standard will generally be to the right of both the distribution peak and the average (Mean).

I thought perhaps to quote the Mean, and the Mean of the heaviest 50% of samples. Or the Mean and to also list the top (20%) of actual coin weights in the sample. Or to take the distribution peak and assume the standard is a notch above. This makes some practical sense as in Roman Republican denarii, I suspect the distribution peak of large samples is probably 3.7 grams, and the Mean a bit less (3.6 grams), but we suspect from completely unworn examples that the weight standard is about 3.9 grams. So, distribution peak + 5% might be reasonable. Or mean + 5%. Or just show the graph and let viewers make up their own mind.

Are there any views out there?

Andrew

[crawforde]

I have weighed way more things than I care to remember, here is my 2 cents (sorry).

It always seems to be better to start without a hypothesis regarding the shape of the expected distribution, or at least not have it well developed and in my thoughts during initial weighing and descriptive statistics.  Stay open there may be surprises.  Wait for the numbers.

Metrology can be a bit of a puzzle, but some work has been done on other series.  have you contacted Rob Tye?

additional data about each coin recorded in a simple categorical format may make " tweaking " the data or doing more involved analysis later easier.  Encrustations, wear, percentage of flan missing,...

Once you have ensured the accuracy and precision of the equipment, recorded all pertinent data, and  weighed all the coins, look at the standard deviation, the mean, and the mode, then at the shape of your curve. 

At this point if the numbers are well grouped I would tend to just show the graph, describe the shape, and then hypothesize about the reasons for that  shape and list some valid assumptions like those you made in the original post. 

If you have a weird distribution, large data set, multiple series to look at, and/or multiple denominations with good descriptions and you think it might be useful I would be willing to look at the numbers and attempt some hypothesis testing (ANOVA etc.) or even some multi-variate statistics and ordination tests.  If nothing else this could give a nice graphical representation of the distribution that may lead to new questions. 
This is not really my specialty, but I do have access to some decent software and someone who crunches numbers for a living in the office next door. .

[Andrew McCabe]

I don't have the coins in hand - the database relates to a manuscript document made in the early 20th century covering a proportion (perhaps 1/3) of Republican bronzes types, but in very great numbers for the series covered. No photos and no-one capable of commenting (long dead) nor is it practical to assess condition etc of coins. But the data set seems to be several times larger than Paris for the selection of issues covered - perhaps five times as big. So it's potentially worth a lot. I'm trying to think how to make sense of it given the limitations I discussed above. No rush, it's a long term project. Rob Tye will have read the message (posted elsewhere in a list he is active in) and I'm sure will reply.

[daverino]

One simple test which can be made is to find out if the data has an approximately normal distribution. This is the Chi -square test. It should show if the data set you are analyzing is very biased or "lumpy". A good and simple explanation of the method is given here

http://www.physics.csbsju.edu/stats/chi-square.html

I think, but am not entirely sure, that if the weight of an individual coin were to depend only on normally distributed variables such as its initial weight and its degree of wear (and these variables are independent of each other) then the observed distribution of weights should also be normal. Trying to separate the contribution of each would  require additional hypotheses.

Regards, Dave

PS. A bias to the lighter side in a normal distribution might indicate that the variables are not independent. Initial weight, or weight generally, will be reflected by the total volume of metal while wear and corrosion is from the surface. Therefore, are not lighter initial weight coins with a higher ratio of surface area to volume likely to have a greater percentage loss over time due to wear etc.?

[Lloyd Taylor]

Metrology studies I have seen in the Greek series present all the data in a frequency table. Then it's a case of letting the Statistical Mode be your guide with due consideration of factors such as the impact of the bias of mints to underweight rather than overweight precious metal coinage relative to the weight standard, plus consideration of loss of metal through wear, corrosion etc, when anything other than pristine hoard material is involved in the sample.

Base metal coinage can be more problematic due to the lower level of care in preparation and adherence to weight standards of base metal flans. Bronze typically had an intrinsic value of roughly one fortieth of silver. As a result,  care in weighing the metal of the base metal flan was not much of a consideration when it came to fiduciary coinage. As a result greater scatter and variance from the weight standard in base metal is usual. Multi-modal distributions are sometimes encountered reflecting different mint practices, or sometimes a reduction in weight standard during an emission.

Let the statistical mode(s) be your guide.

[SC]

I agree that with such information (ie no data on condition of the surveyed coins) your best option - in fact only honest option - is to show the graph.  That is especially important in cases where, like Lloyd notes, you could get multi-modal distiributions since resorting to the use of mean value only will in effect conceal the fact that there are multiple modes and render a the mean at a place where there is in reality no peak.

Base metal coinage can be more problematic due to the lower level of care in preparation and adherence to weight standards of base metal flans. Bronze typically had an intrinsic value of roughly one fortieth of silver. As a result,  care in weighing the metal of the base metal flan was not much of a consideration when it came to fiduciary coinage. As a result greater scatter and variance from the weight standard in base metal is usual.

Lloyd, what era are you talking about re the 40:1 AR:AE value ratio?  In the late Roman period I have seen ratios of around 100:1 and 120:1 and find you can use 100:1 as a good quick norm for the 4th century.

Shawn

[Lloyd Taylor]

Lloyd, what era are you talking about re the 40:1 AR:AE value ratio?  In the late Roman period I have seen ratios of around 100:1 and 120:1 and find you can use 100:1 as a good quick norm for the 4th century.

Shawn

Good point - the Bronze vs silver value varied dramatically through time, consistently decreasing with advances in mining and metallurgy.  The figure that I quoted pertains to the time around the advent of bronze fiduciary coinage in the the fifth century BC (i.e. the earliest Greek AE coinage).  The earliest bronzes of Syracuse etc honored this ratio and thus were huge lumps of metal by comparison to latter AE's which were truly fiat currency by the start of the fourth century BC.