Any collector can tell you that the most often asked question is typically a variant of "what's the most expensive?" and "what's the rarest?" Answering the first is usually pretty easy because the results are widely published. In the realm of numismatics, at the moment it appears the most ever paid for any type of coin was a little over ten million dollars for a U.S. Dollar dating to 1794 sold by auctioneer Stack's. Auction house Baldwin's holds the record for Greek coins which in 2011 sold a Pantikapaion gold stater for $3.8 million while the current titleholder for Roman coins is Numismatica Ars Classica's 59th sale, also in 2011, of a heavy gold medallion of Maxentius realizing a hammer price of $1.4 million. Answering the second question though is considerably more difficult. The main problem is that with ancient coins we're missing a key part of the variable when we have no idea how many of any them were minted in the first place. We can thus never know an ancient coin's absolute rarity. However, using inferential statistics we can develop a sophisticated model that can predict the chances of finding a coin for sale at auction for any Roman emperor (or empress) . This is not the easiest of tasks because there are problems when looking at only one type of source. For example, go to a flea market, whether here in the states or in Europe, and it won't be hard to find Roman coins for sale. But your chances of finding a truly rare and valuable one whose importance has somehow escaped the notice of its owner is more or less equal with that of finding an overlooked Rembrandt right next to it. On the other end of the scale, the prestigious auction houses that sell the most expensive coins are going to skew the results in the other direction for they have no interest in selling ten buck Constantinian bronzes. Therefore, the only way to make the model accurate is to combine the results of the various markets into one big database. Having "the big picture" then gives us a snapshot of the marketplace for any given period that it records. I can say without hesitation that the biggest project I've ever undertaken is putting together such a database. It is not merely sufficient to gather the records for you also have to constantly classify and re-classify them ten different ways, prune out the bad stuff, plenty of old-fashioned manual data entry to get obscure datasets and on and on a tedious list of chores to arrive at a good distribution histogram capable of being tested against.
Now that that part is ready for action all we need is a working system. As noted in an earlier post, rarity as defined by market availability allows the collector to determine an at-a-glance estimate of the difficulty of locating any coin published with that metric. It attempts to shape an approximation of absolute rarity by correlation to polling instances of sales within a time bracket. As an arbitrary formula it makes no claim of passing scientific rigor but is nevertheless a viable system of reference standards using a simple, if informal, logarithm. But given the database work accomplished we now possess the added benefit of having a normally distributed dataset. In addition to estimating true rarity, it also allows us the opportunity to perform comparative analyses with a high degree of accuracy. Without getting bogged down in the abstruse language of statistics (and I barely qualify as more than a layman on the subject myself), the upshot is that when you have a sufficient number of records that can be said to be representative of a whole you can then use mathematical formulas to express probabilites; that is, the chances of finding any one of these records occurring again in a given amount of time. We gain a scientific "key" in the form of standard deviation to tell us what an average is and, so, a means by which to say how much more or less difficult it will be for the collector to get their hands on a very rare ruler.
For the purposes of this exercise, I tallied all the auction records from 1904 through 2014 belonging to the reigns from Augustus through Anastasius I in keeping with the most popular definition of what the coinage of the Roman Imperial covers. Over 800,000 auctions from literally hundreds of different sellers were returned from this query. This number isn't every Roman coin ever sold at auction, of course, but is easily the greater portion and, more importantly, is broadly representative of all types of coins, grades and prices available. Anyway, from this set were stripped out certain categories that we didn't want to measure including barbarous coins, error coins, coins of uncertain identification and so on to come to a "refined" set of 667,076. Of those auctions we now look at them categorized by ruler to get a total of 201, plus one grouping for anonymous, for a total of 202 imperial members whose coins were sold at any point during those 110 years (worth noting that 99% of those took place since 1990) from a total possible roster of 206 possibilities.
Next, we rank these rulers from most to least number of records. Constantine I takes the cake as most common with 37,428, or 5.6% of the total, down to the famously rare usurpers Saturninus and Proculus who are both recorded with but a single sale each representing an infinitesimally small share of just 0.000149908%. Everyone else falls somewhere in between with a mean value of 3,254; what one could call the average as far as Roman rulers go in terms of rarity and which most closely corresponds to the coins of Trebonianus Gallus if one were to plot this out on a graph. But we can't be content with merely dividing everything into just two groups (those being more and those being less rare than T. Gallus). As noted above, we have the benefit of accepted markers in the standard deviation nomenclature, abbreviated with the symbol σ, and which holds that in any given dataset that is normally distributed - meaning truly representative of the whole - about 68% of the records will be within one standard of deviation from the average. In other words, a little over two thirds of the coins sold were about as common as those of Trebonianus Gallus. A value of two standards of deviation account for 95% of the total and by the time we get to three standards of deviation it should encompass 99.7% of all auctions ever held.This way we can say that if we are lucky enough to own a σ3 coin, say a Hannibalianus or Quietus, you can say with confidence that it's offered at auction only about once in every 300 lots offered (or 3 out of 1,000=0.3%) and as such it is categorically rarer than any of the class σ1 and σ2 rulers by n standard of deviations. As the SDs go higher the rarity often tends to be exponentially larger. For example, that relatively rare Hanniballianus is still over ten times more common than a σ3.5 Clodius Macer which in turn is ten times more common than an ultra-rare σ4.5 Olybrius!
The full spreadsheet is available here and you are welcome to use and redistribute it. The Rank column assigns a numerical value from 0 to 9 while Rarity does same but with the names commonly used in numismatics. These are Most Common (MC), Very Common (VC), Common (C), Less Common (LC), Scarce (S), Very Scarce (VS), Rare (R), Very Rare (VR), Extremely Rare (XR) and NA for the four rulers who are known to have coins minted in their name but which have never been offered for sale at auction. Both of these rankings are tied to σ0.5 intervals according to the mean deviation number class under which they fall. Enjoy!
Edit 12/17/2014: The findings have been met with some controversy. In particular, I'd like to stress that the results were a subset of data taken from coinvac.com which as of today counts a total of 1,447,137 records. Out of this total 800,720 have been identified as belonging to the Roman Imperial era. From this number were subtracted those records with an unknown sale price, lots that had more than one coin, records that are known to be Roman but not identified to a specific authority, fakes, barbarous coinage, coins with major damage, brockages and those with a sale price under $5. This left a total of 667,076 which was the number used for the results. Approximately 70% of the records came from eBay and the other 30% from printed and electronic auctions. The study did not take into consideration Roman provincial coinage.