# Morgan Dollar Relative Rarities Based on P.C.G.S. Data Statistics

9^{th} National Silver Dollar Convention

St. Louis, Missouri

November 10 – 13, 1988

**Morgan Dollar Relative Rarities Based on **

**P.C.G.S. Data Statistics**

**By George E. Bodway, Ph.D.**

As discussed frequently over the last year, the Professional Coin Grading Service (P.C.G.S.) data is going to be very useful for understanding more precisely the rarity of different numismatic coins by date and condition. What follows are just a few useful ways of sorting the data for Morgan dollars.

**Rarity by Date and Mintmark**

The qualities of all grades for both uncirculated non-prooflike (NPL) and uncirculated prooflike (PL) Morgan dollars have been summed and then organized from smallest to the largest quantity. The only grading that comes into play in this particular sorting of the data is deciding between uncirculated as well as NPL and PL. The total numbers represent all NPL and PL Morgans submitted to P.C.G.S. less: 1) those returned for quality control reasons, and 2) those determined to be circulated.

The first column in Table 1A is a listing of all NPL Morgan dollars in ascending order while the second column shows the actual quantities, both from the 4/1/87 P.C.G.S. data. Figure 1B lists the same information for PL Morgans and IC for the sum of NPLs plus PLs.

Another step that would be of interest is to list the data in order by rarity by date for each grade – in other words, repeat Table 1A for each grade. I did not do this because there are at least four qualifications that need to be discussed as part of the data. One of them, I believe, would seriously limit the usefulness at this time. I did sum up the data for the total NPL and PL in grades 64 and better, but did not include the data because of the first Qualification 1 below (i.e., a shortage of enough quantities of the tougher dates to be representatives).

**Qualification 1 – **The results assume there is a large enough sample to be statistically significant in representing the entire population. I do not believe there are enough dollars graded yet (i.e., 73,000 of approximately 13,000,000*) to be significant by grade. Although the quantities are marginal, the relative rarity may already be quite useful for the *total by date.* This concern will slowly disappear over time as more and more material is graded by P.C.G.S.

**Qualification 2 – **The results are only accurate to the extent that everyone agrees with the P.C.G.S. grading standards and their interpretations of what is a prooflike. There are concerns raised in this area periodically. Grading is subjective to some extent and contains an element of personal artistic appeal so that there will always be some concerns or disagreement. Only time will tell whether they are serious enough to invalidate the relative rarity implications.

**Qualification 3 – ** This concern can be summarized by discussing how accurately the coins submitted to P.C.G.S. represent the “real” surviving distribution. The most obvious deviation is that, understandably, very few low grade coins are intentionally submitted. In fact, most people probably submit what they think are MS-65+ coins hoping to get MS-66 or better and in reality they come back centered at MS-63 or MS-64 with some higher than MS-65 and some lower than MS-63. This means that the data is heavily skewed toward the high end of the grading scale for any date (see Figure 1). Another possible perturbation is that a lot of great coins may be locked in sets put away years ago and it may be hard to get a grading sampling of these into the data base for a long time (although P.C.G.S. is working on this).

**Qualification 4 – **There are numerous coins that are broken out of the holder and resubmitted to P.C.G.S. So, for example, when it lists, in the future, three MS-64 1897-Os, one or even two of this count may be for grading the same coin over again. This obviously distorts the data. One would guess that this error will show up most often in the MS-64 grade and is likely to get worse with time rather than better.

In summary then, it is not obvious that the problems associated with Figure 1 will be correct any time soon and must be taken into account in certain kinds of comparisons. This should not significantly affect relative rarities within a given grade. It will have some effect on comparisons of the sum across grades, i.e., the data presented in Table 1A, 1B, and 1C, because the actual quantities by date are dominated by the low grades and the relative rarity may be different than the higher grades rarity which is only weakly represented. Relative rarities below MS-64 are of course most impacted by this factor. With these four auctions then the Tables 1A, 1B and 1C are presented for displaying the relative rarities of the Morgan dollars based on the April 1, 1987 P.C.G.S. data.

It will be interesting to compare this data, particularly as the data base builds up, to previous rarity projections by Les Fox, Wayne Miller, Alan Hager and David Hall, which was based on more subjective but great in-depth experience. Over time it is also going to be enlightening to sort the data in different way, as the data base becomes more statistically significant.

**Relative Rarity by Grade**

Another subject I would like to discuss is the relative rarity and possible future availability of high grade Morgan dollars. Table 2 is a presentation of this data with the following assumptions:

- That the P.C.G.S. data for MS-65, MS-66, MS-67, and MS-68 is fairly accurate in representing the true distribution.
- There are about 13 M total Morgan dollars still in existence (2).
- The real quantities of MS-60 through MS-64 exceed the quantities of MS-65 by 20:1
- The qualifications listed previously still apply and may affect accuracy.
- By taking the ratios of MS-66 to MS-65 of 10:1, MS-67 to MS-66 of 17:1, MS-68 to MS-67 of 22:1 all from the P.C.G.S. data and assuming MS-68 to MS-69 and MS-70 to MS-69 of 30:1 and 50:1 respectively

**NOTE** (The MS-69 and MS-70 ratios are based on an assumption on my part, the MS-65, MS-66, MS-67, MS-68 data is as accurate as the P.C.G.S. data truly represents the total relative distribution, between these grades and MS-65s. The ratio of MS-65 to the sum of MS-60, 61, 62, 63 and 64 is an attempt to correct for the phenomenon indicated by Figure 1. The P.C.G.S. ratio is currently about 7:1 while Les Fox’s* data is about 11:1. I believe the grading standards are tougher today, including P.C.G.S. standards, than Les assumed when compiling his estimates and, therefore, put forward my estimate of 20:1. This means that MS-65 and better dollars comprise about 5% of the total Morgan dollar population. The total dollars still in existence of 13 M was taken directly from Les’ book*.)

Using this information, you can then end up with Table 2 as the most likely total availability of Morgan dollars by grade, with 40% of them being in the four common early S-Mints and 60% in the 10 most common dates.

It is now possible to use Table 2 to answer a lot of questions about the potential rarity of individual dates and mintmarks on a statistical basis. I will show some examples of how this can be done. If you assume that the grade of a Morgan dollar is strictly a statistical phenomenon, i.e., all dollars have an equal known probability of getting damaged in a bag of 1000 as it is being jostled around, then one can calculate that the *most probable *number of, say, MS-67 dollars likely to show up of a given date and mintmark. The following formula was derived from the P.C.G.S. data, Les Fox’s* information, and my previous assumptions.

** **

**Formula 1: **MSXX (quantity) = MS-65 (quantity) x 56 x percentage Table 2. Some examples to follow:

**Example 1. – **How many MS-68 1881-Ss will eventually show up if P.C.G.S. graded all 13 million Morgan dollars?

No. 1881-S (quantities MS-68s) = 2,942 x 56 x .0003 = 49

(Note this is almost 1/3 of the total expected supply of MS-68 dollars).

**Example 2. – **How many MS-67 1904-Ss would we expect to statistically find in the eventual distribution?

No. 1904-S (quantities of MS-67s) = 3 x 56 x .006 =

**Example 3. – **How many MS-68 1896-Ss (quantities of MS-68s) = 1 x 56 x .0003 = .02

In other words, the probability of finding a single MS-68 1896-S, if one ever went through all 13 million dollars one by one, would be less than 1 chance in fifty or virtually nonexistent. For example, you would have to have 50 piles of 13 million dollars in order to find a *single* 1896-S in MS-68 condition in *one *of the piles. Such a dollar would be extraordinarily rare, defying the statistics which say it should not exist.

**Example 4. – **How many MS-67 1884-Ss will eventually be recognized? Since no MS-65s have been graded by P.C.G.S., what I will do is take the sum form MS-60 through MS-64, divide by 20, and then multiply by 56 and then .006 for MS-67s

No. 1884-S (quantities MS-67s) = 6/20 x 56 x .006 = .1

The changes of finding a single 1884-S in MS-67 are less than one in ten. A higher probability than the 1896-S in MS-68 but still another extraordinary rare dollar. IN other words, in this case you would have to have 10 piles of 13 million dollars in order to find a *single *1884-S in MS-67 condition in *one *of the piles.

**Example 5. – **This is the last of these examples I will use so you can see how the information and Formula 1 are used. How many 1885s will eventually be uncovered in MS-68 condition?

1885 (MS-68 quantities) = 291 x 56 x .0003 = 5

One has already presumably been found, and this says that statistically 4 more will some day be located.

Obviously, it is a lot of fun to ask such “what if” questions, and this probably gives a pretty reasonable first pass insight. Again, the validity should improve substantially as the P.C.G.S. data base builds up. It will never be perfectly accurate because we are dealing with small numbers calculated from large ones, and the grades are affected by known and unknown *non-random* factors as well, such as striking quality at various mints in given years, quality of planchets which affect the luster, and differences in the handling of the Morgan dollar bags.

I would like to state again that these relative rarities only represent the distribution in coins *submitted* to P.C.G.S. and will accurately represent the “real” distribution with time as the data base builds up.

**Prooflike Rarity**

One other item I would like to cover is the ratio of prooflikes to P.C.G.S. in the distribution. The P.C.G.S. data shows a ratio of PL’s to total PL’s + NPL’s of 8.3%. This compares to a ratio of PL’s to NPL’s + PL’s in Les Fox’s book* of 3.4%. This is a difference of almost 3:1 and may be explained by:

- The qualifications listed distort the data and explain the difference.
- People submit more prooflikes to P.C.G.S. than are in the “real” distribution.
- Les Fox underestimates the available number of prooflike by this factor.

Unfortunately, the P.C.G.S. does not distinguish semiproof coins and lumps them in the NPL dollars. Removing these from Les Fox’s number would lower his percentage by about 2/3 to much less than 2%, widening even further the difference between his estimates and the P.C.G.S.

Another unfortunate practice of the P.C.G.S. although it doesn’t affect the above data, is that they do not recognize *cameo* prooflikes in their own right. These are rare and beautiful dollars and we are not going to get any statistics on their relative rarity with the present practice of the P.C.G.S. Also note in Table 1B the date with the rarest percentage of prooflikes, excluding the first 12 nos., is the 21-D with 0.4% followed closely by the 03-P, 21-S, 21-P, and 99-O at about 1%.