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mickeyman
Penny Pincher Member
Canada
243 Posts |
 Posted - 05/19/2008 : 14:18:12
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I am writing this to try to document the means of using the observations I have been making to calculate changes in the circulating coinage. Part of this is so that others can obtain their own results, and the other reason is so that my reasoning can be inspected as there is one step the validity of which I am not completely sure.
The main difficulty with this type of project is sampling. To get a true estimate of circulating coinage, samples should be random. The problem is that I am not collecting nickels to learn how rapidly .999 Ni coins are disappearing from circulation, I am trying to obtain as many of those nickels as I can. A good sample might be ten or so rolls from each bank, but once I have a good bank, I tend to hit it for more.
A second difficulty is the presence of rolls of 2008s directly from the Mint, along with rolls that have been sorted either by hoarders or by companies such as Coinstar (acting for the Mint?). They are clearly part of the landscape and should be countedbut in a small sample it is easy for them to be either over-represented or under-represented.
The basic idea is that you start off knowing the previous percentages of .999 Ni in circulation, as well as CuNi. If I had completed my last survey at the beginning of 2008 instead of in late 2007, I could have ignored completely the 2007s and concentrated only on 2008 coinshowever, my last sampling ran from late September to early November, and presumably more 2007s entered circulation afterwards.
My last sampling suggested that the fraction of .999 Ni in circulation in late 2007 was .212 (21.2%), for CuNi the fraction was 0.529, and for 2007s the fraction was 0.0496. Since then, an unknown fraction of .999 Ni have been removed (z1), an unknown fraction of CuNi have been removed (z2), an unknown fraction of 2007s have been added (z3) and an unknown fraction of 2008s have been added (z4). How do we use our observations of the current population of nickels to find these variables?
Simplistically, we might say that if we find that 2.5% of nickels are 2008s, then we can say that the number of 2008s added, expressed as a fraction, was 0.025. But this doesnt quite work out as we can see in the following example. Suppose we had 1000 nickels in circulation, and the Mint creates 25 2008s. Now there are 25 2008s in circulation and 1025 nickels in circulation, and the observed proportion is 25/1025 = 0.024. Although this is a small difference, the picture becomes worse if we consider that some pure Ni nickels have been removed as well. Suppose that 35 .999 Ni nickels and 100 CuNi were removed at the same time as the 2008s were added. Now there are 25 2008s in circulation and 890 nickels in circulation, so that observed percentage of 2008s is now 25/890 = 0.028.
Each of the above variables is influenced by all of the others. But all is not lost. It is possible to construct a series of equations relating the four sought variables in terms of our observations.
The proportion of pure .999 Ni in circulation should be:
A = (0.212 z1)/(1 z1 z2 + z3 + z4). The numerator represents the difference between our starting proportion of pure nickels and the unknown number removed. The denominator represents the number of nickels in circulation expressed as a proportion of the total circulation in late 2007. It may be either higher or lower than 1 depending on whether the flood of new nickels released has been higher than the number of nickels removed from circulation. A is our observed proportion of .999 Ni in current sampling. Most postings on this site suggest that A is somewhere between 0.15 and 0.20. My numbers have been around 0.18, but they may be skewed by my preference to buy nickels from banks where the percentages are high.
This equation can be reorganized to:
(1 A) z1 A z2 + A z3 + A z4 = 0.212 A.
Remember that A is an observed number.
Similarly, the proportion of observed CuNi in the current population (B) is:
B = (0.529 z2)/(1 z1 z2 + z3 + z4)
Which can be rearranged to:
- B z1 + (1 B) z2 + B z3 + B z4 = 0.529 B
The proportion of observed 2007s (C) will be:
C = (0.0496 + z3)/(1 z1 z2 + z3 + z4)
And this can be rearranged to:
- C z1 C z2 + (C 1) z3 + C z4 = 0.0496 C
or
C z1 + C z2 + (1 C) z3 C z4 = C 0.0496
The proportion of observed 2008s (D) will be:
D = z4/(1 z1 z2 + z3 + z4)
which can be rearranged to:
D z1 + D z2 D z3 + (1 D) z4 = D
We have a system of four equations and four unknowns
(1 A) z1 A z2 + A z3 + A z4 = 0.212 A.
- B z1 + (1 B) z2 + B z3 + B z4 = 0.529 B
C z1 + C z2 + (1 C) z3 C z4 = C 0.0496
D z1 + D z2 D z3 + (1 D) z4 = D
There will be a unique solution for z1, z2, z3, and z4. The most straightforward way to solve this is to use the method of matrix determinantsthe reason I chose this is because you can have excel do all the work for you.
For a description of other methods to solve this, see You must be logged in to see this link.
The coefficients of the variables in the above four equations give us our first matrix. In comma delimited form, it will be:
1 A, - A, A, A
- B, 1 B, B, B
C, C, 1 C, -C
D, D, - D, 1 D
Remember that A, B, C, and D are all known to youthese are your observed numbers. You can find the determinant of the above matrix using the MDETERM function in excel. If you have the above numbers occupying a 4 x 4 cell square, the MDETERM function will calculate the determinant. Let us call this determinant [I]. There are five determinants we have to find.
The next matrix is found by replacing the first column of the above matrix by the arguments to the right of the equal sign in our system of four equations. In comma delimited form, this matrix will be:
0.212 A, - A, A, A
0.529 B, 1 B, B, B
C 0.0496, C, 1 C, - C
D, D, - D, 1 D
Using MDETERM, we can find the determinant of the above matrix. Let us call this determinant [I1]. Our unknown variable z1 = [I1]/[I].
The next matrix is found by replacing the second column in the first matrix by the arguments to the right of the equal signs in our system of four equations. In comma-delimited form, this matrix is:
1 A, 0.212 A, A, A
- B, 0.529 B, B, B
C, C 0.0496, 1 C, - C
D, D, -D, 1 D
If the determinant of this matrix is [I2], then our variable z2 = [I2]/[I].
Similarly, replace the third column of the first matrix by the above arguments to find [I3]. Then z3 = [I3]/[I].
Similarly, replace the fourth column of the first matrix by the above arguments to find [I4]. Then z4 = [I4]/[I].
The beauty of this method is that you can set the whole thing up in excel, so that these calculations are done automatically as you input numbers.
Of course this assumes that excel calculates the determinant properly. One day I will write my story about why there may be some doubts about that.
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Not all who wander are lost. |
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fb101
Administrator
USA
2856 Posts |
 Posted - 05/19/2008 : 14:22:36
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Now that I understand all that,
I realize I could have hand sorted a box in the time it took to read it.
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HoardCopperByTheTon
Administrator
USA
6807 Posts |
Posted - 05/19/2008 : 14:45:29
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I would assume most of the hoarders reporting on this board would have the same preference for obtaining raw stock with the highest percentage of "real" nickels as you do. So the reported percentage by them, and your 18% are going to be slightly skewed compared to overall nickel population. Over time, all these pockets will be mined out causing a smoothing effect and your percentages will more accurately reflect the true population.
Understanding what is happening and being able to do something about it are two different things. If you had several data points you might be able to do a linear regretian analysis of the decline of the nickel percentage in circulation. Without doing all the math I think we could postulate that the slope of that curve would flatten out over time. No matter how much analysis you do, the conclusion is the same. Sort as much as you can, as fast as you can to achieve the best yields.  |
If your percentages are low.. just sort more. If your percentages are high.. just sort more.
Now selling Copper pennies. 1.6x plus shipping. Limited amounts available. |
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jadedragon
Administrator
Canada
3788 Posts |
Posted - 05/19/2008 : 23:20:22
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I pretty much understand the math, but I am not tracking my own percentages. After sorting a few dozen boxes out, and thinking about the percentages, I think that there is a huge amount of variation between boxes and rolls.
Sometimes I get lots of 2000+, other times plenty of pre-82.
The distribution of US coin is pretty high here in the Fraser Valley, but then we are adjoining the border and par has been a reality for the US/CDN $'s for some months. Pulling US coins from the wild in Canada for fun and profit just is not profitable like it used to be.
Given the huge number of coins out there, I doubt we can make much of a dent in the stock, but I'm trying in my town. Before the long weekend I bought pretty much every nickel roll in a bank on the south side of town. That should cause an influx of coin to be brought in for me to buy. |
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man. George Bernard Shaw.
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Calculating changes in circulation (long)
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