#1




Estimating the value of a mass of US coins
So I was in my cousin's office this weekend and I saw a 5 gallon water bottle that had been emptied of water and filled with spare change. My first thought was that it must be terribly heavy, and when I tried to lift it I discovered I was correct  it's extremely heavy. I was able to tip it to get my hands under it, but I was unable to lift it. Perhaps with help from another person I could even carry it around.
The bottle is 18.93 liters, give or take, which is .667 cubic feet. Combined with this page I have what I consider to be a lower bound for the value of the bottle: $328, if it's full of pennies and not packed optimally. I could probably compute an upper bound as well, assuming it's made of the highest valuepervolume coin (I think that's going to be dimes, but I'm not sure). If I were able to obtain an accurate weight for the full jug (and a reasonably precise estimate for the mass of the empty jug), how closely could I estimate the value of all of the coins inside? I'd like to think that simple linear algebra would do the trick, but I can only think of three equations  one for total mass of the coins, one for volume, and one for total value  and I know I've got at least four unknowns (number of each type of coin). So: .25q + .10d + .05n + .01p = $(some large number) Mq*q + Md*d + Mn*n + Mp * p = (some large number of kilograms), where M(x) is the mass, in kilograms, of coin x. Vq*q + Vd * d + Vn * n + Vp * p = .667 cubic feet, where V(x) is the volume, in cubic feet, of coin x. M(x) and V(x) are almost certainly public information, but that still leaves me with three equations and five unknowns, even if I do know the mass to within a few grams. I think my only hope is to find an equation governing the frequency with which each coin is circulated relative to the others. Assuming that's possible, could one estimate the value of the coins inside without knowing the mass of the full jug at all? 
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#2




Since you know the volume, you could calculate the number of coins of each denomination that fit in that volume, given a less than optimal packing. You could then take the weighted average based on the frequency of each coin. It's not necessarily a great estimate, but as a first approximation, it should be OK.

#3




I don't think you could do it even with a known mass. You have 4 nonuniform parts (coins) with unknown proportions that don't fit together nicely.

#4




I've been wondering the same thing. I have a big pepsi bottle full of coins. It is about 3 feet high. I know that I put more than 100 dollars worth each year and have been doing so for many years. I'm guessing it is more than a thousand dollars worth. (I have a separate place for pennies, so it is all quarter, dimes and nickels.

#5




If you want to cash it in, interesting choice:
A) Spend an entire weekend rolling the coins, or B) Brave the 1015% vigorish that those coin machines typically take. 
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#8




I'd be willing to bet that the dollarsperunitvolume of the average american pocket change bucket is pretty constant, especially if you aren't dealing with someone who intentionally biases their coin collecting habits (e.g. someone who does coinop laundry habitually seperating out their quarters). Now the question is how to figure that out. It would probably be interesting to know the actual distribution of US coins, so that you could predict the value of a randomly chosen sample from that pool, but it's important to note that the contents of a pocket change bucket are not randomly chosen (resulting as they are from transactions where there are 99 core permutations that make the most sense to get), so the "value density" (warning: invented term) of the average bucket is likely different from that of the US coin circulation as a whole.

#9




A glass jar full of coins has another characteristic as well, that should be considered. It is under a very large tension from the "fluid" pressure of the coins. It can sit for decades, with no hint of problems, and then the drop of a single coin, or the torsion of trying to lift it will put some small flaw over the edge.
The failure is catastrophic, startling, and one hell of a mess to clean up. Or, it can be one of those silly twist of fate things. A good friend of mine had a glass carboy full of coins. Someone opened the window just above his desk, and a very cold wind blew in. (VPI, dead of winter, from steam heat to plateau wind chill) There was a sharp crack. He looked at the carboy, and there was a neat crack all the way around the very bottom. He made the mistake of checking it. Then the coins squirted out around the base faster and wider than you would ever imagine. Tris 
#10




This one's got me curious so I did a little exercise.
Assumption 1: change will be given from a particular transaction in the most logical way, e.g. 45 cents will be given as a quarter and two dimes, not as four dimes and a nickel. Obviously, this is not always true, but is probably true enough not to affect the results too much. Assumption 2: all subdollar totals are equally likely (you're just as likely to have a transaction ending in .23 as in .99). This one is probably wrong enough to impact the results, due to retail pricing practices, but I'll continue anyway, and maybe someone can come up with a way to correct this one and weight the results. If you got all 99 different change amounts once each, you would have: 200 pennies, 40 nickels, 60 dimes and 150 quarters. So given the assumptions above, the average change jar should be 44.4% pennies, 8.8% nickels, 13.3% dimes, and 33.3% quarters. Note that this is just the relative number of each coin, not the relative weight, so you need to know how many coins there are total in order to apply them. More calculations are necessary to say that a given jar filled this way is n% pennies by weight, etc. Hang tight... On preview: wow, Tris, I wish I could have seen that. 
#11




Ok, plugging those numbers and weights of coins from here into a spreadsheet, I get:
By weight, a bucket of pocket change (using my assumptions about collection from above) is 29.6% pennies, 11.9% nickels, 8.1% dimes and 50.42% quarters. So a pound (453.6g) of said change contains: 134.5 grams of pennies (= $0.53) 53.8 grams of nickels (= $0.53) 36.6 grams of dimes (= $1.61) 228.7 grams of quarters (= $10.09) ... giving this hypothetical change mixture a value of $12.77 per pound Ok, anyone see anything blatantly wrong with my calculations? 
#12




My change weighs 55 lbs  no pennies.

#13




If you are able to get an accurate total weight and empty container weight, perhaps the best way to estimate the total value is through sampling. Take out a couple to a few pounds of coins at random, count out the value of this sample and weigh it. From this you can obtain a $/lb value that you can extrapolate to the entire container full of coins. The only assumption required is that the composition of the coin mixture is fairly uniform.

#14




Well if you took a sample of my hypothetical standard change mixture and simply threw out all the pennies, it would be (by weight) 16.9% nickels, 11.5% dimes, and 71.7% quarters, putting its value at about $17.39 per pound.

#15




FWIW I keep my pennies, nickles and dimes* in a onequart glass milk bottle. I shake it from time to time to get the coins to settle. When it's full I usually have $45$55 in change.
*I save the quarters for laundry. When I'm home (with my washer and dryer) I just save them. 
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5 gallons * (4 quarts / 1 gallon) * (12.77 with quarters / 5.43 without quarters) * $50 per quartwithoutquarters = about $2,350! Also, for anyone interested in playing around with the numbers, I've put together a Google Spreadsheet here that includes mass estimates for full distributions of coins, and the "no pennies" and "no quarters" mixes. 
#19




For some strange reason I've been tallying the coins in my coffee can before taking them to the bank. It's how I deal with change.
2.5 years worth of coins. 10,843 coins worth $1204.36 (I excluded 253 $1 coins.) Average coin value is $.1111. Quarters at 33.9%. Dimes at 16.5%. Nickels at 12.2%. Pennies at 37.4%. I'm pretty good at emptying the pockets and rarely raid the can for prime coins. Hope this helps. 
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My prediction by coin count was: Quarters at 33.3% Dimes at 13.3% Nickels at 8.8% Pennies at 44.4% My percentages would have come up with a prediction of $1142.74 for 10,843 coins, which is off by 5%. I have to say, I feel like that's not too bad. 
#21




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For example, in my "no pennies" mix, 16.9% of the weight is from nickels, 11.5% of the weight is from dimes, and 71.7% of the weight is from quarters. It appears that you came up with 21.34%, 14.52%, and 90.77%, respectively, which adds up to more than 100%, so I think you made a mistake there. I might be overlooking something obvious, though. Our $/lb numbers agree for the regular mix of all coins ($12.77), but I got $17.39/lb for "no pennies" and $5.43/lb for "no quarters". 
#22




I uploaded my spreadsheet too so you can take a look. Mine's not as pretty.
http://spreadsheets.google.com/ccc?k...LvwFr0KgXDeSbA 
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From Jurph's spreadsheet.
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#26




Whenever I go to either spreadsheet, there's a sign in?

#27




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If we had more data points with known volumes, we'd probably be able to determine a useful (but imperfect) packing factor by which we could multiply a given volume to approximate the mass of normallydistributed coins therein. It would also be interesting to measure the volume of water that could be poured into a container "full" of coins. Assuming the water displaces all the air between the coins, you could get really good data on packing factor from that, too. Of course then you'd want to disperse and dry the coins ASAP to avoid corrosion and ending up with a jarshaped ingot of US currency... 
#28




When I come down to visit you folks, I'll surreptitiously throw a Canadian twodollar coin into your change jar. It'll completely destroy your math.
Bwahahahaha! Seriously, I wonder what the equivalent calculations are for other countries? Canada? Europe? Japan? In Canada, we have twodollar and onedollar coins in common usage, but they tend to be used often and quickly disappear from one's change pouch. Quarters disappear almost as quickly, and the remaining change accumulates pennies disproportionately. Realworld Canadian change jars are mostly pennies, I suspect.
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#29




This thread, right here, is why I love the SDMB!

#30




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One more possible complexity though. Haven't the weights of coins changed over the years. (particularly the quarter) So you'd have to find the distribution of "old" quarters vs "new" quarters, (and nickels too I think) ? Or is it only volume that matters in your calculations??? Just a thought, and now I go and leave it up to all you intelligent industrious folks 
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#32




I had worried about coins' ages affecting their mass, too, which was why I was tempted to go with volume first.
The two most relevant dates for that line of questioning are 1965 and 1982  by 1965, none of the coins we're considering* contained any silver; in 1982 the penny was changed from 3.1g to 2.5g. I suspect that the very small percentage of marginallyheavier "silver" coins is going to disappear in the noise of normal coin distribution; note that the theoretical distribution galt worked out and the observed distribution from Bergebo vary by more than 2% in mass. If we assume that all coins minted in the last fifty years are equally distributed in the pockets of America, then pre1965 "silver" coins would comprise ~20% of the samples, and the average mass of (for example) a dime would shift up from 2.27 grams to 2.32 grams  about a 2% weight gain. The penny is a more severe case; an even distribution over the last fifty years would yield an almost 50/50 distribution of heavier (3.1g) pennies, making the average penny weigh 2.8g, a 12% error. Quarters went from 6.25 grams down to 5.67 grams in 1965, so even distribution would yield about a 2% gain in mass. The mass of the nickel remained unchanged. The effects of such a sample would be interesting; the sheer volume of pre1982 pennies along with minor contributions from dimes and quarters would mean that a large sample would be almost 6% heavier than the same distribution of allmodern coins. Once you add in their relative values**, it turns out that my hypothetical 50year "perfect" sample of coins (galt's distribution) is worth about 5% less by mass: about $26.73 per kilogram. An evenlydistributed twentyyear sample would be much closer to expectations; "silver" coins would have constant mass and pennies would add less than 1% phantom mass (that is, a change of mass with no corresponding change in value). As a curiosity, a sample of coins that were all from before 1965 would be worth $24.64 per kilogram  a 12.5% drop in value (by mass)! I really doubt we'll see an even distribution out there; in fact, I suspect that coins have a "half life" and that as time goes on, fewer and fewer coins from each year are available. Mint production volume also fluctuates: in the last 3 years the number of pennies produced annually varied from 6.1 billion (2006) to 7.7 billion (2005) with 2004 coming in between with 6.8 billion. Other coin production varied similarly. I'm sure somebody out there knows how the coin years are distributed within the set of circulating coins, but if they do, they know it from sampling. If you're aiming for accuracy, and you know that the coin sample contains pre1982 coins, then you should probably shave your value estimate down by maybe one percent, at most. So now we know that the presence of older coins hurts our ability to estimate value by mass alone. Unless we can predict the packing factor for flat cylinders in mostlycylindrical containers, we can't estimate by volume alone except to determine an upper bound. But both estimates together, taken with an assumption of the distribution of change in circulation, can give solid upper and lower bounds for the value of a known mass/volume of coins. *  the circulated half dollar retained its silver content until 1970, but is almost never seen in circulation. **  and ignore the possibility that you've got an ultrarare coin worth $300 in your coin jar. 
#33




The metal in pre1965 silver coins is worth considerably more than their face value, isn't it? It's probably a reasonable assumption to make that they've been nearly entirely removed from the circulating currency. In fact, since they do have a different weight, the chance of them being removed goes up rapdily, since it's easy to mechanically remove them from mixed change.
Based on a brief sampling of websites with relatively absurd investment advice* provided by Google, it seems my assumption above is true. There are likely few to no pre1965 90% silver coins in circulation. *This is not meant as a comment on the validity of buying precious metals as an investment, only as a comment on the logical fallacies displayed at sites like this. 
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us treasury coin bags
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#37




One possibly simplifying fact: If I recall correctly, pennies have the same "value density" as nickels, and dimes have the same as quarters. So we could deal with only two types of coin, "base" coins and "silver" coins.
One possibly complicating fact: Optimum packing density is, in principle, higher for objects of a mix of sizes than for objects of uniform size (the smaller ones can fit into gaps between the bigger ones). So it's conceivable that a container of mixed coins could have a higher value than the bounds calculated. I doubt this would be significant, though, since the packing probably isn't too close to optimal anyway.
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I can't believe I didn't think of this earlier, but Coinstar.com has a utility that will estimate how much change is in your change container. They note that the "actual value may vary based on container and coin mix," and estimate that one gallon of change is worth $228.32. Of course their estimates are done by volume, but they do take packing factors into account. 
#39




Our coins have just changed and the 5cent one is going to become obsolete. Imagine if you had $1000 of five cent coins and you don't take them to the bank in time!

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What?!?!?
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#42




Theft of quarters vs pigguy bank
Recently my husband and I were victims of a theft. The person stole 15 rolls of quarters=$150. We know the individual who stole from us. He said that he was at the coin counting machine turning in his daughters piggy bank, not our quarters...he is obviously full of poop. He is on video unwrapping coins for 6.30 minutes!Number 1: It does not take 6.30 minutes to empty a piggy bank in the machine. Number 2: I was wondering if anyone thought it possible for him to receive $130 from a child's medium piggy bank dimensions 71/2" W x 8" H. I truly want to believe this kid! He was a good worker with a family he needed to support. Please help me by letting me know if #2 is a possibility! Thanks!

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This is a perfectly reasonable question to ask, but please don't ask it at the end of someone else's fouryearold thread. Feel free to start a new thread. I'm closing this one. [/moderating]
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