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Would it be possible to approximate accurately the relative quantities of Loonies, Twoonies, quarters, dimes, nickles (and perhaps the discontinued penny) in circulation from simply obtaining a large enough sample of coins through everyday use? By everyday use I refer to the coins you get back in change when you make a purchase in a grocery store for example.

I suppose this is a 2 part question:

  1. Is the method of sampling sufficient, or is there some kind of bias introduced because you are collecting samples through a deterministic process (of collecting change?) What size of samples would you need?
  2. If the sampling is sufficient for an accurate approximation, can you use it to determine the relative quantities of each coin type in circulation? Or, for example, is it that the sample size necessary to accurately approximate the relative quantities would itself change the relative quantities of each coin type in circulation?
COOLSerdash
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gmatt
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    The population of 'coins in circulation' and 'coins that appear in my change' would need to be the same, at the right relative frequencies. Why would this be the case? – Glen_b Oct 19 '13 at 22:24
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    Related: [Canadian mint circulation currency](http://www.mint.ca/store/mint/learn/circulation-currency-1100028). You can click through to get the quantity minted for each coin by year. – cardinal Oct 19 '13 at 22:40
  • @Glen_b: I don't know why this would or would not be the case that is why I asked! – gmatt Jun 30 '14 at 19:03
  • A substantive issue to settle at the outset concerns the population of coins "in circulation." Is a coin in circulation if it's lying around somewhere and has not been destroyed? Or must it be exchanged periodically between people? If so, how often? Since this population is continually changing as new coins are minted and introduced into circulation and old ones are lost, worn out, destroyed, or hoarded by collectors, how should those processes be accommodated? – whuber Jun 30 '14 at 21:06
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    As far as the possible/impossible issue goes, you might want to settle for practicability instead. It's certainly possible to count the coins in active circulation during a short period of time: perform a [mark and recapture experiment](http://en.wikipedia.org/wiki/Mark_and_recapture). The number of coins to mark would have to be enormous, though, making such an approach theoretically *possible* but not *practicable*. – whuber Jun 30 '14 at 21:09
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    It is possible, indeed likely, that some types of coins circulate faster than others (eg because they are more convenient amounts for typical transactions). If your sample is based on a process such as receipt of change, it may therefore be biased because it over-represents those types of coins that circulate faster. It would be better to take a sample of people and times and infer the relative quantities from the stock of coins that those people held at those times. If possible, although it may be impracticable, coins held by businesses should also be considered. – Adam Bailey Jul 01 '14 at 09:35

2 Answers2

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I would have to say that it would be extremely difficult for you to estimate the relative quantities of coins in circulation through any but an exhaustive (collecting a large portion of those coins simultaneously) survey.

The reason is because most businesses (I believe) hold a reasonably large portion of coins in stock and will only distribute the coins which most efficiently lead to correct change. Thus even if you go into the same store 100 times and collect change each time unless you have exhausted the stock of available coins, the coins that you receive in exchange for your sampling will only be those which correspond only with the least change required to fulfill your needs.

Assuming you draw change requirements uniformly between 1 cent and 499 cents this ratio is:

       200        100         25         10          5          1 
0.13559322 0.06779661 0.25423729 0.13559322 0.06779661 0.33898305

If the store has no shortage of coins then your sampling procedure will automatically return the above ratios which have no correlation between the specific samples and the greater population of coins in circulation. To see how I came up with these numbers see my blog post on the topic.

But this does not account for the oddities of prices which tend to cluster ending in .09 as in .99, .49, or .39 (in the US at least) which will definitely contribute to higher ratio of pennies required for many purchases than in the uniform draw of change. Purchase requirements would need be specified so as to not cause further contamination of the data. Overall, I think it is clear that this is a pretty problematic study design.

If you were forced to do something like this then you might be alt to 1. record change totals for each purchase, 2. calculating efficient coinage selection via the method I propose on my blog for each purchase, 3. record coins actually returned, 4. estimate the different between the optimal returned coin quantities and that actually returned to estimate to what degree coin stocks might be diverging from the optimal quantities. From there I am not sure what to do with it in order to estimate total coins available in the currency.

Good luck and thanks for the interesting question!

Francis Smart
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  • Thank you for the thought experiment/simulation, it was instructive to see that there would be a correlation/sampling bias. – gmatt Jul 07 '14 at 16:50
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The bigger problem is going to be part 1, not part 2.

It will be relatively easy to get a big sample of coins. But how do you know those coins are a random sample? Maybe people where you live use more of a particular coin than people in other parts of Canada. You certainly use money in a way that is not the same as everyone else.

For example, some people will pay for nearly everything with credit or debit cards; some will make even large purchases with cash. If you only buy cheap stuff with cash, you are going to get smaller coins. If you tend to have a lot of small bills and coins in your wallet, you will get smaller coins.

Probably not possible to get a truly random sample, but I'd try to get samples from different people in different parts of the country (rural/urban; west, center, Atlantic, etc.) and different ages, incomes etc.

Peter Flom
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