Approximating the relative quantities of coins in Canada 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: 


*

*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?

*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?

 A: 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!
A: 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. 
