It’s been about a month since my last post–sorry for the radio silence. I recently started as Director of Community for FirstMark Capital, so I’ll probably be writing a little less frequently now.

The inspiration for this newest analysis comes from a good friend of mine, who (for reasons unbeknownst to me) shared a picture of his Coinstar receipt. He turned 27 pounds of change into $256.14. When looking at his receipt, I was surprised by the distribution of different coins. What, this made me wonder, should one expect to make from 27 pounds of coins?

In order to figure this out, we have to make a few assumptions. Let’s start with an easy one–that change is always made with the fewest possible coins. Fortunately, the algorithm for doing so is simple; it’s the greedy one.

Next, we have to figure out how much change is made in an average retail transaction. The answer to this second question is not as obvious. We could perhaps survey the price of hundreds of different consumer goods, but that seems like overkill. So, here’s how we’re going to do it for the purposes of this analysis: we are going to treat each digit of the price differently.

I’m also going to keep in mind my own cash vs. plastic habits. I rarely purchase anything over $20 with cash, save the tab at an occasional cash-only restaurant. And, even in this case, I just include the change in the tip. So, everything in my piggy back will come from smaller transactions.

To start, let’s assume that the last digit of the price is always a 9, as in $1.99 or $1.49 (JC Penney’s new policy notwithstanding). That was easy.

For the second-to-last digit of the price (the “dime” slot, if you will), a 9 should once again be the most common digit, whereas a 0 should be the least-common digit. This sounds like a reverse of Benford’s law, which states that “… in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way.” (You should click that link–it’s seriously interesting. HT to Radiolab.) Yet, a 4 in the dime slot seems almost as likely as a 9, whereas a 6 seems almost as unlikely as a 1. So, here’s what we’ll do: use the same logarithmic distribution for probabilities, but make 9 the most likely digit, 4 the second, 8 the third, 3 the fourth, and so on. No guarantee this is right, but let’s go with it:

Finally, we need to consider the whole dollar amount, because we’ll need to apply a sales tax of 8.875% for New York City and, in the case of my friend, a 6.25% sales tax for Boston. My half-scientific method of choosing this distribution is to look at a gallery of distributions found here and pick the one that looks about right. Why, it’s the Gamma distribution of course. We’ll set alpha to 2 and beta to 1 so that the function maxes out at about $20. Here’s the distribution in action:

Next, this is what 200,000 randomly-drawn prices look like:

For my habits, at least, this looks correct. Now, to finish out the analysis, let’s tie this back to weight. Fortunately, the U.S. Mint standardizes and publishes the weight of each coin here. With that in hand… drumroll please… we’d expect about 34.9 quarters, 19.8 dimes, 11.5 nickels, and 61.2 pennies in a New York pound of coins, for a total value of $12.00. A Boston pound is worth slightly less–$11.81.

So, do these results match up with our single Boston data point? As a gut-check, I also ran the numbers with a completely uniform distribution of prices from $0 to $100:

Clearly, there’s not a match here. I suspect there are a few good reasons for it. I know, for example, that a lot of my friend’s quarters go to his laundry. In addition, it’s possible that quarters, dimes, and nickels are spent before making it to Coinstar–either shortly after they’re received, or in a hotly-contested game of 5/10/25 cent poker. In any case, we’ve demonstrated that he is–quantitatively speaking–different.

Let’s close out with one last question: accepting all of the above, how does the expected value of a pound of change vary depending on sales tax? Here’s the answer for 200,000 randomly-generated transactions, with tax rates ranging between 5% and 10% in increments of 0.1%: