Debra and I have a jar in which we put our spare change. When we fill it, about once every two years or so, we take the jar to the bank, where they pour the contents into their coin counter/sorter. We take the cash and splurge on something. It’s not a huge amount of money: usually around $200.

When the jar fills up, I like to dump it on the living room floor and count the money. Sifting through those coins brings back memories of my grandmother, who was a coin collector. I’m also reminded of watching the pinball machine service guys coming to Dad’s bowling center. And the brief period when I worked for my dad at the vending company, or emptying the coins from the vending machines we owned in our motels. Yeah, sifting through coins is a happy trip down memory lane for me.

While I was counting today, I got to thinking about the expected percentages of the different coins, and whether I could accurately estimate the amount of money in a change jar just by counting a single type of coin. So if I had, say, $100 in quarters, then how much money should I expect to be in the jar? How would I go about estimating that?

First, I have to make some assumptions.

- When I purchase something, I always get the optimum number of coins in change. That is, if I pay 75 cents for something, I’ll get a quarter back: never two dimes and a nickel, etc. This, in my experience, is a pretty safe assumption.
- I always pay for things in even dollar amounts. That is, if something costs $2.03, I give the cashier an even dollar amount and expect change. I never give, say, $5.03, and expect $3.00 back. I actually do that sometimes, but most often I don’t have any change in my pocket.
- I never take money out of the jar except for when I’m emptying it completely. That’s almost certainly true these days, because I don’t use the vending machines at work anymore. There’s no need for me to filch quarters.
- Each change amount is equally likely. That is, I’m just as likely to get 23 cents in change as I am to get 99 cents in change. This is probably a bad assumption. Cashiers often give me a quarter in change when the actual amount due is 24 cents. Rounding up one or two cents is common. Although this assumption doesn’t hold true, the percentage error it causes is likely small.

With those assumptions, how do I estimate how many coins of each type are in the jar?

The standard “counting up” method of making change is proven optimum. That is, if somebody pays $1.00 for an item that costs $0.33, then the change is two pennies, a nickel, a dime, and two quarters. If we write out the coins required to make change for every amount from $0.01 to $0.99, we end up with:

200 pennies (42.55% of coins)

40 nickels (8.51%)

80 dimes (17.02%)

150 quarters (31.91%)

$100 is 400 quarters. Given 400 quarters, I would expect the jar to hold (400/0.3191), or 1,253 coins, with this distribution:

533 pennies ($5.33)

106 nickels ($5.30)

213 dimes ($21.30)

400 quarters ($100.00)

(You’ll notice there’s a rounding error: the count of coins above is only 1,252.)

That gives me a total of $131.93. So I should be able to count the value of the quarters, add 30%, and have a pretty decent estimate of the amount of money in the jar.

I just happened to write down the contents of our change jar:

840 pennies ($8.40) (38.85% of coins)

330 nickels ($16.50) (15.26%)

402 dimes ($40.20) (18.59%)

590 quarters ($147.50) (27.29%)

That’s 2,162 coins, with a total value of $212.60. If I add 30% to the value of the quarters, I get $191.75, which is about 10% below the actual amount of money in the jar. The percentages of dimes and pennies are pretty close. The jar is light on quarters, and heavy on nickels.

A 10% error by counting only 27% of the coins, though, isn’t bad.

If you have a change jar and are willing to play along, I’d like to know your results. You don’t have to give me actual dollar amounts: just tell me the percentage error, and whether it’s high or low.

Something else I just noticed. The theoretically perfect jar has 1,252 coins, with a total value of $131.93. Our jar had 2,162 coins with a total value of $212.60. In both cases, if you divide the number of coins by 10, you come very close to the total value in the jar. $125.20 is only about 5% shy of the $131.93 actual total. And $216.20 is only 1.5% higher than the actual total in of $212.60 in our jar. Freaky.