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December 15, 2011 6:06 AM

Is there a way for me to estimate the total value of a certain weight of random mixed us coins?

I have a water cooler bottle that I have used to collect my loose change for about the last 10 years. It is about 3/5s - 2/3s full and weighs, literally, as of last night, 95 pounds. It is truly a random mix of whatever change I had in my pocket at the end of the day. It is mostly nickels, dimes, pennies, quarters with a few dollar coins (thanks Metro North ticket machines).

I have drawn a line on the bottle and set a date that which ever comes first, at that point I am cashing it in. I have already spoken to a bank that will accept it without a service charge.

Is there any good rule of thumb, assuming it is truly a random mix of US change from purchases, to estimate value based on weight or volume? People who have seen the container have estimated anywhere from $400 to $2,000. My guess would be closer to $750, but I truly have no friggin clue.

I came up with my guess by estimating based on amount I think I averaged putting away per day. Specifically, I took 10 years times 365 days times $0.20/day average put in and came up with close to $750 (730). Obviously, this estimate is only as accurate as me knowing the actual time frame for saving (I could be off by a year or two) and picking the appropriate amount I averaged putting in each day. There were days I put in none (vacation, no loose change, gave it to homeless guy asking for spare change, etc) and days where I put in $8. Most days it is less than a dollar in that I try to maximize the efficiency of my spending change when paying and there is really no reason to ever have more than a dollar in change if you do that. Getting dollar coins back from vending affects that, but it was a true rarity, maybe once a year.

I do not want to empty it and count it either manually (NO!) or mechanically until the day I cash out. Suggestions? Looking to either refine my method of estimate or come up with another such as based on weight or volume.

posted by JohnnyGunn to Work & Money (26 answers total)

Its hard to know what random change you tend to collect but if it is about constant, then do the following:

1. For the next week or two use another container (small bottle) to store your change.
2. Manually total it at the end.
3. Take the weight of your change in the large bottle and divide by the weight in the small bottle and multiply by the manual total.

Alternately, if your large bottle is an open container, just scoop out a small amount now and do as above.
posted by vacapinta at 6:16 AM on December 15, 2011 [4 favorites]

The always awesome notmartha already did this. The answer is roughly $12/lb.
posted by phunniemee at 6:17 AM on December 15, 2011 [5 favorites]

I've been doing this for years using coffee cans ( before I got hooked on Starbucks). I used to guestimate that a regular can of coins ( plus whatever else was in my pocket ) was about $50 after sorting, and this held pretty much true +/- $5.00.
posted by lobstah at 6:34 AM on December 15, 2011

The US Mint has the weight of all coins on their website.

Obviously, this can't tell us the average, but it made for some interesting equations. (Bear in mind that the tool I used to convert grams to ounces certainly rounded things in the process, but still, this is interesting):

95 pounds of dimes = $1900
95 pounds of quarters = $1900
95 pounds of half-dollars = $1900

95 pounds of pennies = $190 (You can see the rounding at play here. A penny and dime weigh close to the same, but not exactly. It would have been more accurate to do all the math in grams, then convert to pounds at the end.)

95 pounds of nickels = $447
95 pounds of dollar coins (presidential or Native American) $6080

If phunniemee's link is a good estimate, 95 x 12 = $1140...that might not be too far off, unless you have way more pennies than anything else.
posted by The Deej at 6:41 AM on December 15, 2011

One more interesting thing to add to Deej's post. The value / volume of dimes and quarters is also roughly the same. I found:

Quarter 4.4 cents/gram 0.03 cents/ml
Dime 4.4 cents/gram 0.03 cents/ml
Nickel 1 cent/ gram 0.007 cents/ml
Penny .4 cents/gram 0.002 cents/ml

As far as value per mass and value per volume Quarters and Dimes are the same.

If you assume only these coins, you could weight the coins and estimate the total volume. This would give you 2 equations (one for mass and one for volume) and 3 unknowns (# Quarters+Dimes, # Nickels, # Pennies). Use one of the statistical estimates given in phunniemee's link for the ratio of pennies to nickels (this won't effect the outcome much I would think) and you can solve the equations for your estimate...
posted by NoDef at 6:51 AM on December 15, 2011

So just to take this to completion:

95 pounds of dimes = $1900
95 pounds of quarters = $1900
95 pounds of pennies = $190
95 pounds of nickels = $447

Assuming an even mix of coins: $1900 + $1900 + $190 + $447 = $4437

$4437 / 4 = $1109

The always awesome notmartha already did this. The answer is roughly $12/lb.

$12 * 95 = $1140

Sounds like these are all in the same ballpark. You're looking at about $1100 - $1150 in change right now.
posted by NotMyselfRightNow at 6:52 AM on December 15, 2011

Some more accurate numbers, with less rounding:

95 pounds of:

Pennies = $173.28
Nickels = $433.20
Dimes = $1909.50
Quarters = $19.09.50
Half-Dollars = $1900.00
Dollars = $5320.00
posted by The Deej at 6:54 AM on December 15, 2011

(Oops, obviously the quarters are $1909.50. Got decimal happy there.)
posted by The Deej at 6:56 AM on December 15, 2011

the composition of the coins makes a huge difference.

you should grab a handful or two from the top, calculate the value, then use something like a postage scale to calculate the weight. then scale up accordingly. you're not really getting a random sample, but it's probably as close as you're going to get without expending more energy than just taking it all to a bank.
posted by cupcake1337 at 7:49 AM on December 15, 2011 [1 favorite]

Response by poster: Thank you phunniemee for the link(s) and Deej (and NMRN) for the calculations.

The opening to the jug is such that I cannot just scoop out a random amount to use as a sample.

If you take the $1100 estimate and back into an average savings per day, you get around $0.30/day. That seems reasonable, but a little high, although when you think about $110 per year, that seems about right.

As for drinks at the next meetup, if I had a nickel for every time someone said that...nevermind.

I have told my kids that if they guess within $50 of the actual amount when we turn it in, they can have 2x the amount they miss it by. (puts my max exposure per kid @ $100). I am debating whether or not to tell them how much it weighs. If I do tell, it would be simply as a test to see if they are resourceful enough to go on the internet and find this thread or the linked sites.
posted by JohnnyGunn at 8:07 AM on December 15, 2011

Volumetric answer: I filled a five gallon water jug with mixed change. Was nearly $1600.
posted by Area Control at 8:17 AM on December 15, 2011 [1 favorite]

This handy calculator does the math for you. Weigh the coins and grab a handful to estimate the relative mix of quarters/dimes/nickels/pennies and it will do the math for you. I have charted mixed groups of coins in past puzzlers and have determined that the average mix that seems to get the most accurate results is 40% pennies, 9% nickels, 18% dimes and 33% quarters. So if I can't get a random handful I enter 40/9/18/33/0/0 in the squares and I usually get a very accurate result.
posted by Lame_username at 8:23 AM on December 15, 2011 [1 favorite]

The question and most of the answers assume an even distribution of coins (except the empirical methods like the one from notmartha). But most cashiers will hand back properly calculated change, which does not have an even distribution.

If you calculate the amount of change required for every amount from 1 cent to 99, you get an average of 1.48 quarters, .79 dimes, .4 nickels, and 2.98 pennies. Multiplying by the weight of each coin means a random transaction will generate, on average, 8.42g of quarters, 1.79g of dimes, 2.02g of nickels, and 7.45g of pennies.

The average value is thus 50 cents and weighs 19.68g. Your 95lbs is 43,091g, which suggests an expected value of $2,189.
posted by jedicus at 8:28 AM on December 15, 2011 [2 favorites]

Whoops, forgot to convert from units of 50 cents to dollars. That should be $1,094.
posted by jedicus at 8:29 AM on December 15, 2011

mefi's own madcaptenor wrote about this once. If every transaction is equally likely to end in each possible number of cents, and cashiers always hand back the smallest possible number of coins, then it's $28.58 per kilogram, or $12.96 per pound.

My change at that period of my life usually came out a little lower in value than that, because I was raiding it for quarters to do laundry.
posted by madcaptenor at 8:51 AM on December 15, 2011 [3 favorites]

then it's $28.58 per kilogram

Ack, my change counting algorithm had an off-by-one error, leading to excess pennies and thus undervaluing the result. Your numbers are correct.
posted by jedicus at 8:58 AM on December 15, 2011

Yeah, that's what I figured. It's hard to figure out the dime and nickel numbers in your head, but any number of pennies between 0 and 4, and any number of quarters between 0 and 3, are equally likely.
posted by madcaptenor at 9:05 AM on December 15, 2011

jedicus: which then assumes an equal distribution in payments (total prices plus tax) whose change portion is evenly distributed between 1-99 cents, which seems unlikely.

I mean, as long as we're being pedantic and all ...
posted by spaceman_spiff at 10:02 AM on December 15, 2011

Well, yeah, but do you have actual data?
posted by madcaptenor at 10:06 AM on December 15, 2011

it weights 95 pounds. i don't know how strong you are, but instead of scooping out a handful you could tip it over and dump a little out.
posted by cupcake1337 at 10:32 AM on December 15, 2011

I have told my kids that if they guess within $50 of the actual amount when we turn it in, they can have 2x the amount they miss it by. (puts my max exposure per kid @ $100).

Wouldn't that mean that the more accurate they are, the less of a payout they get? Miss it by 20 bucks, you get 40; miss it by 2 bucks, you get 4. Seems like you'd want to give them double of (50-[how much they miss it by]), so if they miss it by 25, they get 50, and if they miss it by 2, they get 96.

I'm just thinking of the children!
posted by FatherDagon at 11:32 AM on December 15, 2011 [4 favorites]

Quarters, Dimes, and Half-Dollars have the same mass/value ratio because when they were first minted, they were minted in silver with their face value equal to their metal value. The other coins are "tokens," inasmuch as their face value has never been intentionally related to their scrap-metal value.

A few years ago I did this calculation to work out how best to fill my piggy bank, and I doubled its monetary capacity by eliminating tokens from the contents.
posted by Sunburnt at 11:39 AM on December 15, 2011 [1 favorite]

Response by poster: FatherDagon: Yes and No. Assuming they have confidence in their ability to guess, they should just adjust that by $49 to maximize their return. Guessing the exact will not give you the highest payout (I should give something for getting with $5 or so) but you also have the ability to miss on both sides. If you guess $1100 for instance and then adjust to $1051 you can maximize you payout but not the odds of getting that payout. If you guess the $1100, you can miss up to $49 on each side. Look at expected return not just payout.

But, fear not for the children, as I say to them, if I die, I want to be reincarnated as one of my kids, they have it so good. (Grr...get offa my lawn.)
posted by JohnnyGunn at 12:55 PM on December 15, 2011

If you guess $1100 for instance and then adjust to $1051 you can maximize you payout but not the odds of getting that payout.

that's right if you are assuming the guesses are independent of the actual amount in the jar. since your kids have some information about the amount in the jar there is at least some dependence.

since there are many coins they could use a central limit theorem. they could make some assumptions about the composition of the coins to estimate the average $/weight density, and then scale. the problem is really estimating the true $/weight density. the maximum likely hood estimate would be the sample density: (total $ observed)/(total weight observed). they can maximize their expected payoff by offsetting their estimate. they would be indifferent between going up or down unless they were close to an edge case (all pennies, all silver). also, they could improve their estimate by each independently making a guess, then averaging their guesses.

you should take FatherDagon's advice and amend your deal.
posted by cupcake1337 at 2:20 PM on December 15, 2011

Response by poster: Cupcake1337: The jar in question is translucent. They can see the coins and probably make some sort of pretty good calculation on composition. One semi wild card is that there are at least 25 dollar coins in the container that I cannot see from just looking into it. I am also always willing to buy change at 100 cents on the dollar for bills. Whenever they save their coins, I cash them out like a bank would.

The truth is if any of my kids took the time to understand what you wrote and what was written above and made an informed guess, I would be very happy and find a way to to reward them for that. I know they are capable of it, I am not as sure about the motivation. They are primarily motivated to work on improving their athletic skills more than their academic ones. As they are great kids, pretty good students and good to very good high school athletes, I let them be.
posted by JohnnyGunn at 5:09 PM on December 15, 2011

No idea how much it weighed but my dad just cashed in a gallon-sized container (although not a milk jug) and got $150 out of it.
posted by IndigoRain at 10:43 PM on December 16, 2011