Sunday, June 5, 2011

The Distribution of Loose Change - A Thought Experiment

Well here I am at our Alabama retreat.  I can't call it our home anymore as Catherine, Mya, Lucas, Ella, Russel, Fluff, JJ, SkyHigh, and Skyla, are in Canada and settling in well. 

Packing is monotonous, mind numbing work, especially with the additional paperwork required detailing the contents and estimated value of every single box for the pointy haired types of Canada Customs.  This job is almost as mind numbing as working for my former company, but alas, with a much improved management structure.  Yes, Catherine now wears the pants, and is the main, or should I say sole, bread winner.  She likes to call it crumb winner, but we all make our own beds, so we'd better lay in it; crumbs and all.

This post is more of a distraction, although, granted, it may sound like a crying out for help.  "Dogbert, please save me!".  Any-hoo,  I was emptying the piggy banks for packing and was curious what the typical distribution of coinage was.  From this and the weight of each individual coin, I could determine what I should expect per pound.  This is for US coinage only. Obviously every currency has different weights and in Canada for instance, we need to account for Loonies and Toonies.  I can attest that here in Alabama, there is no shortage of loonies, especially on Keel Mountain, yet, I digress, not the coin variety.

So lets assume that coinage is only provided when it is less than $1.  You might rarely receive quarters if the manager snagged all the dollar bills for a night on the town, but let's affirm that in the bible belt, this is a rare event.  Along the same reasoning, your not going to receive 2 dimes and a nickle, when a quarter will do.  Also lets assume the median value of all possible choices.  It would be interesting to see if the distribution was different if all your shopping was at Walmart and you only purchased things that ended in 86, but I like to keep my little OCD secrets private.  If your not following, you may not have received cashier training, or your mind is no longer capable of critical thinking.  I'm sure there are some great new YouTube videos of cute little kittens...

Ok, now that the Creationists are gone, lets get down to business.  I derived all this from here, but dumbed it down to my way of thinking - maybe the Creationists CAN follow along.  For quarters, we know that we'll receive less than 4.  So we should expect a median frequency of 1.5 (3/2) quarters.  For pennies, less than 5, so the median is 2 (4/2).  Now we only need to deal with the values of 20, 15, 10, 5, and 0.
  • 0 cents - no coins
  • 5 cents - one nickel
  • 10 cents - one dime
  • 15 cents - one dime, one nickel
  • 20 cents - two dimes
So out of the 5 possibilities, we see 4 dimes, and two nickels.  So, 4/5 for dimes and 2/5 for nickels.  In summary, for any transaction we should expect a medium frequency of 1.5 (37.5c) quarters, 0.8 (8c) dimes, 0.4 (2c) nickels and 2 pennies in change, or 49.5 cents.  Of course your not going to get change in half quarters or 2 fifths of a nickel, any more than you'll give birth to 1.2 children.  This is explained better in the source link.  It would be fun though to ask a teller for 8 half quarters to feed the parking meter :)  Now how much does a pound of flesh weigh?
  • quarter - 0.200 oz
  • dime - 0.080 oz
  • nickel - 0.176 oz
  • penny - 0.088 oz
Are you still with me?  As attentive as with a dominatrix, stiletto on your nutters?  Alrighty then!  So 1.5@0.2 oz, 0.8@0.08 oz, 0.4@0.176 oz, and 2@0.088 oz, or an expected weight of 0.611 oz at a value of 49.5 cents.  The equation 0.495 * 16/ 0.611 gives us the median value per pound of change at $12.96.

Now I have 21.188 lbs of change, so I should expect  $274.60 US or about $500 Canadian.  Don't you just love sticking it to Americans?!  I plan on taking the coins over to the local Coinstar.  Interestingly, Coinstar provides an online record of the amount of coins counted of each type.  I'll update this post with the results.

Mya and Lucas will enjoy the gift certificate for Amazon.com, but for now, I'm heading back to the mindless drudgery of packing. Ugh!



So the grand total is dun-dun-du-da, $265.31.  That is pretty darn close!  But wait, there's more.  If I subtract the 15.6 oz of Euros, Francs, Pounds, Kroners, and Loonies the machine rejected, plus one Alberta 4-H 75th Anniversary Commemorative coin and a marble, the original estimate is $274.60 - $12.64 = $261.96.  A difference of $3.35.  Amazing huh!  But that's not all.  If I further subtract the 3 dollar coins and 1 half dollar coin found, but not included in the experiment since they are fairly rare in circulation, 4 coins out of 2651, we arrive at $265.31 - $3.5 = $261.81.  This is a difference of only 15c! $261.96 expected and $261.81 received.  Unbelievable!

Now what do you say we factor out the weight of the dollar and half dollar coins?  Do we risk it?  Do we risk it?  Of course we do!  Unless, of course, you bet on this horse, then I'm truly sorry, but we must make sacrifices for science.
  • Half Dollar 0.400 oz
  • Dollar 0.286 oz
So 3@0.286 oz and 1@0.4 oz equals 1.285 oz.  This means a correction of $12.96 *1.285 / 16 = $1.04.  Subtracting this from our adjusted estimate of $261.96 brings us to $260.92. Compared to our actual of $261.81.  A difference of only 89c.  Sound and true, but that horse is dead.

This is a small experiment, but with over 2500 coins in the population, never the less, uncannily valid.

I'm interesting in knowing what others get using the $12.96 or $13/lb rule.  So you folks who've been hoarding your coins waiting for Armageddon, or making yourself look scary and unattractive by stuffing your braziers, get that change back into circulation.  Better yet, donate it to me, I'll take good care of it.



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