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Expected vs. Real -- Coin Recovery Ratios
Posted by Johnnyanglo
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Expected vs. Real -- Coin Recovery Ratios August 02, 2020 07:00AM |
There should be relationships between coins lost to the environment and coins held by people. Since reception of coinage is typically the result of merchandise transactions at the register, the distribution of change should be a major determinate of coins lost in the ground.
It is assumed that a store clerk will return coinage economically, with the least level of thought and work possible. The largest denominations should be given first (quarter), then smaller value coins next (dime, nickel, penny). If you pay $10 for a $9.26 purchase, you'll receive 74 cents in change. Logically, it won't be as 74 pennies if other options are available. Your 74 cents in change is unlikely to be 7 dimes and 4 pennies and even much less likely to be 14 nickels and 4 pennies. The most efficient method will require the least number of coins (2 quarters, 2 dimes, 4 pennies). Therefore, the Optimal #1 change distribution for 99 cents (if it were $1 it would be paper money) is:
Optimal #1
Efficiency in change production stipulates the use of up to 3 quarters, 2 dimes, 1 nickel, and 4 pennies. That is the maximums that can be used. Picking 3 dimes is less efficient than picking 1 quarter and a nickel. Under this scheme, the calculated ratio of coin types received per 100 transactions is 150 quarters, 80 dimes, 40 nickels, and 200 pennies. This provides a ratio of Q, D, N, P of 3.75 : 2 : 1 : 5. This means, that for every 100 nickels received, a person should receive 200 dimes, 500 pennies, and about 375 quarters.
It is possible to have change returned in less than optimal manner, using odd combinations of coinage, such as 100 pennies for $1.00. Unlikely but possible. Under this scheme, the teller has a choice of providing a variety of coins, as long as they all total $1.00. This is different than the arrangement above because this method assumes any change combination is possible just as long as it equals $1. The Optimal #1 above limits the denominations used so big coins get used first. The "any way to make 1$" method provides the following ratios:
Here the coin returned ratio is Q, D, N, P of 0.137 : 0.458 : 1 : 5. Which means for every 100 nickels a person receives they receive about 46 dimes, 500 pennies, and 14 quarters. Of note, there were 242 possible coin combinations to reach exactly $1 using four coins, no more and no less. Forcing the exact change to equal $1 limits the use of quarters and dimes, so their ratios are much less. The larger denominations cannot be divided in as many ways to make change for $1. The nickel and penny ratios remain unchanged compared to the expected ratios for the optimal coin distribution (where the fewest coins are being chosen to make change).
In the following example, the same procedure is used as above (any combination of coins to reach $1) but it includes both dollar and half-dollar coins. This yields:
Now, even fewer quarters and dimes are chosen in favor of making change using a dollar or half-dollar. The use of 6 coins vs. 4 coins to reach $1 allows a few more coin combinations (293 ways to make $1 using 6 coins as compared to 242 combinations with 4 coins). Overall, the final ratios are about the same (4 vs. 6 coins) and underestimate the larger denominations. With six denominations, for every 100 nickels received, you'd get 200 dimes, 500 pennies, 133 quarters, 34 half-dollars, and 1 dollar coin.
The Optimal #1 ratios at the top should be fairly realistic to the coins in a person's pocket. A cashier's preference will be to return 3 quarters in change and not 5 dimes or a quarter and 75 pennies (though it depends on coin availability in the till). I calculated a second Optimal #2 that also uses the fewest coin combinations. It is really just to double-check that all combinations are used to provide change. Rather than matching $1 exactly, any possible change combination is allowed, using the minimum coinage. Just as in the Optimal #1 a cashier is allowed to use up to 3 quarters, 2 dimes, 1 nickel, and 4 pennies. I found that there were 116 possible coin combinations for change that would yield < $1 using a total of 570 coins. Here is the result:
Optimal #2
Turns out, Optimal #2 found 18% fewer quarter combinations and 18% fewer penny combinations. I believe Optimal #2 to probably be more accurate. Every possible change scenario < $1 is accounted. It has Q, D, N, P ratios of 3.1 : 2.0 : 1.0 : 4.1. This means, for every 100 nickels received in change, you'd get about 200 dimes, 410 pennies, and 310 quarters.
Next, I went ahead and searched the internet for coin find results as given by detectorists and tabulated them Now it is possible to compare actual recovered coin ratios from the ground with ratios based on coinage carried by people. The postulate here is that there should be a close relation to the ratio of coins recovered to coins gained in merchant transactions at the register. Not all the detectorists included data on the ubiquitous memorial cent. So, I tabulated coin recovery ratios for those that did report memorial cent data (on the RIGHT). The table on the LEFT did not always report memorial cents. Here are the results:
Detector coin recovery numbers on the LEFT (above) include data that is missing some of the recovery data for memorial cents, but is more accurate in non-penny numbers. Data on the RIGHT includes only data where memorial cent counts are present (to more accurately represent penny finds). There were 198 data points on the LEFT and 145 data points on the RIGHT. The ratio of recovered coins by detectorists are very similar. The data on the LEFT is more accurate for all coins, except pennies. The data on the RIGHT is more accurate in regards to recovered pennies (because it included every penny type). Thus, the best ratios for detectorist recovered coins, is:
This means for every 100 nickels recovered from the ground, you'd get 262 dimes, 690 pennies, and 233 quarters. You'd also get 3.2 dollar coins and maybe 0.68 half-dollars (these are not necessarily silver coins) with your 100 nickels. Comparing the various notional ratios based on cash register coinage with the actual probabilities of detector coin recovery, there is general agreement, but also some differences.
Coin ratios - Notional vs. Actual
There are more pennies recovered than would be expected (penny ratio of 6.9 vs. 4.11 to 5.0 handed out compared to each nickel received). It could be inferred, when a penny is lost it more often stays lost. Perhaps not enough value to try and find it or it hides better. It is likely the dark copper color blends well with dirt and makes pennies hard to find. If someone kept all the pennies from 100 cashier transactions, they'd have collected between 411 and 500 pennies depending on how the values are calculated, along with 100 nickels. When a detectorist has collected 100 nickels from the ground, they will also have gathered about 690 pennies to boot. It appears that detectorists recover from the ground about 38% to 68% more pennies relative to nickels than are circulating in the pockets of people. Pennies are lost more often. This means the average loss rate for pennies is probably about 50% greater than for nickels.
For every 100 cashier transactions, a person collects 200 dimes in change and 100 nickels, hence twice as many dimes. A detectorist who recovers 100 nickels also gathers 262 dimes. The detector recovery rate for dimes is about 31% higher than the coins available for loss in the pockets of people, that is, there are 31% more lost dimes in the ground than there should be based on the amount of dimes in people's pockets. If the same percentage of coins fall out a person's pocket, they should loose 2x more dimes than nickels. But the ground contains 2.6x more dimes than nickels. This indicates that perhaps a rather smallish dime is more easily lost from a trouser pocket than the larger nickel. Perhaps also a dime is not terribly valuable to spend the effort hunting for it and its size makes it harder to find. There are about 2.5 times more pennies in people's pockets than dimes but there is also about 2.6 times more pennies than dimes found in the ground, which is a commensurate loss to holdings. The loss rate of dimes and pennies are fairly similar. Both have about 31 - 35% more losses than expected by their actual numbers in people's pockets. Not so with quarters.
For every 100 cashier transactions, a person receives from 307 to 375 quarters and 100 nickels. A detectorist will recover 233 quarters for every 100 nickels. The ground contains 24-38% fewer quarters as compared to the amount carried in people's pockets. Apparently, people do not want to lose quarters and will find a dropped quarter. A quarter is also easier to spot in the grass. Perhaps quarters are also less likely to fall out of a pocket due to their larger size and weight.
There about 3x more pennies lost than quarters, about 13% more dimes lost than quarters, about 2.3x more quarters lost than nickels, about 2.6x more dimes lost than nickels, and about 6.9x more pennies lost than nickels. The dollar coin is about 4.7x more likely to be found than a half-dollar. You'll dig 218x more pennies than dollar coins and 990x more pennies than half-dollars. You'll find 3x more pennies than quarters and 7x more pennies than nickels. You have a chance of getting one dollar coin out of every 402 coins recovered and of getting a half-dollar coin out of every 1867 coins recovered. You should expect one quarter for every 5th or 6th coin dug and a dime every 5th coin dug. You will find a nickel for every 13th coin dug and a penny every 2nd coin dug, on average.
-- Johnnyanglo
It is assumed that a store clerk will return coinage economically, with the least level of thought and work possible. The largest denominations should be given first (quarter), then smaller value coins next (dime, nickel, penny). If you pay $10 for a $9.26 purchase, you'll receive 74 cents in change. Logically, it won't be as 74 pennies if other options are available. Your 74 cents in change is unlikely to be 7 dimes and 4 pennies and even much less likely to be 14 nickels and 4 pennies. The most efficient method will require the least number of coins (2 quarters, 2 dimes, 4 pennies). Therefore, the Optimal #1 change distribution for 99 cents (if it were $1 it would be paper money) is:
Optimal #1
Efficiency in change production stipulates the use of up to 3 quarters, 2 dimes, 1 nickel, and 4 pennies. That is the maximums that can be used. Picking 3 dimes is less efficient than picking 1 quarter and a nickel. Under this scheme, the calculated ratio of coin types received per 100 transactions is 150 quarters, 80 dimes, 40 nickels, and 200 pennies. This provides a ratio of Q, D, N, P of 3.75 : 2 : 1 : 5. This means, that for every 100 nickels received, a person should receive 200 dimes, 500 pennies, and about 375 quarters.
It is possible to have change returned in less than optimal manner, using odd combinations of coinage, such as 100 pennies for $1.00. Unlikely but possible. Under this scheme, the teller has a choice of providing a variety of coins, as long as they all total $1.00. This is different than the arrangement above because this method assumes any change combination is possible just as long as it equals $1. The Optimal #1 above limits the denominations used so big coins get used first. The "any way to make 1$" method provides the following ratios:
Here the coin returned ratio is Q, D, N, P of 0.137 : 0.458 : 1 : 5. Which means for every 100 nickels a person receives they receive about 46 dimes, 500 pennies, and 14 quarters. Of note, there were 242 possible coin combinations to reach exactly $1 using four coins, no more and no less. Forcing the exact change to equal $1 limits the use of quarters and dimes, so their ratios are much less. The larger denominations cannot be divided in as many ways to make change for $1. The nickel and penny ratios remain unchanged compared to the expected ratios for the optimal coin distribution (where the fewest coins are being chosen to make change).
In the following example, the same procedure is used as above (any combination of coins to reach $1) but it includes both dollar and half-dollar coins. This yields:
Now, even fewer quarters and dimes are chosen in favor of making change using a dollar or half-dollar. The use of 6 coins vs. 4 coins to reach $1 allows a few more coin combinations (293 ways to make $1 using 6 coins as compared to 242 combinations with 4 coins). Overall, the final ratios are about the same (4 vs. 6 coins) and underestimate the larger denominations. With six denominations, for every 100 nickels received, you'd get 200 dimes, 500 pennies, 133 quarters, 34 half-dollars, and 1 dollar coin.
The Optimal #1 ratios at the top should be fairly realistic to the coins in a person's pocket. A cashier's preference will be to return 3 quarters in change and not 5 dimes or a quarter and 75 pennies (though it depends on coin availability in the till). I calculated a second Optimal #2 that also uses the fewest coin combinations. It is really just to double-check that all combinations are used to provide change. Rather than matching $1 exactly, any possible change combination is allowed, using the minimum coinage. Just as in the Optimal #1 a cashier is allowed to use up to 3 quarters, 2 dimes, 1 nickel, and 4 pennies. I found that there were 116 possible coin combinations for change that would yield < $1 using a total of 570 coins. Here is the result:
Optimal #2
Turns out, Optimal #2 found 18% fewer quarter combinations and 18% fewer penny combinations. I believe Optimal #2 to probably be more accurate. Every possible change scenario < $1 is accounted. It has Q, D, N, P ratios of 3.1 : 2.0 : 1.0 : 4.1. This means, for every 100 nickels received in change, you'd get about 200 dimes, 410 pennies, and 310 quarters.
Next, I went ahead and searched the internet for coin find results as given by detectorists and tabulated them Now it is possible to compare actual recovered coin ratios from the ground with ratios based on coinage carried by people. The postulate here is that there should be a close relation to the ratio of coins recovered to coins gained in merchant transactions at the register. Not all the detectorists included data on the ubiquitous memorial cent. So, I tabulated coin recovery ratios for those that did report memorial cent data (on the RIGHT). The table on the LEFT did not always report memorial cents. Here are the results:
Detector coin recovery numbers on the LEFT (above) include data that is missing some of the recovery data for memorial cents, but is more accurate in non-penny numbers. Data on the RIGHT includes only data where memorial cent counts are present (to more accurately represent penny finds). There were 198 data points on the LEFT and 145 data points on the RIGHT. The ratio of recovered coins by detectorists are very similar. The data on the LEFT is more accurate for all coins, except pennies. The data on the RIGHT is more accurate in regards to recovered pennies (because it included every penny type). Thus, the best ratios for detectorist recovered coins, is:
This means for every 100 nickels recovered from the ground, you'd get 262 dimes, 690 pennies, and 233 quarters. You'd also get 3.2 dollar coins and maybe 0.68 half-dollars (these are not necessarily silver coins) with your 100 nickels. Comparing the various notional ratios based on cash register coinage with the actual probabilities of detector coin recovery, there is general agreement, but also some differences.
Coin ratios - Notional vs. Actual
There are more pennies recovered than would be expected (penny ratio of 6.9 vs. 4.11 to 5.0 handed out compared to each nickel received). It could be inferred, when a penny is lost it more often stays lost. Perhaps not enough value to try and find it or it hides better. It is likely the dark copper color blends well with dirt and makes pennies hard to find. If someone kept all the pennies from 100 cashier transactions, they'd have collected between 411 and 500 pennies depending on how the values are calculated, along with 100 nickels. When a detectorist has collected 100 nickels from the ground, they will also have gathered about 690 pennies to boot. It appears that detectorists recover from the ground about 38% to 68% more pennies relative to nickels than are circulating in the pockets of people. Pennies are lost more often. This means the average loss rate for pennies is probably about 50% greater than for nickels.
For every 100 cashier transactions, a person collects 200 dimes in change and 100 nickels, hence twice as many dimes. A detectorist who recovers 100 nickels also gathers 262 dimes. The detector recovery rate for dimes is about 31% higher than the coins available for loss in the pockets of people, that is, there are 31% more lost dimes in the ground than there should be based on the amount of dimes in people's pockets. If the same percentage of coins fall out a person's pocket, they should loose 2x more dimes than nickels. But the ground contains 2.6x more dimes than nickels. This indicates that perhaps a rather smallish dime is more easily lost from a trouser pocket than the larger nickel. Perhaps also a dime is not terribly valuable to spend the effort hunting for it and its size makes it harder to find. There are about 2.5 times more pennies in people's pockets than dimes but there is also about 2.6 times more pennies than dimes found in the ground, which is a commensurate loss to holdings. The loss rate of dimes and pennies are fairly similar. Both have about 31 - 35% more losses than expected by their actual numbers in people's pockets. Not so with quarters.
For every 100 cashier transactions, a person receives from 307 to 375 quarters and 100 nickels. A detectorist will recover 233 quarters for every 100 nickels. The ground contains 24-38% fewer quarters as compared to the amount carried in people's pockets. Apparently, people do not want to lose quarters and will find a dropped quarter. A quarter is also easier to spot in the grass. Perhaps quarters are also less likely to fall out of a pocket due to their larger size and weight.
There about 3x more pennies lost than quarters, about 13% more dimes lost than quarters, about 2.3x more quarters lost than nickels, about 2.6x more dimes lost than nickels, and about 6.9x more pennies lost than nickels. The dollar coin is about 4.7x more likely to be found than a half-dollar. You'll dig 218x more pennies than dollar coins and 990x more pennies than half-dollars. You'll find 3x more pennies than quarters and 7x more pennies than nickels. You have a chance of getting one dollar coin out of every 402 coins recovered and of getting a half-dollar coin out of every 1867 coins recovered. You should expect one quarter for every 5th or 6th coin dug and a dime every 5th coin dug. You will find a nickel for every 13th coin dug and a penny every 2nd coin dug, on average.
-- Johnnyanglo