Where is my Dollar?
This is an essay about counting. Simple counting, you know, like the kind that the puppet vampire on Sesame Street does. You remember “The Count,” right? One, Two, Three, Four, blah, blah, blah, blah. I just did a little research, and I discovered his full name is Count Von Count. Who knew?
I have sat through a lot of lectures in my life, so many that I couldn’t begin to count (get it?). This may sound surprising, but one of the best lectures I ever attended was about the proper way to count. This was in a graduate statistics class. This seemingly straightforward process is a lot more complicated than you might imagine.
To set the stage, I must tell you that in what feels like a previous lifetime, I was an archaeologist. I worked in The Bahamas at the Island of first contact, the one made famous by Christopher Columbus and his crew. For those of you who do not remember, the name of the Island is San Salvador, and it is beautiful.
There are numerous ways we could begin our journey into the strange world of simple counting. We could quickly start with a discussion of how easy it seems to be to count, say, artifacts excavated from an archaeological site. If you do not “dig” that we could talk about how scientists count the number of genes in the human genome. Instead, and by explicit design, we will start with a fictitious handyman. This man works at a non-existent motel located in a town that you will not find on any map. His story is much more fascinating and informative than anything else I might dream up.
The story begins like this: there was a man named Ichabod who worked at a campground as a general laborer. Ichabod spent most of his time doing odd jobs, usually something different every day. On a random Friday, the 13th three men came in needing a cabin. The manager took their information, charged them $60, and sent them on their way. A few hours after they left, the manager realized he made an error and called Ichabod into his office. He told Ichabod that he had mistakenly overcharged the three guys at campsite number 234 by $5 and gave Ichabod the money to hand over to them.
Ichabod worked his way down a winding dirt road to cabin number 234 with 5 one-dollar bills in his pocket. As he approached their camp, he realized that he had no way to divide the $5 evenly by three. He thought that there would be no harm in just giving each of the three guys one dollar and pocketing the extra two dollars for himself. That is precisely what he did.
A few hours later, Ichabod was bored, his work for the evening was done, and there were no pressing emergencies. He decided that he wanted to figure out why the three guys got a discount. He quickly found out that they had a AAA coupon for $5 off for one night’s stay. Something didn’t seem quite right, so Ichabod got the manager’s calculator and multiplied $19 by three. After Ichabod gave them each a dollar back, that is what each of them paid. He came up with $57. He then looked in his pocket and found the two dollars he decided to keep for himself. He multiplied 19 by three again and once again came up with 57. He added the two in his pocket to get $59. Wait a minute, he thought, they initially paid $60.
Ichabod mulled this over for a few more minutes. Let’s see, each guy ended up paying $19 for the campsite. The three of them together paid $57. I kept two. Ichabod had a perplexed and slightly angry look on his face when he looked up at the stars and yelled: “Where is my dollar?”
Do not bother racking your brain just yet. The answer will be as curious at the end of this essay as it is right now. The point to consider at this juncture is simply the slippery nature of straightforward counting. And yes, the Twilight Zone feel of the end of the last paragraph was put there on purpose.
One of the little known aspects of science is the serious thought that scientists have to put into the seemingly artless process of counting. Counting should be straightforward, shouldn’t it? It should be easy regardless of whether you have kids learning sums or if you have archaeologists counting projectile points. Unfortunately, this is not the case. Simple counting is anything but.
Consider the dilemma of the archaeologist studying regional settlement patterns. This form of archaeology had become more commonplace in recent decades. When studying regional settlement patterns, archaeologists do not look at a single site, they consider the relationship between all the sites in a given area. Typically, the scientist would be interested in the differences in the artifacts found at the various locations. Different artifacts found at a particular site would imply different uses for that site, and this is what studying non-site level archaeology is all about. How exactly to do this, though? Simple counts are virtually useless because Site A may have twice as many projectile points as Site B, but maybe the site itself is 10 times as large. The naive investigator might think that Site A was an area where the use of projectile points was much more important than it actually was.
What to do then? There are nearly as many strategies employed as there are archaeologists to do the counting. Many people find percentages to be very powerful. If projectile points are found to be 85% of the total number of artifacts found at Site A, then this is a very important observation, especially if Site B has three times as many projectile points, but they only represent 2% of the total excavated from that site.
Some archaeologists prefer to work with volumes. They weigh the artifacts and then calculate volumes with respect to all other excavated material. You can even consider the weight of individual types of artifacts vis-à-vis the total amount of dirt dug up. This can lead to percentages or even ratios. The point being that simple counts are almost always deceptive and are rarely useful.
Animal bones, including those of humans, present more issues. A simple, yet potent technique, called Minimum Number of Individuals, or MNI, is often used. If you find 15 sheep vertebra at your site, they may have come from 15 sheep. That is not a whole lot of help. But what happens if you are studying mass burials in the U.S. Southwest and you discover 35 human vertebrae, 8 finger bones, 22 foot bones, and 5 right clavicles? The minimum number of individuals required to explain that group of artifacts is 5. Why 5? Because humans have only one right clavicle each. That might prove to be something worth knowing.
The types of issues we have been talking about are fundamental and might even be surprising to someone who thinks only of Indiana Jones whenever archaeology is mentioned. Once the archaeologist decides how they want to count, the problem with counting is just beginning. Then issues of a statistical nature raise their ugly heads.
It might well be the case that some aspects of the size, e.g., length, of the projectile points that have been excavated are diagnostic. In other words, they can tell us something vital if we are smart enough to tease the information out of the material. You might think that it would be a good idea to get an average length of the points so that you could compare this number with a similar amount from other sites. Let’s see if we can illustrate just how tricky that is.
Determining the average length of points could not, on the face of it, be more simple. Just add all the individual lengths up and then divide by the total number of points to get an average. No calculus involved here. The problem is that it might very well be the case that you will end up with an average size that is not representative of anything in existence. What if you found 6 points 1 centimeter in length and 5 that were 4 centimeters in length? You will calculate an average projectile point length of around 2.5 centimeters. This 2.5-centimeter long point does not exist, you did not find even one that is that length. It should be clear that this is a terrible idea.
The solution to the averaging problem is to split the points up into two separate “batches” of numbers that will be considered separately. It is pretty obvious that the larger points were used for different purposes than the smaller ones. Reviewed independently, two distinct averages can be calculated that will prove very informative when compared to comparable data from other sites.
This essay has been about simple counting. I am sure that you now appreciate how much care goes into adding, subtracting, multiplying, and dividing. Truth be known, we have only scratched the surface of what is an important scientific issue. This topic becomes even more outrageous when advanced statistical techniques are employed. This is true for all scientists, not just archaeologists.
I am sure that there are many clever ways to end an essay of this nature. Ridiculous puns about counting ways to make mistakes come readily to mind. Or I could remind you to count yourself lucky that you do not have to think deeply about such nonsense. You and I both know, though, that I must end with a simple, elegant, yet slightly disturbing question. Tell me, exactly where is Ichabod’s dollar?
