Playing With Data - Quadratic Cameos

I haven't been blogging much the past few years in large part due to being somewhat removed from the science/skeptic/education scene. The past 3 years I've been working in estate jewelry and am currently functioning as an inventory manager. While it is certainly far afield of my main background, I do frequently find ways where I'm applying my scientific background.

Today was a good example of that. One of our buyers has a fondness for cameos. Unfortunately, cameos just don't sell well anymore. There's some rare exceptions, such as the one pictured here. This one happened to be an 1800's piece in a 22k gold mounting that was in spectacular condition.

But most aren't. The most common ones are cameos carved from shell that I refer to as the "profile of the homely young lady". They often come in a 10k gold mounting and if we try to sell them at auction, they often sell for roughly the value of the gold in the mounting. Then, after auction fees, we've made less money than if the cameo was simply pulled out of the mounting and the mounting melted. Thus, when buying, it's important for our buyers to know roughly how much of the weight of a piece is shell, and how much is gold.

I've been collecting the cameos that we've pulled out of the mountings for several months now and have a good collection, so I put some data together today and figured that this could be a good project for a math class, looking at a few types of functions.

From each piece in my collection, I took 3 pieces of information: The height, the width, and the weight. Ideally, I'd have taken another, the thickness, but this is somewhat harder to get at since, in a real world application our buyers would be facing, they would likely not be able to easily measure this. Additionally, I make a weak assumption that this doesn't really change much. After all, even for small cameos, they'll still need to be fairly thick, or risk breaking. So I felt ok leaving this out.

My first pass I tried putting together equations from just single pairings of the height vs weight, or the width vs weight. Before graphing it and letting Excel do the fit for me, it bears some thinking about what the plot might look like. It certainly wouldn't be a linear equation because what we're really looking at is an increase in volume which is length x width x thickness, which would mean it should scale towards the 3rd power. But because the thickness probably stays more or less constant, ad the length and width increase proportionally to one another, this means I should be looking for the data to fit a second degree polynomial, or a quadratic function.

And sure enough, when I plotted everything up, it ended up coming out pretty well.

The thing I really like about data like this is that there's lots of little things you can see by looking at it. The first thing that I noticed (I actually noticed it while taking the data) is that there seems to be several somewhat standard sizes. You can see this borne out on the graph because there's several little vertical groups. I hadn't really considered this before, but there's probably a good reason for this. As with many things in jewelry, there's often a sort of "mix and match" that goes on. Customers could pick the carved cameo they wanted, and then separately pick out the mounting they liked. If there were standardized sizes, this means that jewelers can insert them fairly easily.

Another thing that jumped out at me, this time from the graph, is that there is more scatter towards the larger cameos. If this were something like astronomical data where this was a plot of the recession velocity of galaxies as a function of distance, I would expect that the larger scatter would be due to larger uncertainties in the measurement at larger distances. It would look about the same. But that's not the case here. In fact, the uncertainty in measurement should actually go down as you get to larger heights. I was measuring in mm, so if I were 1mm off, this would be a large error for the small cameos, but becomes rather insignificant towards the larger ones.

So where is the breakdown? It's likely based on the assumption I called out earlier; the cameos aren't all a consistent thickness. This gets magnified as you get towards larger cameos because the variation in thickness is getting amplified by the rapidly growing surface area.

Which brings up another question. I first did this in just one dimension - the height. I had another hidden assumption in there, is that all the cameos are essentially the same overall shape. I only selected the oval ones. None of the ones with clipped corners or heart shapes. But do they really all have the same ratio of major and minor axes? If they don't, then perhaps I'm missing something and that could be the reason for the scatter on the right of the above graph.

To try to minimize that difference, I looked at the area. Kind of. Instead of going through the full calculation to find the actual area of an oval (A = pi x (major axis)/2 x (minor axis)/2), I figured the pi and the "over 2"'s would be common factors, so I simplified this down to just the height x width. Plotting those up vs the weight gave another graph. Again, before looking at the next graph, consider what sort of fit this should be.

If you guessed linear, you guessed right! So what are we learning from this graph?

We still see the large scatter towards the high end. Similarly, if you look at the R^2 value (the residuals), you see that it's only slightly lower than for the simple one dimensional plot. This is a good indication that the scatter on the previous graphs is not caused by significantly different shapes. But aside from looking at the graph, there's a better way to check. The best way is to simply divide the height of each one by the width and see if there's much difference. I did this and found it was very consistent, right around a ratio of 1.3:1.

So how will I use this from here? Probably as a tool to give my buyers in the future. I'll likely spare them all the math that I used to come up with this, but giving them the final graph, they should be able to fairly easily use this to estimate how much of the weight of an item is gold and how much is shell that will get stuck in my giant bag and isn't being turned into money. Perhaps then they can stop paying too much.