model is to data as chicken is to egg
- Author:
- Judah L Schwartz
If we have a collection of pieces of aluminum and a collection of pieces pine and a collection of pieces of polystyrene most people would be willing to agree that there is some property of aluminum that is “larger” than that of pine. Suppose, for the moment, we call this property “oomph”.
Is there some way we can pin down what such a property called “oomph” might be? More to the point, is there some way we can say whether the “oomph” of of some third material like polystyrene is closer to the “oomph” of aluminum than it is to the “oomph” of pine? And if so, how much closer? More generally, if we a bunch of different materials, can we find a way to order them according to the “oomph” property – from the “largest” to the “smallest”? First thoughts would suggest that the relevant property of the aluminum pieces, the pine pieces and the polystyrene pieces that we want to pay attention to is how “big” they are. But by “big” we might mean their physical size (volume – amount of space they take up) or we might mean their weight (or their mass). Clearly each piece of aluminum, pine and polystyrene has both a volume and a mass. Assuming we have the proper instruments for measuring volume and mass, we could measure both the volume and the mass for each of the pieces of aluminum, pine and polystyrene. If we do so, we will quickly discover that our mystery property, “oomph”, cannot be mass and it cannot be volume. We can find some pieces of pine with more mass than pieces of aluminum and some pieces of pine with less mass than pieces of aluminum. If we try to organize our data we might make a table something like this:
material 1 |
aluminum |
material 2 |
pine |
material 3 |
polystyrene |
||
volume (cc) |
mass (gm) |
volume (cc) |
mass (gm) |
volume (cc) |
mass (gm) |
||
0 |
0 |
0 |
0 |
0 |
0 |
||
120 |
324 |
15 |
8.3 |
41 |
43.1 |
||
73 |
197 |
62 |
34.1 |
103 |
108.2 |
||
31 |
83.7 |
127 |
69.9 |
79 |
83 |
||
110 |
297 |
90 |
49.5 |
67 |
70.4 |
mass is proportional to volume.
This is the model that we will assume. The model in symbols If we try to express this mathematical model in symbols we can write for aluminum mass (in gm) = 2.7 x volume (in cc) and for pine mass (in gm) = 0.55 x volume (in cc) and for polystyrene mass (in gm) = 1.05 x volume (in cc) This pattern suggests that the quantities 2.7 gm/cc, 0.55 gm/cc and 1.05 gm/cc somehow characterize the materials aluminum, pine and polystyrene respectively. Could these quantities be a measure of the mystery property “oomph”? Suppose we use a symbol like the letter d to stand for the quantity that characterizes the material. Then we can write our mathematical model in symbols more generally asmass = d x volume or m = d x v
If mass is measured in grams and volume is measured in cc then d must be in gm/cc. The model in images We can also try to express our model in images as suggested by the graph representation of our data. Our plotted data suggest that object made of a given material will have masses and volumes that are related in such a way that they will lie on a straight line [that goes through the point mass = 0, volume = 0 ] when plotted in a Mass vs. Volume graph. Clearly what characterizes a given material is the slope of that line. The line for aluminum climbs faster than the line for pine – and we say that that line has a greater slope. People who build roofs on houses, install plumbing pipes or handicap access ramp all measure slope in the same way. They ask “how much does the {roof, pipe, ramp} rise for every unit of horizontal run of the {roof, pipe, ramp}? The amount of rise divided by the amount of run is a measure of the slope. Suppose we have two aluminum objects [the green dot and the red dot in the graph], one having a volume that is 100 cc greater than the other. Our model tells us that because they are made of the same material their data points will lie on the same straight line. To get from the green dot to the red dot we have to run horizontally 100 cc and then rise vertically 270 gm. Another way of saying this is that we have to rise vertically 2.7 gm for every horizontal run of 1cc. The slope of our aluminum line is 2.7 gm/ccComing from our model expressed in graphical form, the quantity that expresses the characteristic property of the material is the slope of the line that corresponds to that material. How does our model expressed in graphical form correspond to our model expressed in symbolic form? In the symbolic form of our model, which says “mass is proportional to volume” , the quantity that characterizes the material we have in mind is the quantity that multiplies the volume to yield the mass. In the graphical form of our model, the quantity that characterizes the material we have in mind is the quantity that measures the slope of the line for that material. These are two representations of the same quantity – the density of the material. The Bottom Line We start with some initial notions about what quantities might be important enough in a given situation to warrant our collecting data. We can organize our collected data in different ways. Moreover, the collected data may lead us to propose a model which allows us to predict the result of as yet unmade measurements. Needless to say, we must then collect more data to see if the predictions of our model are upheld. This activity may lead us to refine our model. So here is something to think about – do you really believe that no matter how small a piece of aluminum you take the mass of the piece of aluminum in grams is 2.7 times the volume of that piece of aluminum in cc? It really seems to work over a very wide range of values – but suppose we start to get to sizes that are comparable to the sizes of atoms? Or smaller?