Creating a Mathematical Model

Every mathematical model begins with a real world observation. Someone says, "Hmmm... I think I notice a pattern here!" If the pattern is relatively simple, coming up with a mathematical model could be straight-forward...

For instance, imagine that a biologist is observing a newly discovered type of (very simple) alien organism: Slowly, the organism begins to multiply. Being an intelligent observer, the biologist notes the time of each spawn:

Translating what she sees into mathematics, the thoughtful biologist makes a table of observed outcomes:

"Hmmmm...," she thinks. "The alien doubles every hour. That sure does remind me of something!" And, without a second to lose, the Biologist scribbles a formula onto a sheet of paper:

So, now the biologist has a mathematical model. The next step for her is to use her model to make a prediction, "I think that in the fourth hour there will be 16 aliens!" Before she can be sure that her model is accurate, the biologist will then have to wait until she has verified a sufficient number of predictions with expected outcomes.

Now, what if the alien starts acting a little funny. What if during the fifth hour, the alien were to only sort of double...

Would the biologist be completely wrong in her mathematical model? Well, it's hard to say... but statistics is one of the tools by which we can gauge the accuracy of a mathematical model. If the alien creatures continue to spawn erratically, the biologist would have no choice but to refine her model.

Try this exercise to test your understanding of how a mathematical model is developed!