Section 5.6 Why Model?
At the beginning of this chapter, I said that many real world phenomena can be modeled with theoretical distributions. But it might not have been clear why we should care.
Like all models, theoretical distributions are abstractions, which means they leave out details that are considered irrelevant. For example, an observed distribution might have measurement errors or quirks that are specific to the sample; theoretical models ignore these idiosyncrasies.
Theoretical models are also a form of data compression. When a model fits a dataset well, a small set of numbers can summarize a large amount of data.
It is sometimes surprising when data from a natural phenomenon fit a theoretical distribution, but these observations can provide insight into physical systems. Sometimes we can explain why an observed distribution has a particular form. For example, in the previous section we found that adult weights are well-modeled by a lognormal distribution, which suggests that changes in weight from year to year might be proportional to current weight.
Also, theoretical distributions lend themselves to mathematical analysis, as we’ll see in Chapter 14.
But it is important to remember that all models are imperfect. Data from the real world never fit a theoretical distribution perfectly. People sometimes talk as if data are generated by models; for example, they might say that the distribution of human heights is normal, or the distribution of income is lognormal. Taken literally, these claims cannot be true — there are always differences between the real world and mathematical models.
Models are useful if they capture the relevant aspects of the real world and leave out unneeded details. But what is relevant or unneeded depends on what you are planning to use the model for.
