Probability Theory and Data Analysis

In the prior chapters we've discussed ways to summarize collections of data visually and numerically. At this point our attention turns from the analysis of empirical information—data collected through observation—to the consideration of some useful theoretical constructs and models. Probability theory is a large and useful body of material and is the subject of entire courses in its own right. In an elementary applied statistics course, and in this book, probability is especially germane in two respects. First, sometimes the goal of data analysis is to approximate the likelihoods of events or outcomes that are important to a decision-maker. For example, people who plan communications or transportation networks want to know the likelihood of traffic at different network locations during the day.

Second, probability theory provides the bridge between purely descriptive analysis and statistical inference, or the practice of drawing general conclusions from a single sample. Whenever we rely on one sample to represent a population, we need to be aware that our sample is one of many possible samples we might have taken. Each sample imperfectly represents the parent population, and samples vary in their representativeness. Because it's impossible to know for certain just how representative any sample is, sampling creates an unavoidable risk of drawing erroneous conclusions about a population. To manage the risk, we'll (eventually) use probability theory.

In this chapter we'll confine our attention to a few probability topics that pertain to these two uses. If you are taking an introductory statistics course, your principal text might include additional topics. The next section of this chapter selectively reviews some of the elementary vocabulary and rules of probability theory. It does not provide an exhaustive treatment but recapitulates some of the terms and ideas covered by your major text.