Understanding Statistical Analysis and Modeling is a text for graduate and advanced undergraduate students in the social, behavioral, or managerial sciences seeking to understand the logic of statistical analysis. Robert Bruhl covers all the basic methods of descriptive and inferential statistics in an accessible manner by way of asking and answering research questions. Concepts are discussed in the context of a specific research project and the book includes probability theory as the basis for understanding statistical inference. Instructions on using SPSS® are included so that readers focus on interpreting statistical analysis rather than calculations. Tables are used, rather than formulas, to describe the various calculations involved with statistical analysis and the exercises in the book are intended to encourage students to formulate and execute their own empirical investigations.

# Chapter 8: Coexisting Properties and Joint Probability Models

### 8.0 Learning Objectives

When a phenomenon can be described as having two properties, those properties are said to coexist in that phenomenon. In this chapter, we have the following objectives:

• to describe the construction of a probability model involving two coexisting properties;
• to describe the construction of a probability model involving two coexisting properties in which the probabilities of occurrence of one property are unrelated to—or independent of—the probabilities of occurrence of the other property; and
• to describe a measure of covariability—said to be the covariance—in a probability model in which the two coexisting properties both are quantitative in nature.

### 8.1 Motivation

In an association study, the investigator is interested in the extent to which one property of a phenomenon might be “related” to ...