Chapter 2: Examining Relationships

Introduction

We are often interested in comparisons among several distributions or relationships among several variables. A study of data often leads us to ask whether there is a correlation between two variables that are closely linked in the data.

Types of variables

To study a relationship between two variables, we need to measure both variables on the same individuals. But we need to be cautious of possible lurking variables, i.e., other variables not being studied by which may influence a possible relationship between the explanatory and response variables.

Another caution: a relationship between two variables may not be a causal one.

General procedure for studying possible relationships between two variables:

1. Plot the data and add numerical summaries
2. Look for overall pattern and deviations from those patterns.
3. Use mathematical models to describe regular patterns

Section 2.1: Scatterplots

The most common way to display the relation between two quantitative variables. The explanatory–response pair is graphed on a set of axes analogous to graphing the point (x, y). If there is no explanatory-response distinction, either variable can go on the horizontal axis.

Here is how to construct a scatterplot on the TI-83.

Interpreting scatterplots

The overall pattern of scatterplots by examining the form, direction and strength. Thus, to describe the overall pattern of scatterplots look for:

• Form, e.g.,
  • clusters – groups of data clumped together with gaps in between
  • outliers – an individual value that falls outside the pattern
  • linear, quadratic, exponential, logarithmic, etc.?
• Direction, e.g.,
  • Positively associated – high values of one variable correspond to high values of the other variable
  • Negatively associated – high values of one variable correspond to low values of the other variable
• Strength, i.e.,
  • how closely the points follow a clear pattern. We'll learn a method to quantify the strength in section 2.2

Adding categorical variables to scatterplots

We can add a categorical variable to a scatterplot by using different colors or symbols to plot points. See examples 2.5 and 2.6 on pp. 87–89 of the text.