# Chapter 23: Correlation

The statistical procedures we have described thus far have all focused on the comparison of a measured dependent variable across categories of an independent variable. These procedures are generally applied to experimental and quasi-experimental designs for the purpose of group comparisons. We will now begin to examine procedures for exploratory analyses, where the purpose of the research question is to evaluate the relationship between two measured variables. Where statistical tests of group differences address the question "Is group A different from group B?" or "Does this treatment cause this outcome?", measures of **correlation** ask, "What is the relationship between A and B?" or "Does variable A increase with variable B?"

The concept of correlation is, by and large, a familiar one. Pairs of observations, *X* and *Y*, are examined to see if they tend to "go together." For instance, we generally accept that taller people tend to weigh more than shorter people, that children resemble their parents in intelligence, and that heart rate increases with physical exertion. These variables are correlated, in that the value of one variable (*X*) is associated with the value of the other variable (*Y*). With a strong correlation, we can infer something about the second value by knowing the first. Correlation can be applied to paired observations on two different variables, such as heart rate and level of exertion, or to one variable measured on two occasions, such as intelligence of a parent and child.

**Correlation coefficients** are used to quantitatively describe the strength and direction of a relationship between two variables. The purpose of this chapter is to introduce several types of correlation coefficients that can be applied to a variety of exploratory research designs and types of data. The most commonly reported measure is the Pearson product-moment coefficient of correlation, for use when both *X* and *Y* are on the interval or ratio scales. We include procedures for correlating ranked data using the Spearman rho (*r*_{s}) and several correlation methods for use with data in the form of dichotomies.

It is often useful to examine a statistical relationship by first creating a **scatter diagram** or **scatter plot**, as shown in Figure 23.1. In a scatter plot each point (dot) represents the intersection of a pair of related observations. With a sufficient number of data points, a scatter plot can visually clarify the strength and shape of a relationship. For instance, the points in Figure 23.1A show a pattern in which the values of *Y* increase in exact proportion to the values of *X*. This is considered a perfect positive relationship, with data points falling on a straight line. In Figure 23.IB, the data demonstrate a negative slope in a perfect negative relationship, with lower values of *Y* associated with higher values of *X*.