## Introduction

Correlation is, by and large, a familiar concept. Pairs of observations, X and Y, are examined to see if they “go together.” For instance, we generally accept that heart rate increases with physical exertion, that weight loss will be related to caloric intake, or that weight loss will increase as the frequency of exercise increases. These variables are correlated in that the value of one variable is systematically related to values of the second variable, although not perfectly and to differing degrees. With a strong correlation, we can infer something about the second value by knowing the first. In Chapter 28, chi-square was used to look at the association between nominal variables. The purpose of this chapter is to introduce several correlation statistics for use with ordinal and ratio/interval data that can be applied to a variety of exploratory research designs.

## Concepts of Correlation

The basic purpose of correlation is to describe the association between two variables. These bivariate relationships are defined by the shared pattern within the two sets of data. For example, if we measure heart rate and physical exertion in several individuals, we would expect to see higher scores in heart rate with higher scores in exertion, and the same for low scores. This is a positive relationship. If we look at the association between weight and exercise duration, we might expect to see lower scores in weight with higher exercise time in a negative relationship.

### Scatter Plots

Visualizing these data can help to clarify patterns using a scatter plot, as shown in Figure 29-1. Each dot represents the intersection of a pair of related observations. The points in Plot A show a pattern in which the values of Y increase in exact proportion to the values of X, in a perfect positive relationship with data points falling on a straight line. In Plot B the data demonstrate a perfect negative relationship, with lower values of Y associated with higher values of X.

###### Figure 29–1

Examples of scatterplots with different degrees of correlation. Plots A, B, C and D represent strong positive correlations. Plot E shows a moderately strong negative correlation. Plot F shows a scatterplot with a random pattern and a correlation close to zero. More realistic results are shown in the other plots. The closer the points approximate a straight line, the stronger the association. Plots C and D show a positive relationship, but the correlation in Plot D is not as strong. Plot E shows a negative relationship with greater variance from a straight line. In Plot F the points have a seemingly random pattern, reflecting no relationship. Positive correlations are sometimes called direct relationships. Negative correlations may also be described as indirect or inverse relationships.

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