Scatterplot

Correlation and Regression

Correlation is a good way to examine the linear relationships between two variables, for example, a person’s weight and height, or a student’s high school grade point average, and his SAT/ACT score. The strength of the relationship between two variables is usually described in terms of the correlation coefficient (also known as the Pearson correlation coefficient), which ranges from −1 to 1. A scatterplot is often used to provide researchers the graphic view of what tends to happen to one score when another score increases/decreases.

Figure 1 Bivariate Scatterplot

Figure 2 Trivariate Scatterplot

When a set of variables is at hand, it is fairly easy to draw the scatterplot by plotting one score on the vertical axis and the other on the horizontal axis. Figures 4 through 7 provide examples of variable pairs with correlations of −0.8, −0.3, 0.3, and 0.8, respectively.

A positive correlation describes the situation in which an increase in variable X is associated with an increase in variable Y, whereas a negative correlation implies that an increase in variable X is associated with a decrease in variable Y. But a correlation of 0.8 is not stronger than a correlation of −0.8. It is the magnitude that matters. They simply work in opposite directions.

In certain extreme situations, all of the dots fall on a straight line. This is called a perfect correlation. Figures 8 and 9 show perfect positive and perfect negative correlations, respectively.

The line interpolating all the dots in Figures 8 and 9 is considered as the line of best fit. In a real-world data analysis, the line of best fit does not necessarily go through all the dots in a scatterplot. Essentially, the line of best fit means that a line is closest to most of the dots or a line is as close to most of the dots as [Page 1468]possible. The vertical distances between the line and those dots are called residuals. In statistics, the least squares method is used to get a regression line, which minimizes the sum of the squared residuals. Generally, the line of best fit is also referred to as the regression line.

...