Correlation

Scatterplots A scatterplot is a graph that shows the relationship between two quantitative variables. The one here shows the relationship between gross domestic product per person and the life expectancy for many countries. In this chapter we will investigate scatterplots and the patterns that prevail.

Our previous example did not show a linear pattern Our previous example did not show a linear pattern. We will concentrate more on linear relationships. The pattern of the points will form a cigar shape like on the left below. The left shows a linear relationship. We say the variables are linearly related or linearly correlated. The scatterplot on the right does not show a linear pattern; the points have a curved pattern.

What is the strength and direction of these two plots? When we look at scatterplots, we will look for 1.) overall pattern 2.) deviations or outliers 3.) direction and strength Let’s discuss strength and direction in more detail. What is the strength and direction of these two plots?

Strength describes how well the points follow the straight line pattern. Notice both of these plots follow the straight line pretty well, so we’d say these linear relationships are strong. Direction tells us if the points are going up or down as you go from left to right on the graph. The left plot has positive correlation (slopes up) and the right has negative correlation (slopes down).

Correlation coefficient The correlation coefficient, denoted by r, describes how well the straight line fits the pattern of the points. Compare the three scatterplots below. Notice the first plot’s points follow the line closely. The second plot’s points are linear but do not follow the line as closely. The third plot’s points are not in any linear pattern whatsoever.

r = .7 r = .9 r = 0 If r = 1, then the points fall exactly on a straight line sloping up, left to right. If r = 0, then the points do not follow a straight-line relationship at all. If r is somewhere between 0 and 1, then the points form an upward sloping pattern. The closer r is to 1, the closer the points follow a straight line.

r = -.9 r = -.7 r = -.5 If r = -1, then the points fall exactly on a straight line sloping down, left to right. If r = 0, then the points do not follow a straight-line relationship at all. If r is somewhere between -1 and 0, then the points form a downward sloping pattern. The closer r is to -1, the closer the points follow a straight line.