4.2 Correlation The Correlation Coefficient r Properties of r 1

Correlation  We can often see the strength of the relationship between two quantitative variables in a scatterplot, but be careful. The two figures here are both scatterplots of the same data, on different scales. The second seems to be a stronger association…  So we need a measure of association independent of the graphics…

3 A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. Linear relations are important because a straight line is a simple pattern that is quite common. Our eyes are not good judges of how strong a relationship is. Therefore, we use a numerical measure to supplement our scatterplot and help us interpret the strength of the linear relationship. The correlation r measures the strength of the linear relationship between two quantitative variables. Measuring Linear Association

4 We say a linear relationship is strong if the points lie close to a straight line and weak if they are widely scattered about a line. The following facts about r help us further interpret the strength of the linear relationship. Properties of Correlation  r is always a number between –1 and 1.  r > 0 indicates a positive association.  r < 0 indicates a negative association.  Values of r near 0 indicate a very weak linear relationship.  The strength of the linear relationship increases as r moves away from 0 toward –1 or 1.  The extreme values r = –1 and r = 1 occur only in the case of a perfect linear relationship. Properties of Correlation  r is always a number between –1 and 1.  r > 0 indicates a positive association.  r < 0 indicates a negative association.  Values of r near 0 indicate a very weak linear relationship.  The strength of the linear relationship increases as r moves away from 0 toward –1 or 1.  The extreme values r = –1 and r = 1 occur only in the case of a perfect linear relationship. Measuring Linear Association

5 Correlation

The correlation coefficient r Time to swim: = 35, s x = 0.7 Pulse rate: = 140 s y = 9.5

r does not distinguish between x & y The correlation coefficient, r, treats x and y symmetrically "Time to swim" is the explanatory variable here, and belongs on the x axis. However, in either plot r is the same (r=-0.75). r = -0.75

Changing the units of measure of variables does not change the correlation coefficient r, because we "standardize out" the units when getting z-scores. r has no unit of measure (unlike x and y ) r = z-score plot is the same for both plots z for time z for pulse

9 Cautions:  Correlation requires that both variables be quantitative.  Correlation does not describe curved relationships between variables, no matter how strong the relationship is.  Correlation is not resistant. r is strongly affected by a few outlying observations.  Correlation is not a complete summary of two-variable data. Cautions:  Correlation requires that both variables be quantitative.  Correlation does not describe curved relationships between variables, no matter how strong the relationship is.  Correlation is not resistant. r is strongly affected by a few outlying observations.  Correlation is not a complete summary of two-variable data.

10 HW: Read section 4.2 on the Correlation Coefficient. Pay particular attention to the Figure 4.12… Work the following exercises: # , , HW: Read section 4.2 on the Correlation Coefficient. Pay particular attention to the Figure 4.12… Work the following exercises: # , ,