4.2 Correlation

• The Correlation Coefficient r• Properties of r

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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…

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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.The correlation r measures the strength of the linear relationship between two quantitative variables.

Measuring Linear Association

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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

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Correlation

The correlation coefficient r

Time to swim: = 35, sx = 0.7

Pulse rate: = 140 sy = 9.5

r does not distinguish between x & yThe 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 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 = -0.75

r = -0.75

z-score plot is the same for both plots

z for time z for pulse

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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.

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HW: Read section 4.2 on the Correlation Coefficient. Pay particular attention to the Figure 4.12…

Work the following exercises: #4.36-4.38, 4.41-4.44, 4.47-4.49

HW: Read section 4.2 on the Correlation Coefficient. Pay particular attention to the Figure 4.12…

Work the following exercises: #4.36-4.38, 4.41-4.44, 4.47-4.49