Copyright © 2011 Pearson Education, Inc. Association between Quantitative Variables Chapter 6

6.1 Scatterplots Is household natural gas consumption associated with climate?  Annual household natural gas consumption measured in thousands of cubic feet (MCF)  Climate as measured by the National Weather Service using heating degree days (HDD) Copyright © 2011 Pearson Education, Inc. 3 of 30

6.1 Scatterplots Association between Numerical Variables  A graph displaying pairs of values as points on a two-dimensional grid  The explanatory variable is placed on the x-axis  The response variable is placed on the y-axis Copyright © 2011 Pearson Education, Inc. 4 of 30

6.1 Scatterplots Scatterplot of Natural Gas Consumption (y) versus Heating Degree-Days (x) Copyright © 2011 Pearson Education, Inc. 5 of 30

6.2 Association in Scatterplots Visual Test for Association  Compare the original scatterplot to others that randomly match the coordinates  If you can pick the original out as having a pattern, then there is an association Copyright © 2011 Pearson Education, Inc. 6 of 30

6.2 Association in Scatterplots Describing Association 1. Direction. Does it trend up or down? 2. Curvature. Is the pattern linear or curved? 3. Variation. Are the points tightly clustered around the trend? 4. Outliers. Is there something unexpected? Copyright © 2011 Pearson Education, Inc. 7 of 30

6.2 Association in Scatterplots Gas Consumption vs. Heating Degree Days 1. Direction: Positive. 2. Curvature: Linear. 3. Variation: Considerable scatter. 4. Outliers: None apparent. Copyright © 2011 Pearson Education, Inc. 8 of 30

6.3 Measuring Association Covariance  A measure that quantifies the linear association  Depends on units of measurement and is therefore difficult to interpret Copyright © 2011 Pearson Education, Inc. 9 of 30

6.3 Measuring Association Correlation (r)  Standardized measure of the strength of the linear association (has no units)  Always between -1 and +1  Easy to interpret Copyright © 2011 Pearson Education, Inc. 10 of 30

6.3 Measuring Association Gas Consumption and Heating Degree Days Cov (HDD, Gas) = 63,357 HDD X MCF Corr (HDD, Gas) = 0.55 The association is positive and moderate. Copyright © 2011 Pearson Education, Inc. 11 of 30

6.3 Measuring Association Scatterplot for r = 1 Copyright © 2011 Pearson Education, Inc. 12 of 30

6.3 Measuring Association Scatterplot for r = Copyright © 2011 Pearson Education, Inc. 13 of 30

6.3 Measuring Association Scatterplot for r = 0.75 Copyright © 2011 Pearson Education, Inc. 14 of 30

6.3 Measuring Association Scatterplot for r = Copyright © 2011 Pearson Education, Inc. 15 of 30

6.3 Measuring Association Scatterplot for r = 0 Copyright © 2011 Pearson Education, Inc. 16 of 30

6.3 Measuring Association Correlation Matrix A table showing all of the correlations among a set of numerical variables. Copyright © 2011 Pearson Education, Inc. 17 of 30

6.4 Summarizing Association with a Line Expressed using z-scores Slope-Intercept Form Copyright © 2011 Pearson Education, Inc. 18 of 30

6.4 Summarizing Association with a Line Line Relating Gas Consumption (y) to Heating Degree Days (x) Copyright © 2011 Pearson Education, Inc. 19 of 30

6.4 Summarizing Association with a Line Lines and Prediction  Use the correlation line to customize an ad for estimated savings from insulation based on climate.  For a home in a cold climate (HDD = 8,800), the predicted gas consumption is 154 MCF.  At $10 / MCF, the predicted cost is $1,540.  Assuming that insulation saves 30% on gas bill, estimated savings is $462. Copyright © 2011 Pearson Education, Inc. 20 of 30

6.5 Spurious Correlation Lurking Variables  Scatterplots and correlation reveal association, not causation  Spurious correlations result from underlying lurking variables Copyright © 2011 Pearson Education, Inc. 21 of 30

6.5 Spurious Correlation Checklist: Covariance and Correlation  Numerical variables  No obvious lurking variables  Linear  Outliers Copyright © 2011 Pearson Education, Inc. 22 of 30

4M Example 6.1: LOCATING A NEW STORE Motivation Is it better to locate a new retail outlet far from competing stores? Copyright © 2011 Pearson Education, Inc. 23 of 30

4M Example 6.1: LOCATING A NEW STORE Method Is there an association between sales at the retail outlets and distance to nearest competitor? For 55 stores in the chain, data are gathered for total sales in the prior year and distance in miles from the nearest competitor. Copyright © 2011 Pearson Education, Inc. 24 of 30

4M Example 6.1: LOCATING A NEW STORE Mechanics Copyright © 2011 Pearson Education, Inc. 25 of 30

4M Example 6.1: LOCATING A NEW STORE Mechanics Compute the correlation between sales and distance to be r = Copyright © 2011 Pearson Education, Inc. 26 of 30

4M Example 6.1: LOCATING A NEW STORE Message The data show a strong, positive linear association between distance to the nearest competitor and sales. It is better to locate a new store far from its competitors. Copyright © 2011 Pearson Education, Inc. 27 of 30

Best Practices  To understand the relationship between two numerical variables, start with a scatterplot.  Look at the plot, look at the plot, look at the plot.  Use clear labels for the scatterplot. Copyright © 2011 Pearson Education, Inc. 28 of 30

Best Practices (Continued)  Describe a relationship completely.  Consider the possibility of lurking variables.  Use a correlation to quantify the association between two numerical variables that are linearly related. Copyright © 2011 Pearson Education, Inc. 29 of 30

Pitfalls  Don’t use the correlation if data are categorical.  Don’t treat association and correlation as causation.  Don’t assume that a correlation of zero means that the variables are not associated.  Don’t assume that a correlation near -1 or +1 means near perfect association. Copyright © 2011 Pearson Education, Inc. 30 of 30