Lecture 9 Correlation i Question For each of the following scatterplots state whether There is a weak or strong association between the variables and The association is positive negative or neither ii A D B E C F Try this online game http guessthecorrelation com Question For each of the following scatterplots state whetheri There is a weak or strong association between the variables andii The association is positive negative or neither Correlation and Line Fitting Which association is stronger Actually they re two plots of the same data on different scales It s hard to assess the strength of an association by eye from a scatterplot The correlation between two quantitative variables is a number between 1 and 1 denoted r that tells us the direction and strength of the linear straight line association between them Correlation and Line Fitting Exercise Find r x 3 5 7 y 4 5 3 Use Examples of correlations Examples of correlations Properties of correlation The sign of r indicates the direction of the association assuming it is linear r is between 1 and 1 A value of r near 1 or 1 indicates a strong linear association points lie close to a straight line A value of r near 0 indicates a weak linear association or no association but may also arise from a nonlinear association r will not change if we exchange explanatory and response variables express the values using different units Correlation is not resistant to outliers Properties of correlation How an outlier can distort an association Correlation Outlier Example Make scatterplot Correlation is about 0 5 Why not closer to 1 when there s such a good association for most of the data x 1 2 3 4 10 10 y 1 3 3 5 1 11 12 10 8 6 4 2 0 y 0 2 4 8 10 12 6 x Correlation Outlier Example r 0 48107 r 0 99254 x 1 2 3 4 10 10 y 1 3 3 5 1 11 10 5 0 0 10 5 0 0 10 5 0 0 10 5 0 0 5 10 5 10 r 0 25355 r 0 94868 5 10 5 10 Correlation doesn t work with nonlinear associations r has to do with how well the points fit to a straight line and with the direction of that line Correlation doesn t work with nonlinear associations Practice question Which scatterplot has the largest correlation closest to 1 A Scatterplot A B Scatterplot B C Scatterplot C D It is impossible to tell A B C Exercise couple woman s height man s height Shown Heights of 6 women who date men and of the men they date 1 2 3 4 5 6 66 64 66 65 70 65 72 68 70 68 71 65 a Make scatterplot done below Guess r is it positive or negative near 1 or 1 b Find r c If all the men were 6 inches shorter would correlation change Does the correlation tell us about whether women tend to date men taller than themselves If heights were in centimeters would correlation change If each woman dated a man exactly 3 in taller than herself what would be the correlation d e men s heights 74 72 70 68 66 64 62 60 63 64 65 66 67 68 69 70 71 women s heights Practice test question The two scatterplots show the same data except that the point at A has been added to the dataset at the bottom What can you say about the correlation coefficient of the dataset with A added A It is greater than 0 85 B It is less than 0 85 C It equals 0 85 D It is impossible to tell without computing the correlation Practice test question Shown is the calculation of the correlation of x and y with some numbers missing i 1 2 3 4 xi 0 2 3 6 yi 5 2 2 4 x 2 75 y 3 25 sx 2 5 sy 1 5 xi x sx yi y sy product 1 1 T 0 1 1 3 1 17 0 83 0 83 0 65 1 28 0 25 0 08 0 65 r Sum 0 46 1 What is T A 0 3 B 0 3 C 0 75 D 0 75 E None of the above 0 46 2 What is r A B C D E 0 115 0 153 0 None of the above