Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Correlation Lecture 48 Section 13 7 Robb T Koether Hampden Sydney College Summary Mon Apr 21 2008 Outline Correlation Robb T Koether 1 Introduction 2 The Correlation Coefficient 3 Calculating the Correlation Coefficient 4 Calculating r on the IT 83 5 Hidden Variables 6 Summary Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary Introduction Correlation Robb T Koether Introduction The Correlation Coefficient So far our only way to judge the strength and direction of a linear relationship has been to look at a scatterplot Calculating the Correlation Coefficient Now we will quantify those concepts in a quantify called the correlation coefficient However there are some dangers present Calculating r on the IT 83 Hidden Variables Summary 1 2 Correlation does not imply causation There may be hidden variables at work The Correlation Coefficient Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary The correlation coefficient r is a number between 1 and 1 It measures the direction and strength of the linear relationship If r 0 then the relationship is positive If r 0 then the relationship is negative The closer r is to 1 or 1 the stronger the relationship The closer r is to 0 the weaker the relationship Strong Positive Linear Association Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary In this display r is close to 1 Strong Negative Linear Association Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary In this display r is close to 1 Almost No Linear Association Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary In this display r is close to 0 Interpretation of r Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary 1 0 1 Interpretation of r Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary Strong negative correlation 1 0 8 Strong positive correlation 0 0 8 1 Interpretation of r Correlation Robb T Koether Introduction The Correlation Coefficient Weak negative correlation Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary 1 0 8 Weak positive correlation 0 2 0 0 2 0 8 1 Interpretation of r Correlation Robb T Koether Introduction The Correlation Coefficient No significant correlation Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary 1 0 8 0 2 0 0 2 0 8 1 Correlation vs Cause and Effect Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary If the value of r is close to 1 or 1 that indicates that x is a good predictor of y It does not indicate that x causes y The correlation coefficient alone cannot be used to determine cause and effect Calculating the Correlation Coefficient Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary There are many formulas for r The basic formula is P r pP x x y y P x x 2 y y 2 which can be expanded to P P P n xy x y q r q P P 2 P 2 P 2 2 n x x n y y Calculating the Correlation Coefficient Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 If we define SSX x x 2 X SSY y y 2 X SSXY x x y y then the formula becomes Hidden Variables Summary X r SSXY SSX SSY Example Correlation Robb T Koether Consider again the data Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary x 1 3 4 5 8 9 11 15 y 8 12 9 14 16 20 17 24 Example Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary We found earlier that SSX 150 SSY 206 SSXY 165 Example Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary Then compute r 165 0 9387 r p 150 206 Calculating r on the IT 83 Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary To calculate r on the TI 83 Be sure that Diagnostic is turned on Press CATALOG and select DiagnosticsOn Then follow the procedure that produces the regression line In the same window the TI 83 reports the values of r 2 and r Calculating r on the IT 83 Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Use the TI 83 to calculate r in the preceding example Find r for English SOL Passing Rate vs Graduation Rate Find r for the Free Lunch vs Graduation Rate Calculating r on the IT 83 Find r for the Teachers Salary vs Graduation Rate Hidden Variables Find r for the S T Ratio vs Graduation Rate Summary Calculating r on the IT 83 Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary For the Free Lunch vs Graduation Rate we get r 0 8544 For English SOL Passing Rate vs Graduation Rate we get r 0 7500 For the Teachers Salary vs Graduation Rate we get r 0 0817 For the S T Ratio vs Graduation Rate we get r 0 0022 Calculating r on the IT 83 Correlation Robb T Koether Introduction The Correlation Coefficient Calculating the Correlation Coefficient Calculating r on the IT 83 Hidden Variables Summary For the Free Lunch vs Graduation Rate we get r 0 8544 For English SOL Passing Rate vs Graduation Rate we get r 0 7500 For the Teachers Salary vs Graduation Rate we get r 0 0817 For the S T Ratio vs
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