IntroductionThe Correlation CoefficientCalculating the Correlation CoefficientCalculating r on the IT-83Hidden VariablesSummaryCorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCorrelationLecture 48Section 13.7Robb T. KoetherHampden-Sydney CollegeMon, Apr 21, 2008CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryOutline1Introduction2The Correlation Coefficient3Calculating the Correlation Coefficient4Calculating r on the IT-835Hidden Variables6SummaryCorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryIntroductionSo far our only way to judge the strength and directionof a linear relationship has been to look at a scatterplot.Now we will quantify those concepts in a quantify calledthe correlation coefficient.However, there are some dangers present.1“Correlation does not imply causation.”2There may be hidden variables at work.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryThe Correlation CoefficientThe correlation coefficient r is a number between −1and +1.It measures the direction and strength of the linearrelationship.If r > 0, then the relationship is positive. If r < 0, thenthe relationship is negative.The closer r is to +1 or −1, the stronger therelationship.The closer r is to 0, the weaker the relationship.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryStrong Positive Linear AssociationIn this display, r is close to +1CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryStrong Negative Linear AssociationIn this display, r is close to −1CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryAlmost No Linear AssociationIn this display, r is close to 0CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryInterpretation of r-1 0 1CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryInterpretation of r-1 0 10.8-0.8StrongnegativecorrelationStrongpositivecorrelationCorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryInterpretation of r-1 0 10.8-0.8WeaknegativecorrelationWeakpositivecorrelation0.2-0.2CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryInterpretation of r-1 0 10.8-0.8Nosignificantcorrelation-0.2 0.2CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCorrelation vs. Cause and EffectIf the value of r is close to +1 or −1, that indicates thatx is a good predictor of y.It does not indicate that x causes y.The correlation coefficient alone cannot be used todetermine cause and effect.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCalculating the Correlation CoefficientThere are many formulas for r.The basic formula isr =P(x − x)(y − y )pP(x − x)2P(y − y)2which can be expanded tor =nPxy −PxPyqnPx2− (Px)2qnPy2− (Py)2.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCalculating the Correlation CoefficientIf we defineSSX =X(x − x)2,SSY =X(y − y)2,SSXY =X(x − x)(y − y ),then the formula becomesr =SSXY√SSX · SSY.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryExampleConsider again the datax y1 83 124 95 148 169 2011 1715 24CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryExampleWe found earlier thatSSX = 150SSY = 206SSXY = 165CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryExampleThen compute r:r =165p(150)(206)= 0.9387.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCalculating r on the IT-83To 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 regressionline.In the same window, the TI-83 reports the values of r2and r.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCalculating r on the IT-83Use the TI-83 to calculate r in the preceding example.Find r for English SOL Passing Rate vs. GraduationRate.Find r for the Free Lunch vs. Graduation Rate.Find r for the Teachers’ Salary vs. Graduation Rate.Find r for the S/T Ratio vs. Graduation Rate.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCalculating r on the IT-83For the Free Lunch vs. Graduation Rate, we getr = −0.8544.For English SOL Passing Rate vs. Graduation Rate, weget r = 0.7500.For the Teachers’ Salary vs. Graduation Rate, we getr = 0.0817.For the S/T Ratio vs. Graduation Rate, we getr = 0.0022.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the IT-83HiddenVariablesSummaryCalculating r on the IT-83For the Free Lunch vs. Graduation Rate, we getr = −0.8544.For English SOL Passing Rate vs. Graduation Rate, weget r = 0.7500.For the Teachers’ Salary vs. Graduation Rate, we getr = 0.0817.For the S/T Ratio vs. Graduation Rate, we getr = 0.0022.CorrelationRobb T.KoetherIntroductionTheCorrelationCoefficientCalculatingtheCorrelationCoefficientCalculating ron the
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