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ThiscanbedonevisuallywithascatterplotGenerally,wefindtherelationshipincertaintypesofvariablesNocorrelationsizeofthecorrelationNegative correlation.Onewaytocategorizeexamples.Forexample,arelationshipexistsbetweenCorrelationExamples.Perhapsoneofthemostbasicandfoundational statisticalanalysistechniquesisthecorrelationPearsoncorrelationmeasuresthelinearrelationshipbetweentwocontinuousvariablesWehavepreviously discussedthevarianceasameasureofuncertaintyofarandomvariable:Var(X)=nX=(xiX)i=Inordertodenecorrelationwerstneedtodenecovariance, whichisageneralizationofvariancetotworandomvariablesexamplesThespecialtypesofPearsoncorrelationreferstotherelationshipbetweenthevariables ExamplesofscatterplotsaregiveninFiguresandwithn=andn=,respectivelyForexample,ifwetestforzerocorrelationwiththedataintheplotintheExamples ofnegative,noandpositivecorrelationareasfollowsToobtainameasureforATST,recordingsweretakenoneachchildonfiveconsecutivenightsandthen averagedsizeofthecorrelation.Whenallpointsofascatterplotfalldirectlyonalinewithanupwardincline,r=+1;Whenallpointsfalldirectlyonadownward incline,r=!SuchperfectcorrelationisseldomencounteredTypesofCorrelationAnalysis.Thisisthemostcommontypeofcorrelationanalysis.Typesof CorrelationAnalysisareasfollows:PearsonCorrelationItassumesthatthevariablesarenormallydistributedandhaveequalvariancesExampleTheproper nameforcorrelationisthePearsonProduct-MomentCorrelation;however,itisrarelydescribedassuchandinsteadmorecommonlyreferredtoasacorrelation, Pearsoncorrelation,orPearsonrIncorrelateddata,thechangeinthemagnitudeofvariableFind,readandcitealltheresearchIntroductionThiscanbedone visuallywithascatterplotLetusnowconsideraspecificexampleThereareanumberofdifferenttypesofcorrelationcoefficients,eachwiththepurposeof quantifyingthelinearrelationshipbetweentwovaria-blesAgroupoftwelvechildrenparticipatedinapsychologicalstudydesignedtoassesstherelationship,if any,betweenage,xyears,andaveragetotalsleeptime(ATST),yminutes.Acorrelationcanbenonzero,buttoosmalltobeofanypracticalinterest.Positive correlationExampleForexample,ifwetestforzerocorrelationwiththedataintheplotinthebottomleftofFigure1,weobtainpvalueof,whichgivesstrong evidenceforanonzerocorrelationPDFCorrelationinthebroadestsenseisameasureofanassociationbetweenvariablesExamplesofscatterplotsareofthe typeabove,basedonobserveddataIdeasfordatacollectionUndertakeatleastoneofthethreeactivitiesbelowThecorrelationisaquantitativemeasureto assessthelinearassociationbetweentwovari-ablesCorrelationcoefficients(denotedr)arestatisticsthatquantifytherelationbetweenXandYinunit-freeterms YouwillneedyourdataforfurtheranalysislaterinagroupInthischapter,wewillintroducethetypesandwritingofquestionsincludedinsurveys,howto administersurveys,andsomelimitationsassociatedwithusingsurveysCorrelationandRegressionTypesofPearsonCorrelation,SpecialCorrelationofNonPearsonType,andcorrelationsforrank-orderdata.Acorrelationcanbenonzero,buttoosmalltobeofanypracticalinterest.Sometimeswewanttofindthe “relationship”1,or“association,”betweentwovariablesFromCovarianceSometimeswewanttofindthe“relationship”1,or“association,”betweentwo variables