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1、1COVARIANCE,VARIANCE,ANDCORRELATIONThischapterintroducescovarianceandcorrelation,twoconceptsthatwillpreparethewayforthetreatmentofregressionanalysistocome.Asecondandequallyimportantobjectiveistoshowyouhowtomanipulateexpressionsinvolvingsamplevarianceandcovariance.Severa
2、ldetailedexamplesareprovidedtogiveyoupractice.Theyareusedveryextensivelyinfuturechaptersanditisvitalthattheybecomesecondnaturetoyou.Theysimplifythemathematicsandmaketheanalysismucheasiertofollow.1.1SampleCovarianceSamplecovarianceisameasureofassociationbetweentwovariabl
3、es.Theconceptwillbeillustratedwithasimpleexample.Table1.1showsyearsofschooling,S,andhourlyearningsin1992,indollars,Y,forasubsetof20respondentsfromtheUnitedStatesNationalLongitudinalSurveyofYouth,thedatasetwhichisusedformanyofthepracticalillustrationsandexercisesinthiste
4、xt.Sisthehighestgradecompleted,inthecaseofthosewhodidnotgoontocollege,and12plusthenumberofyearsofcollegecompleted,fortheothers.Figure1.1showsthedataplottedasascatterdiagram.Youcanseethatthereisa(weak)positiveassociationbetweenthetwovariables.TABLE1.1ObservationSYObserva
5、tionSY11517.24111715.3821615.00121212.703814.91131226.00464.501497.5051518.0015155.006126.29161221.6371219.23171612.1081818.6918125.559127.2119127.50102042.0620148.00ÓC.Dougherty1999.Allrightsreserved.Copiesmaybemadeforpersonaluse.Versionof1August1999COVARIANCE,VARIANCE
6、,ANDCORRELATION245403530252015Hourlyearnings($)105002468101214161820HighestgradecompletedFigure1.1.Hourlyearningsandschooling,20NLSYrespondentsThesamplecovariancestatisticenablesyoutosummarizethisassociationwithasinglenumber.Ingeneral,givennobservationsontwovariablesxan
7、dy,thesamplecovariancebetweenxandyisgivenby1Cov(x,y)=[(x1-x)(y1-y)+...+(xn-x)(yn-y)]n(1.1)1n=å(xi-x)(yi-y)ni=1Note:InSection1.4wewillalsodefinethepopulationcovariance.Todistinguishbetweenthetwo,wewilluseCov(x,y),withacapitalC,torefertothesamplecovarianceandpop.cov(x,y)t
8、orefertothepopulationcovariancebetweenxandy.Occasionallyitwillbeconvenienttorefertothelatterassxy.Thisconventi
