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1、Skewness,Kurtosis,andtheNormalCurveãCopyright2005,KarlL.Wuensch-Allrightsreserved.SkewnessIneverydaylanguage,theterms“skewed”and“askew”areusedtorefertosomethingthatisoutoflineordistortedononeside.Whenreferringtotheshapeoffrequencyorprobabilitydistributions,“skewness”referstoasymmetryofthe
2、distribution.Adistributionwithanasymmetrictailextendingouttotherightisreferredtoas“positivelyskewed”or“skewedtotheright,”whileadistributionwithanasymmetrictailextendingouttotheleftisreferredtoas“negativelyskewed”or“skewedtotheleft.”Skewnesscanrangefromminusinfinitytopositiveinfinity.KarlP
3、earson(1895)firstsuggestedmeasuringskewnessbystandardizingthedifferencebetweenthemeanandthemode,thatis,.Populationmodesarenotwellestimatedfromsamplemodes,butonecanestimatethedifferencebetweenthemeanandthemodeasbeingthreetimesthedifferencebetweenthemeanandthemedian(Stuart&Ord,1994),leading
4、tothefollowingestimateofskewness:.Manystatisticiansusethismeasurebutwiththe‘3’eliminated,thatis,.Thisstatisticrangesfrom-1to+1.Absolutevaluesabove0.2indicategreatskewness(Hildebrand,1986).Skewnesshasalsobeendefinedwithrespecttothethirdmomentaboutthemean:,whichissimplytheexpectedvalueofthe
5、distributionofcubedzscores.Skewnessmeasuredinthiswayissometimesreferredtoas“Fisher’sskewness.”Whenthedeviationsfromthemeanaregreaterinonedirectionthanintheotherdirection,thisstatisticwilldeviatefromzerointhedirectionofthelargerdeviations.Fromsampledata,Fisher’sskewnessismostoftenestimated
6、by:.Forlargesamplesizes(n>150),g1maybedistributedapproximatelynormally,withastandarderrorofapproximately.Whileonecouldusethissamplingdistributiontoconstructconfidenceintervalsforortestsofhypothesesaboutg1,thereisrarelyanyvalueindoingso.Themostcommonlyusedmeasuresofskewness(thosediscussedh
7、ere)mayproducesurprisingresults,suchasanegativevaluewhentheshapeofthedistributionappearsskewedtotheright.Theremaybesuperioralternativemeasuresnotcommonlyused(Groeneveld&Meeden,1984).8Itisimportantforbehavioralresearcherstonoticeskewnesswhenitappearsintheirdata.Great