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1、__________________________________________________Skewness,Kurtosis,andtheNormalCurveSkewnessIneverydaylanguage,theterms“skewed”and“askew”areusedtorefertosomethingthatisoutoflineordistortedononeside.Whenreferringtotheshapeoffrequencyorprobabilitydistri
2、butions,“skewness”referstoasymmetryofthedistribution.Adistributionwithanasymmetrictailextendingouttotherightisreferredtoas“positivelyskewed”or“skewedtotheright,”whileadistributionwithanasymmetrictailextendingouttotheleftisreferredtoas“negativelyskewed”
3、or“skewedtotheleft.”Skewnesscanrangefromminusinfinitytopositiveinfinity.KarlPearson(1895)firstsuggestedmeasuringskewnessbystandardizingthedifferencebetweenthemeanandthemode,thatis,.Populationmodesarenotwellestimatedfromsamplemodes,butonecanestimatethed
4、ifferencebetweenthemeanandthemodeasbeingthreetimesthedifferencebetweenthemeanandthemedian(Stuart&Ord,1994),leadingtothefollowingestimateofskewness:.Manystatisticiansusethismeasure_________________________________________________________________________
5、___________________________butwiththe‘3’eliminated,thatis,.Thisstatisticrangesfrom-1to+1.Absolutevaluesabove0.2indicategreatskewness(Hildebrand,1986).Skewnesshasalsobeendefinedwithrespecttothethirdmomentaboutthemean:,whichissimplytheexpectedvalueofthed
6、istributionofcubedzscores.Skewnessmeasuredinthiswayissometimesreferredtoas“Fisher’sskewness.”Whenthedeviationsfromthemeanaregreaterinonedirectionthanintheotherdirection,thisstatisticwilldeviatefromzerointhedirectionofthelargerdeviations.Fromsampledata,
7、Fisher’sskewnessismostoftenestimatedby:.Forlargesamplesizes(n>150),g1maybedistributedapproximatelynormally,withastandarderrorofapproximately.Whileonecouldusethissamplingdistributiontoconstructconfidenceintervalsforortestsofhypothesesaboutg1,thereisrare
8、lyanyvalueindoingso.Themostcommonlyusedmeasuresofskewness(thosediscussedhere)mayproducesurprisingresults,suchasanegativevaluewhentheshapeofthedistributionappearsskewedtotheright.There_____________________________________________________
