skewnesskurtosis峰度,偏度介紹

Skewness,Kurtosis,andtheNormalCurve?

Skewness

Ineverydaylanguage,theterms“skewed”and“askew”areusedtorefertosomethingthatisoutoflineordistortedononeside.Whenreferringtotheshapeoffrequencyorprobabilitydistributions,“skewness”referstoasymmetryofthedistribution.Adistributionwithanasymmetrictailextendingouttotherightisreferredtoas“positivelyskewed”or“skewedtotheright,”whileadistributionwithanasymmetrictailextendingouttotheleftisreferredtoas“negativelyskewed”or“skewedtotheleft.”Skewnesscanrangefromminusinfinitytopositiveinfinity.

KarlPearson(1895)firstsuggestedmeasuringskewnessbystandardizingthedifferencebetweenthemeanandthemode,thatis,sk=—_mode.Populationmodeso

,_3(M一median)

sk=

est s

arenotwellestimatedfromsamplemodes,butonecanestimatethedifferencebetweenthemeanandthemodeasbeingthreetimesthedifferencebetweenthemeanandthemedian(Stuart&Ord,1994),leadingtothefollowingestimateofskewness:

Manystatisticiansusethismeasurebutwiththe‘3’eliminated,

(M-median)

thatis,sk= .Thisstatisticrangesfrom-1to+1.Absolutevaluesabove

s

0.2indicategreatskewness(Hildebrand,1986).

Skewnesshasalsobeendefinedwithrespecttothethirdmomentaboutthe

mean:y=?(X-~^)3-,whichissimplytheexpectedvalueofthedistributionofcubedz1 no3

scores.Skewnessmeasuredinthiswayissometimesreferredtoas“Fisher’sskewness.”Whenthedeviationsfromthemeanaregreaterinonedirectionthanintheotherdirection,thisstatisticwilldeviatefromzerointhedirectionofthelargerdeviations.Fromsampledata,Fisher’sskewnessismostoftenestimatedby:

Forlargesamplesizes(n>150),g1maybedistributed

nEz3g=

(n-1)(n-2)

approximatelynormally,withastandarderrorofapproximatelyc6/n.Whileonecouldusethissamplingdistributiontoconstructconfidenceintervalsforortestsofhypothesesabouty1,thereisrarelyanyvalueindoingso.

Themostcommonlyusedmeasuresofskewness(thosediscussedhere)mayproducesurprisingresults,suchasanegativevaluewhentheshapeofthedistribution

appearsskewedtotheright.Theremaybesuperioralternativemeasuresnotcommonlyused(Groeneveld&Meeden,1984).

Itisimportantforbehavioralresearcherstonoticeskewnesswhenitappearsintheirdata.Greatskewnessmaymotivatetheresearchertoinvestigateoutliers.Whenmakingdecisionsaboutwhichmeasureoflocationtoreport(meansbeingdrawninthedirectionoftheskew)andwhichinferentialstatistictoemploy(onewhichassumesnormalityoronewhichdoesnot),oneshouldtakeintoconsiderationtheestimatedskewnessofthepopulation.Normaldistributionshavezeroskewness.Ofcourse,adistributioncanbeperfectlysymmetricbutfarfromnormal.Transformationscommonlyemployedtoreduce(positive)skewnessincludesquareroot,log,andreciprocaltransformations.

AlsoseeSkewnessandtheRelativePositionsofMean,Median,andMode

Kurtosis

KarlPearson(1905)definedadistribution’sdegreeofkurtosisas^=P2-3,

E(X一u)4

whereP=— -,theexpectedvalueofthedistributionofZscoreswhichhave

no4

c_ n(n+1)ZZ4 _ 3(n-1)2

g2—(n-1)(n-2)(n-3)-(n-2)(n-3)

beenraisedtothe4thpower.P2isoftenreferredtoas“Pearson’skurtosis,”andP2-3(oftensymbolizedwithy2)as“kurtosisexcess”or“Fisher’skurtosis,”eventhoughitwasPearsonwhodefinedkurtosisasP2-3.Anunbiasedestimatorfory2is

Forlargesamplesizes(n>1000),g2maybe

distributedapproximatelynormally,withastandarderrorofapproximately<24/n(Snedecor,&Cochran,1967).Whileonecouldusethissamplingdistributiontoconstructconfidenceintervalsforortestsofhypothesesabouty2,thereisrarelyanyvalueindoingso.

Pearson(1905)introducedkurtosisasameasureofhowflatthetopofasymmetricdistributioniswhencomparedtoanormaldistributionofthesamevariance.Hereferredtomoreflat-toppeddistributions(y2<0)as“platykurtic,”lessflat-toppeddistributions(y2>0)as“l(fā)eptokurtic,”andequallyflat-toppeddistributionsas“mesokurtic”(y2六0).Kurtosisisactuallymoreinfluencedbyscoresinthetailsofthedistributionthanscoresinthecenterofadistribution(DeCarlo,1967).Accordingly,itisoftenappropriatetodescribealeptokurticdistributionas“fatinthetails”andaplatykurticdistributionas“thininthetails.”

Student(1927,Biometrika,19,160)publishedacutedescriptionofkurtosis,whichIquotehere:“Platykurticcurveshaveshorterftails'thanthenormalcurveoferrorandleptokurticlongerftails.’Imyselfbearinmindthemeaningofthewordsbytheabovememoriatechnica,wherethefirstfigurerepresentsplatypusandthesecondkangaroos,notedforlepping.”Pleasepointyourbrowsertomembers.aol./jeff570/k.html,scrolldownto“kurtosis,”andlookatStudent’sdrawings.

Moors(1986)demonstratedthatp=Var(Z2)+1.Accordingly,itmaybebesttotreatkurtosisastheextenttowhichscoresaredispersedawayfromtheshouldersofadistribution,wheretheshouldersarethepointswhereZ2=1,thatis,Z=±1.BalandaandMacGillivray(1988)wrote“itisbesttodefinekurtosisvaguelyasthelocation-andscale-freemovementofprobabilitymassfromtheshouldersofadistributionintoitscentreandtails.”Ifonestartswithanormaldistributionandmovesscoresfromtheshouldersintothecenterandthetails,keepingvarianceconstant,kurtosisisincreased.Thedistributionwilllikelyappearmorepeakedinthecenterandfatterinthetails,likea

6、Laplacedistribution(y2=3)orStudentstwithfewdegreesoffreedom(y2=-f~4).

Startingagainwithanormaldistribution,movingscoresfromthetailsandthecentertotheshoulderswilldecreasekurtosis.Auniformdistributioncertainlyhasaflattop,withy=-1.2,buty2canreachaminimumvalueof-2whentwoscorevaluesareequallyprobablyandallotherscorevalueshaveprobabilityzero(arectangularUdistribution,thatis,abinomialdistributionwithn=1,p=.5).OnemightobjectthattherectangularUdistributionhasallofitsscoresinthetails,butcloserinspectionwillrevealthatithasnotails,andthatallofitsscoresareinitsshoulders,exactlyonestandarddeviationfromitsmean.Valuesofg2lessthanthatexpectedforanuniformdistribution(-1.2)maysuggestthatthedistributionisbimodal(Darlington,1970),butbimodaldistributionscanhavehighkurtosisifthemodesaredistantfromtheshoulders.

OneleptokurticdistributionweshalldealwithisStudent’stdistribution.Thekurtosisoftisinfinitewhendf<5,6whendf=5,3whendf=6.Kurtosisdecreasesfurther(towardszero)asdfincreaseandtapproachesthenormaldistribution.

Kurtosisisusuallyofinterestonlywhendealingwithapproximatelysymmetricdistributions.Skeweddistributionsarealwaysleptokurtic(Hopkins&Weeks,1990).Amongtheseveralalternativemeasuresofkurtosisthathavebeenproposed(noneofwhichhasoftenbeenemployed),isonewhichadjuststhemeasurementofkurtosistoremovetheeffectofskewness(Blest,2003).

Thereismuchconfusionabouthowkurtosisisrelatedtotheshapeofdistributions.Manyauthorsoftextbookshaveassertedthatkurtosisisameasureofthepeakednessofdistributions,whichisnotstrictlytrue.

Itiseasytoconfuselowkurtosiswithhighvariance,butdistributionswithidenticalkurtosiscandifferinvariance,anddistributionswithidenticalvariancescandifferinkurtosis.Herearesomesimpledistributionsthatmayhelpyouappreciatethatkurtosisis,inpart,ameasureoftailheavinessrelativetothetotalvarianceinthedistribution(rememberthe"0/inthedenominator).

Table1.

Kurtosisfor7SimpleDistributionsAlsoDifferinginVariance

X

freqA

freqB

freqC

freqD

freqE

freqF

freqG

05

20

20

20

10

05

03

01

10

00

10

20

20

20

20

20

15

20

20

20

10

05

03

01

Kurtosis

-2.0

-1.75

-1.5

-1.0

0.0

1.33

8.0

Variance

25

20

16.6

12.5

8.3

5.77

2.27

Platykurtic

Leptokurtic

WhenIpresentedthesedistributionstomycolleaguesandgraduatestudentsandaskedthemtoidentifywhichhadtheleastkurtosisandwhichthemost,allsaidAhasthemostkurtosis,Gtheleast(exceptingthosewhorefusedtoanswer).ButinfactAhastheleastkurtosis(-2isthesmallestpossiblevalueofkurtosis)andGthemost.Thetrickistodoamentalfrequencyplotwheretheabscissaisinstandarddeviationunits.InthemaximallyplatykurticdistributionA,whichinitiallyappearstohaveallitsscoresinitstails,noscoreismorethanoneoawayfromthemean-thatis,ithasnotails!IntheleptokurticdistributionG,whichseemsonlytohaveafewscoresinitstails,onemustrememberthatthosescores(5&15)aremuchfartherawayfromthemean(3.3o)thanarethe5’s&15,sindistributionA.Infact,inGninepercentofthescoresaremorethanthreeofromthemean,muchmorethanyouwouldexpectinamesokurticdistribution(likeanormaldistribution),thusGdoesindeedhavefattails.

IfyouwereyoutoaskSAStocomputekurtosisontheAscoresinTable1,youwouldgetavaluelessthan-2.0,lessthanthelowestpossiblepopulationkurtosis.Why?SASassumesyourdataareasampleandcomputestheg2estimateofpopulationkurtosis,whichcanfallbelow-2.0.

SuneKarlsson,oftheStockholmSchoolofEconomics,hasprovidedmewiththefollowingmodifiedexamplewhichholdsthevarianceapproximatelyconstant,makingitquiteclearthatahigherkurtosisimpliesthattherearemoreextremeobservations(orthattheextremeobservationsaremoreextreme).Itisalsoevidentthatahigherkurtosisalsoimpliesthatthedistributionismorefsingle-peaked)(thiswouldbeevenmoreevidentifthesumofthefrequencieswasconstant).Ihavehighlightedtherowsrepresentingtheshouldersofthedistributionsothatyoucanseethattheincreaseinkurtosisisassociatedwithamovementofscoresawayfromtheshoulders.

Table2.

KurtosisforSevenSimpleDistributionsNotDifferinginVariance

X

Freq.A

Freq.B

Freq.C

Freq.D

Freq.E

Freq.F

Freq.G

-6.6

0

0

0

0

0

0

1

-0.4

0

0

0

0

0

3

0

1.3

0

0

0

0

5

0

0

2.9

0

0

0

10

0

0

0

3.9

0

0

20

0

0

0

0

4.4

0

20

0

0

0

0

0

5

20

0

0

0

0

0

0

10

0

10

20

20

20

20

20

15

20

0

0

0

0

0

0

15.6

0

20

0

0

0

0

0

16.1

0

0

20

0

0

0

0

17.1

0

0

0

10

0

0

0

18.7

0

0

0

0

5

0

0

20.4

0

0

0

0

0

3

0

26.6

0

0

0

0

0

0

1

Kurtosis

-2.0

-1.75

-1.5

-1.0

0.0

1.33

8.0

Variance

25

25.1

24.8

25.2

25.2

25.0

25.1

Whileisunlikelythatabehavioralresearcherwillbeinterestedinquestionsthatfocusonthekurtosisofadistribution,estimatesofkurtosis,incombinationwithotherinformationabouttheshapeofadistribution,canbeuseful.DeCarlo(1997)describedseveralusesfortheg2statistic.Whenconsideringtheshapeofadistributionofscores,itisusefultohaveathandmeasuresofskewnessandkurtosis,aswellasgraphicaldisplays.Thesestatisticscanhelponedecidewhichestimatorsortestsshouldperformbestwithdatadistributedlikethoseonhand.Highkurtosisshouldalerttheresearchertoinvestigateoutliersinoneorbothtailsofthedistribution.

TestsofSignificance

Somestatisticalpackages(includingSPSS)providebothestimatesofskewnessandkurtosisandstandarderrorsforthoseestimates.Onecandividetheestimatebyit’sstandarderrortoobtainaztestofthenullhypothesisthattheparameteriszero(aswouldbeexpectedinanormalpopulation),butIgenerallyfindsuchtestsoflittleuse.Onemaydoan“eyeballtest”onmeasuresofskewnessandkurtosiswhendecidingwhetherornotasampleis“normalenough”touseaninferentialprocedurethatassumesnormalityofthepopulation(s).Ifyouwishtotestthenullhypothesisthatthesamplecamefromanormalpopulation,youcanuseachi-squaregoodnessoffittest,comparingobservedfrequenciesintenorsointervals(fromlowesttohighestscore)withthefrequenciesthatwouldbeexpectedinthoseintervalswerethepopulationnormal.Thistesthasverylowpower,especiallywithsmallsamplesizes,wherethenormalityassumptionmaybemostcritical.Thusyoumaythinkyourdatacloseenoughtonormal(notsignificantlydifferentfromnormal)touseateststatisticwhichassumesnormalitywheninfactthedataaretoodistinctlynon-normaltoemploysuchatest,thenonsignificanceofthedeviationfromnormalityresultingonlyfromlowpower,smallsamplesizes.SAS’PROCUNIVARIATEwilltestsuchanullhypothesisforyouusingthemorepowerfulKolmogorov-Smirnovstatistic(forlargersamples)ortheShapiro-Wilksstatistic(forsmallersamples).Thesehaveveryhighpower,especiallywithlargesamplesizes,inwhichcasethenormalityassumptionmaybelesscriticalfortheteststatisticwhosenormalityassumptionisbeingquestioned.Thesetestsmaytellyouthatyoursampledifferssignificantlyfromnormalevenwhenthedeviationfromnormalityisnotlargeenoughtocauseproblemswiththeteststatisticwhichassumesnormality.

SASExercises

GotomyStatDatapageanddownloadthefileEDA.dat.GotomySAS-Programspageanddownloadtheprogramfileg1g2.sas.EdittheprogramsothattheINFILEstatementpointscorrectlytothefolderwhereyoulocatedEDA.datandthenruntheprogram,whichillustratesthecomputationofg1andg2.Lookattheprogram.TherawdataarereadfromEDA.datandPROCMEANSisthenusedtocomputeg1andg2.ThenextportionoftheprogramusesPROCSTANDARDtoconvertthedatatozscores.PROCMEANSisthenusedtocomputeg1andg2onthezscores.Notethatstandardizationofthescoreshasnotchangedthevaluesofg1andg2.Thenextportionoftheprogramcreatesadatasetwiththezscoresraisedtothe3rdandthe4thpowers.Thefinalstepoftheprogramusesthesepowersofztocomputeg1andg2usingtheformulaspresentedearlierinthishandout.Notethatthevaluesofg1andg2arethesameasobtainedearlierfromPROCMEANS.

GotomySAS-ProgramspageanddownloadandrunthefileKurtosis-Uniform.sas.Lookattheprogram.ADOloopandtheUNIFORMfunctionareusedtocreateasampleof500,000scoresdrawnfromauniformpopulationwhichrangesfrom0to1.PROCMEANSthencomputesmean,standarddeviation,skewness,andkurtosis.Lookattheoutput.Comparetheobtainedstatisticstotheexpectedvaluesforthefollowingparametersofauniformdistributionthatrangesfromatob:

Parameter

ExpectedValue

Parameter

ExpectedValue

Mean

a+b

2

Skewness

0

StandardDeviation

.,(b-a)2

12

Kurtosis

-1.2

GotomySAS-Programspageanddownloadandrunthefile“Kurtosis-T.sas,”whichdemonstratestheeffectofsamplesize(degreesoffreedom)onthekurtosisofthetdistribution.Lookattheprogram.WithineachsectionoftheprogramaDOloopisusedtocreate500,000samplesofNscores(whereNis10,11,17,or29),eachdrawnfromanormalpopulationwithmean0andstandarddeviation1.PROCMEANSisthenusedtocomputeStudent’stforeachsample,outputtingthesetscoresintoanewdataset.Weshalltreatthisnewdatasetasthesamplingdistributionoft.PROCMEANSisthenusedtocomputethemean,standarddeviation,andkurtosisofthesamplingdistributionsoft.Foreachvalueofdegreesoffreedom,comparetheobtainedstatisticswiththeirexpectedvalues.

Mean

StandardDeviation

Kurtosis

0

:df

6

\df-2

df-4

DownloadandrunmyprogramKurtosis_Beta2.sas.Lookattheprogram.EachsectionoftheprogramcreatesoneofthedistributionsfromTable1aboveandthenconvertsthedatatozscores,raisesthezscorestothefourthpower,andcomputesP2asthemeanofz4.Subtract3fromeachvalueofP2andthencomparetheresultingy2tothevaluegiveninTable1.

DownloadandrunmyprogramKurtosis-Normal.sas.Lookattheprogram.DOloopsandtheNORMALfunctionareusedtocreate10,000samples,eachwith1,000scoresdrawnfromanormalpopulationwithmean0andstandarddeviation1.PROCMEANScreatesanewdatasetwiththeg1andtheg2statisticsforeachsample.PROCMEANSthencomputesthemeanandstandarddeviation(standarderror)forskewnessandkurtosis.Comparethevaluesobtainedwiththoseexpected,0forthemeans,and<6/nand<24/nforthestandarderrors.

References

Balanda&MacGillivray.(1988).Kurtosis:Acriticalreview.AmericanStatistician,42:111-119.

Blest,D.C.(2003).Anewmeasureofkurtosisadjustedforskewness.Australian&NewZealandJournalofStatistics,45,175-179.

Darlington,R.B.(1970).Iskurtosisreally“peakeTheAmericanStatistician,24(2),1922.

DeCarlo,L.T.(1997).Onthemeaninganduseofkurtosis.P