正態(tài)分布檢驗(yàn)的3大步驟及結(jié)果處理spss

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Statistics–Spring2008

Lab#1–DataScreening

Normality

Below,Idescribefivestepsfordetermininganddealingwithnormality.However,thebottomlineisthatalmostnoonecheckstheirdatafornormality;insteadtheyassumenormality,andusethestatisticalteststhatarebaseduponassumptionsofnormalitythathavemorepower(abilitytofindsignificantresultsinthedata).

First,whatisnormality?Anormaldistributionisasymmetricbell-shapedcurvedefinedbytwothings:themean(average)andvariance(variability).

Second,whyisnormalityimportant?Thecentralideabehindstatisticalinferenceisthatassamplesizeincreases,distributionswillapproximatenormal.Moststatisticaltestsrelyupontheassumptionthatyourdatais“normal.Tests”thatrelyupontheassumptionornormalityarecalledparametrictests.Ifyourdataisnotnormal,thenyouwouldusestatisticalteststhatdonotrelyupontheassumptionofnormality,callnon-parametrictests.Non-parametrictestsarelesspowerfulthanparametrictests,whichmeansthenon-parametrictestshavelessabilitytodetectrealdifferencesorvariabilityinyourdata.Inotherwords,youwanttoconduct

parametrictestsbecauseyouwanttoincreaseyourchancesoffindingsignificantresults.

? Third,howdoyoudeterminewhetherdataare“Therenormalare”?threeinterrelatedapproachestodeterminenormality,andallthreeshouldbeconducted.

First,lookatahistogramwiththenormalcurvesuperimposed.Ahistogramprovidesusefulgraphical

representationofthedata.SPSScanalsosuperimposethetheoretical “normal”distributio

ofyourdatasothatyoucancompareyourdatatothenormalcurve.Toobtainahistogramwiththesuperimposednormalcurve:

1.SelectAnalyze-->DescriptiveStatistics-->Frequencies.

2.Moveallvariablesintothe “Variable(s) ”window.

ClickCharts“”,andclickHistogram,“withnormalcurve”.

ClickOK.

Outputbelowisfor“system1”.Notice-shapedthebellblacklinesuperimposedonthedistributionAll.samplesdeviatesomewhatfromnormal,sothequestionishowmuchdeviationfromtheblacklineindicates“non-normality”?Unfortunately,graphicalrepresentationslikehistogramprovideno-hardand-fastrules.Afteryouhaveviewedmany(many!)histograms,overtimeyouwillgetasenseforthenormalityofdata.Inmy

view,thehistogramfor “system1”showsafairlynormaldistribution.

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Second,lookatthevaluesofSkewnessandKurtosis.Skewnessinvolvesthesymmetryofthedistribution.Skewnessthatisnormalinvolvesaperfectlysymmetricdistribution.Apositivelyskeweddistributionhasscoresclusteredtotheleft,withthetailextendingtotheright.Anegativelyskeweddistributionhasscoresclusteredtotheright,withthetailextendingtotheleft.Kurtosisinvolvesthepeakednessofthedistribution.Kurtosisthatisnormalinvolvesadistributionthatisbell-shapedandnottoopeakedorflat.Positivekurtosisisindicatedbyapeak.Negativekurtosisisindicatedbyaflatdistribution.DescriptivestatisticsaboutskewnessandkurtosiscanbefoundbyusingeithertheFrequencies,Descriptives,orExplorecommands.IliketousetheExplore“”commandbecauseitprovidesotherusefulinformationaboutnormality,so

1.

SelectAnalyze-->DescriptiveStatistics-->Explore.

2.

Moveallvariablesintothe

“Variable(s)”window.

ClickPlots“”,andunclickStem“-and-leaf”

ClickOK.

Descriptivesboxtellsyoudescriptivestatisticsaboutthevariable,includingthevalueofSkewnessandKurtosis,withaccompanyingstandarderrorforeach.BothSkewnessandKurtosisare0inanormal

distribution,sothefartherawayfrom0,themorenon-normalthedistribution.Thequestionis“howmuch”skeworkurtosisrenderthedatanon-normal?Thisisanarbitrarydetermination,andsometimesdifficulttointerpretusingthevaluesofSkewnessandKurtosis.Luckily,therearemoreobjectivetestsofnormality,describednext.

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Third,thedescriptivestatisticsforSkewnessandKurtosisarenotasinformativeasestablishedtestsfornormalitythattakeintoaccountbothSkewnessandKurtosissimultaneously.TheKolmogorov-Smirnovtest(K-S)andShapiro-Wilk(S-W)testaredesignedtotestnormalitybycomparingyourdatatoanormal

distributionwiththesamemeanandstandarddeviationofyoursample:

1.

SelectAnalyze-->DescriptiveStatistics-->Explore.

2.

Moveallvariablesintothe

“Variable(s)”window.

ClickPlots“”,andunclickStem“-and-leaf”,andclickNormality“plotswithtests”.

ClickOK.

“TestofNormality”boxgivesthe-KSandS-Wtestresults.IfthetestisNOTsignificant,thenthedataarenormal,soanyvalueabove.05indicatesnormality.Ifthetestissignificant(lessthan.05),thenthedataarenon-normal.Inthiscase,bothtestsindicatethedataarenon-normal.However,onelimitationofthenormalitytestsisthatthelargerthesamplesize,themorelikelytogetsignificantresults.Thus,youmaygetsignificantresultswithonlyslightdeviationsfromnormality.Inthiscase,oursamplesizeislarge(n=327)sothesignificanceoftheK-SandS-Wtestsmayonlyindicateslightdeviationsfromnormality.Youneedtoeyeballyourdata(usinghistograms)todetermineforyourselfifthedatarisetothelevelofnon-normal.

“NormalQ-QPlot”providesagraphicalwaytodeterminethelevelofnormality.Theblacklineindicatesthevaluesyoursampleshouldadheretoifthedistributionwasnormal.Thedotsareyouractualdata.Ifthedotsfallexactlyontheblackline,thenyourdataarenormal.Iftheydeviatefromtheblackline,yourdataarenon-normal.Inthiscase,youcanseesubstantialdeviationfromthestraightblackline.

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Fourth,ifyourdataarenon-normal,whatareyouroptionstodealwithnon-normality?Youhavefourbasicoptions.

Option1istoleaveyourdatanon-normal,andconducttheparametricteststhatrelyupontheassumptionsofnormality.Justbecauseyourdataarenon-normal,doesnotinstantlyinvalidatetheparametrictests.Normality(versusnon-normality)isamatterofdegrees,notastrictcut-offpoint.Slightdeviationsfromnormalitymayrendertheparametrictestsonlyslightlyinaccurate.Theissueisthedegreetowhichthedataarenon-normal.

Option2istoleaveyourdatanon-normal,andconductthenon-parametrictestsdesignedfornon-normaldata.

Option3istoconduct“robust”tests.Thereisagrowingbranchofstatisticscalledarejustaspowerfulasparametrictestsbutaccountfornon-normalityofthedata.

Option4istotransformthedata.Transformingyourdatainvolvingusingmathematicalformulasto

modifythedataintonormality.

?Fifth,howdoyoutransformyourdatainto

“normalThere”aredata?differenttypesoftransformationsbased

uponthetypeofnon-normality.Forexample,seehandout

“Figureonth8last.1page”ofthisdocumentthat

showssixtypesofnon-normality(e.g.,3positiveskewthataremoderate,substantial,andsevere;3negative

skewthataremoderate,substantial,andsevere).Figure8.1alsoshowsthetypeoftransformationforeach

typeof