Butler, Christopher. 1985. Statistics in Linguistics.

Contents Prefacevii 1Somefundamentalconceptsinstatistics1 2Frequencydistributions14 3Measuresofcentraltendencyandvariability27 4Thenormaldistribution44 5Samplestatisticsandpopulationparameters:estimation53 6Projectdesignandhypothesistesting:basicprinciples65 7Parametrictestsofsignificance78 8Someusefulnon-parametrictests98 9Thechi-squaretest112 10Thetdistributionanditsuses127 11Correlation137 12Statistics,computersandcalculators154 Appendix1Statisticaltables168 Appendix2Answerstoexercises182 References211 Index212 Preface Inlanguagestudy,asinthenaturalsciences,sociologyorpsycho-logy,manykindsofworkrequirethecollectionofquantitative data.Theliterarystylisticianmaywishtocounttherelative numbersofvariouscolourterms,tenseforms,alliterativesounds orsomeotherlinguisticfeatureofthetextsinwhichheisinter-ested.Thelanguageteacherorcoursedesignermaywishtoobtain andcomparemeasuresofstudents'performancesundertwo teachingmethods.Thetheoreticianmaywishtocounthowmany wordsinacorpusoftextsoccuronce,howmanytwiceandsoon, andtocomparetheseobserveddatawiththosepredictedbya theoreticalmodelofvocabularydistributionintexts.Theseare justafewexamplesofthemanypossiblekindsofquantitative investigationintolanguage.Inallofthem,weneedwaysof makingsenseofthedata,andthisisthepurposeofstatistical methods. Inmanyquantitativestudies,wecannotinvestigateeverypos-sibleexampleofthephenomenonweareinterestedin.Insome casesexhaustiveinvestigationis/..//,impossible;for example,ifwewerestudyingthetimetakenbyinformantsto utteraparticularsentence,thenumberofpossiblereadingsis infinite.Inothercases,exhaustiveexaminationistheoretically possiblebutimpracticable;forinstance,ifwewereexamining somephonologicalfeatureoftheEnglishspokeninBirmingham, wecouldintheoryobtaindatafromeveryBirminghamresident (or,better,fromeveryresidentsatisfyingasetofpredetermined criteriaforqualifyingasa' Birminghamspeaker' );butthiswould beextremelytime-consuminganddifficulttoorganise,sothat weshouldalmostcertainlybecontentwitha-,/.fromthe totalpopulationweareconcernedwith.Oneimportantpartof viiit./. statisticsisconcernedwithmethodsofsampling,andwiththe relationshipsbetweenmeasurementsmadeonsamples,andthe propertiesofthepopulationsthesesamplesareintendedto represent. Oncewehaveasetofdata,eitherforeveryoccurrenceofour chosenphenomenonorforasampleofit,weusuallyneedto summariseitinsuchawaythatwecandiscernitsgeneralcharac-teristics.Thetoolsavailableforthistaskconstitute..,.. Presentedwithalonglistofnumbersrepresentingour observations,itisoftennoteasytosee,ataglance,anygeneral trendsinthedata.Suchtrendsbecomemoreobviouswhenwe lookatthedistributionofthedata.Forinstance,inalanguage proficiencyteston100learners,wemightrecordmarksoutof20. Wecandeterminehowmanylearnersscore0,howmanyscore1, howmany2andsoon,upto20,therebydrawingupa/...-, ./.-forthedata,whichmaybeconvertedt ographical form,andwhichgivesanindicationofthemosttypicalscore aswellasthespreadofmarks.Moreprecisemeasuresofthese propertiescanbeobtainedbyperformingcertainstatistical calculationsonthedata. Veryoften,weareconcernednotwiththecharacteristicsof justonesetofdata,butwiththecomparisonoftwo(ormore) sets.Forexample,wemightbeinterestedintestingthehypothesis thattheperformanceoftwogroupsoflearners,taughtbydiffer-entmethods,willdifferinalanguageproficiencytest;orwemay wishtoinvestigatewhethertheproportionsoftwopronunciations differinthecasualandformalspeechofinformants.Insuchcases wefacetheproblemofdesigningourstudyinsuchawaythatit willisolatejustthosephenomenawewishtotest.Thesampleswe usemustbechosensoastominimisevariationarisingfromun-wantedcomplicatingfactors,sothatwecanbereasonablycon-fidentthatanyeffectsowingtoourchosenphenomenonarenot swampedbyother,' irrelevant' ,effects.Experimentaldesignis, orshouldbe,inseparablefromstatisticalwork:noamountof sophisticatedstatisticscancompensateforabadlydesigned investigation. Wherecomparisonsareinvolved,weneedtoknownotonly thegeneralcharacteristicsofeachsample(suchasthemosttypical valueandthespreadofvalues)butalsowhetherthecharacteristics ofthetwosamplesaresufficientlydifferentforustoconclude thatthereisarealeffectthatisduetothefactorweareinvesti-gating.Wecanneverbeabsolutelysurethata differencebetween t./.ix twosetsofobservationshasnotarisen' bychance' ,owingto inherentvariabilityinourmaterial.Wecan,however,carryout testswhichmayallowustoclaim'real'differences withaspecifi-ablemarginoferror,say5percent,or1percent,or0.1percent. Thatis,wemay,asaresultofourcalculations,claimtobe95per centsure,or99percentsure,oreven99.9percentsure,thatwe havefounda'real'difference.Thisarea,knownas/,,/. .-,,isanimportantpartof-/..-/ Insummary,then,wheneverwewisht ocollectquantitative dataonlanguage,weneedtopaycarefulattentiontothedesign ofourstudy,andtotheselectionofappropriatestatistical methodsforsummarisingthedata,andfortestinghypotheses concerningdifferencesbetweensetsofdata.Alltheseaspectsof statisticsarediscussedinthisbook.However,sincethebookis introductoryinscope,sometechniquesofinterestt olinguists, suchasmultiplecorrelationandregression,clusteranalysis,and analysisofvariancewithmorethanoneindependentvariable,are excluded.Inordertodealadequatelywiththesemoreadvanced techniques,atleastonefurt hervolumewouldberequired. Manycoursesonapplicationsofstatisticsconcentratefartoo heavilyonthemethodsthemselves,anddonotpaysufficient attentiontothereasoningbehindthechoiceofparticular methods.Ihavetriedtoavoidthispitfallbydiscussingthe' why' aswellasthe' how'ofstatistics.Adifficultproblemforthe writerofanytextonmathematicaltopicsfornon-mathematicians ishowfartogointothederivationofformulae.Whilerecognising thatmostlinguists(includingmyself)willhaveneitheraninterest inthemoretheoreticalsideofthesubjectnorthemathematical backgroundnecessaryforafulldiscussion,Ifeelthatitishighly unsatisfactoryforreadersorstudentssimplytobepresented withaformula,withnoexplanationwhateverofhowitisarrived at.WhereIthoughtitappropriate,Ihaveattemptedtogive anideaoftherationalebehindthevariousmethodsdiscussedin thebook.Nevertheless,readersshouldfindthattheirschoolarith-meticandalgebrawillseethemthroughquiteeasily. Ishouldliketoexpressmythankstothevariousgroupsof studentswhohaveworkedthroughthematerialpresentedhere, andtoTimGibson,whocheckedmanyoftheexercises.My thanksgoalsotothefollowing,forpermissiont ousecopyright orunpublishedmaterial: Statisticaltablesinappendix1:DrH. R.Neaveandhispublishers GeorgeAllen&Unwin,fororiginaloradaptedversionsoftables xt./. 2.1(a),3.1,3.2,3.3,5.1,5.3,6.2,6.4and7.1froms ://.//.--t-,-..t---./.o./../-.-,.-.-s.-.(1978);question7ofchapter9 exercises:DrJ.Connolly,fordatafromhisarticle,' Quantitative analysisofsyntacticchange',-,/-t-,.c./8/2, 108-18(1979);question6ofchapter9exercises:DrJ.Coates andProfessorG.Leech,fordatafromtheirarticle,' Themeanings ofthemodalsinmodernBritishandAmericanEnglish',:/ t,.-t-,.8,23-34(1980);question4ofchapter11 exercises:ProfessorG.Wellsandthepublishersofs../- t..-,fordatafromthearticle,'Languageuseandeduca-tionalsuccess:aresponsetoJoanTough' s:/.t.../,-.-/ .--,(1977)' ,s../-t..-18,9-34;questions2and 3ofchapter2exercises,question3ofchapter3exercises,ques-tion5ofchapter9exercises,questions1and3ofchapter10 exercises:DrA.S.Crompton,fordatafromhisworkonstress andpauseinFrench. IshouldalsoliketothankProfessorD.Crystalforadviceonthe productionofthebook. 1Somefundamental conceptsin statistics 1.1Introduction Inthischapter,someideasoffundamentalimportanceinstatistics areintroduced.Wefirstconsidertheconceptsofpopulationand sample.Wethendifferentiatebetweenthetwobroadareasof statisticalinvestigation:descriptiveandinferentialstatistics. Finally,weconsidersomeimportantwaysinwhichstatistical variablescanbeclassified. 1.2Populationsandsamples Ineverydayspeech,theterm' population'isnormallytakento meanacollectionofhuman,oratleastanimate,entities,asin ' thepopulationofGreaterLondon' ,or' theratpopulationofthe Londonsewers'.Instatistics,however,thetermisusedmore generally,torefertoanycollectionofentities,ofwhateverkind, thatistheobjectofinvestigation.Thus,wemayspeakofthe populationofwordsinagiventext,thepopulationofnounsin Dickens'st..c,,././.orthepopulationofmeasurements ofthetimetakentoutteraparticularsentenceondifferent occasions.Itiswiththecharacteristicsofpopulations,oraggre-gatesofindividualentities,thatstatisticsismostfundamentally concerned. Wemaydrawadistinctionbetween/-.and-/-.popula-tions.Thepopulationofnounsint..c,,././.orofpast historicverbsinCamus' stt..isfinite:thenumberofentities isfixedandcountable.Otherkindsofpopulation,however,are potentiallyinfinite:intheory,atleast,wecouldrepeatourmeas-2s-./.-.-.-/-.,- urementofutterancetimesforasentenceaninfinitenumberof times. Withafinitepopulationwhichisnottoolarge,wemaybeable toinvestigatethewholepopulation.Butifourpopulationis potentiallyinfinite,orifitisfinitebutverylarge,weshallhave tobecontentwith-,/.drawnfromthepopulationconcerned. Theuseofsamples,evenforthestudyoffinitepopulations,cuts downthelabourandcostinvolvedinobtainingresults,andmini-misestheerrorswhichcaneasilybemadeduringtheprocessing oflargeamountsofstatisticaldata.Letusimaginethatweare interestedintheacceptabilityjudgementsofnativespeakersof ParisianFrenchwhenpresentedwithaseriesofFrenchsentences. Clearly,itwouldbeimpossible,inpractice,touseeveryspeakerof ParisianFrenchasaninformant(quiteapartfromtheconsiderable difficultiesinvolvedindecidingwhatcountsas'ParisianFrench' ). Instead,wetakeasampleofthespeakersconcerned,inthehope thatourresultswillbegeneralisabletothepopulationofsuch speakersasawhole. Clearly,greatcaremustbeexercisedinselectingsamplesif generalisationtothepopulationistobevalid.Statisticalmethods ofinferringthepropertiesofpopulationsfromthoseofsamples arebasedontheassumptionthatthesamplesare-.-This doesnotmeanthattheyarechoseninahaphazardmanner: rather,itmeansthateveryunitinthepopulationhasanequal chanceofbeingrepresentedinthesample. Inordertoselectatrulyrandomsamplefromafinitepopula-tion,weneedtohaveacompletelistoftheunitsinthepopulation (oftencalledthe'samplingframe' ).Eachunitinthesampling frameisallocatedanumber.Onewayofdoingthisistoputinto aboxasetofpaperslips,eachbearinganumber,from1tothe totalsizeofthepopulation,mixuptheslipsthoroughly,and withdrawanumberofslipsequaltothedesiredsamplesize.This method,thoughsimple,istedious,andaratherbetterwayisto useatableofrandomnumbers,inwhichthedigits0to9are arrangedinaverylongsequence,inrandomorder.Setsofstatis-ticaltablesincorporatingsuchinformationarereadilyavailable, andanexcerptfromonesuchtableis givenas tableA1 ofappendix 1.Thedigitsaregroupedintofives,butthisismerelyforcon-venienceinreading.Letusimaginethatwewishtodrawasample of50linesofpoetryfromatextcontaining800lines.Sinceour populationsizeis800,athree-digitnumber,weneedtotake :s-./.-.-.-/-.,- groupsofthreedigitsfromthetableofrandomnumbers.Wemay beginanywhereinthetable;forconvenience,letusstartwith thefirstthreedigitsincolumn1oftableAl .Readingdownthe column,weselectitems494,294,252,24,694,772and528as thefirstseven' members'ofoursample.Thenexttwofiguresare 987and850,whicharetoolargeforourpopulationof800,sowe ignorethem.Thenextmembersofthesamplearelines177,637, 616,422,452,403,540,491,76,andsoon. Avariationofsimplerandomsamplingisknownas,.- or.-.-sampling.Here,wedecidethefirstunitinthe samplebytrulyrandommethods,andthentakeunitsatequal intervalsthroughoutthenumberedpopulation,theintervalsbeing chosentogivethedesirednumberinthefinalsample.Letus consideragaintheselectionofasampleof50linesofpoetryfrom atextof800lines.Oursamplingintervalwillbe800/50,or16. Wechoosesomepointintherandomnumbertables,andfindthe firsttwo-digitcombinationwhichissmallerthanorequalto16. Ifthiswas,say,12,weshouldtakelines12,28,44,60,76,and soon,uptoandincludingline796.Suchasampleisnottruly random,becausethesecondandsubsequentunitsarenotchosen independentlyofthefirst.Thisdoesnotmatterseriously,pro-videdthatthereisnoperiodicityinthepopulationsuchthatunits withcertainpropertiestendtorecuratregularintervals.For instance,iftheworkofaparticularprosewritershowedarather regularsentencelengthofabout20words,andthesampling intervalforselectionofasampleofwordsfromhisworkwasalso 20,thesamplemightcontainapreponderanceofwordsfromthe beginningsorendsofsentences,andthismightseriouslyaffectthe representativenessofthesample. Avariationwithinsystematicsamplingis ///-,/-,where thepositionofthebeginningofthesampleisdeterminedran-domly,butthenextitemsaretaken,whereisthesizeofthe sample.Forinstance,inselecting500wordsfromatextof10000 words,onemightstartatword4 581(determinedfromrandom numbertables),andtakewords4 581to5 080inclusive.The chancesofsuchapassagebeingunrepresentativeareconsiderable, andobviouslyvaryaccordingtothenatureofthepopulationand itsstructure. Itisimportanttorealisethattheselectionofasampleby methodsdesignedtoachievetruerandomness(suchasusinga tableofrandomnumbers)doesnotguaranteethatthesample 4s-./.-.-.-/-.,- arrivedat willindeedberepresentativeofthepopulation.Consider, forexample,theselectionofasampleoftenpeoplefromthe adultpopulationofavillageforlinguistictesting,andassume,for simplicity,thatthereareequalnumbersofmenandwomeninthe village.Tocalculatethechancesofgettingvariouspossiblemixes ofmenandwomeninthesample,wemustmakeashortdigression intoelementaryprobabilitytheory. Tosimplifymattersstillfurther,letusfirstconsidertheselec-tionofjustthreepeoplefromthevillageadultpopulation.Since thereareequalnumbersofmenandwomen,theprobabilitythat thefirstpersonselectedwillbeamanis0.5(asis also,ofcourse, theprobabilitythatthispersonwillbeawoman).Now,thelaws ofprobabilitystatethattheprobabilityoftwoindependentevents bothoccurringisequaltotheproductoftheprobabilitiesofthe individualevents.Sothechancesofboththefirstandthesecond membersofoursamplebeingmen(orofbothbeingwomen)is 0.5 X0.5,or0.25,andtheprobabilityofallthreebeingmen(or allbeingwomen)is0.5X0.5X0.5,or0.125.Nowletuslookat thepossibilityofgettingtwomenandonewomaninoursample. Therearethreewaysinwhichthiscouldhappen: Samplememberno.1 man man woman 2 woman man man 3 man woman man Theprobabilityofeachoftheseoutcomesis0.125,sothatthe totalprobabilityofgettingtwomenandonewomanis3 X0.125, or0.375.Similarly,wecangettwowomenandonemaninany ofthreeways: Samplememberno.1 woman woman man 2 man woman woman 3 woman man woman Again,thetotalprobabilityis0.375.Wethusarriveatthesitua-tionshownintable1.1.(Actually,thisisanoversimplification, sincethesimpletheoryholdsgoodonlyifeachunitselectedis putbackintothepopulationbeforethenextchoiceis made,and thisisclearlynotpossibleinoursituation,aswemustavoidthe possibilityofthesamepersonbeingchosentwice.Thiscan, however,beignoredforourpresentpurposes,providedthatthe Table1.5 s-./.-.-.-/-.,-5 OutcomeProbability 3men 2men+1 woman 2women+1 man 3women 0.125 0.375 0.375 0.125 populationislargeihrelationtothesamplesize.)Notethatthe probabilitiesaddupto1.000,asindeedtheymust. Wecouldgothroughthesamekindofreasoningformore complicatedcases,butfortunatelythereisageneralformula whichwillsaveusthistrouble.If,andaretheprobabilitiesof twomutuallyexclusive,independentevents(suchaschoosinga manandchoosingawoman,inourexample),thenthepossible outcomesof-selectionsaredeterminedbyexpandingtheformula ;, .-Forthecasewhere-=2,wehave , .: ,: , , : ,: : ,: For-=3,wehave ;,.:;, .;,::,:. ,: : ,: ,: ,: : ,:: ,:+3 ,:+3 ,:: Forthecasewhere,0.5,thetermsevaluateto0.125 +0.375 +0.375+0. 125,andthisistheresultweobtainedfromfirst principlesabove. This typeofdistributionis knownasthe/--/ ./.-Weshallnotdiscussitfurtherhere:detailscanbe foundinanycomprehensivestatisticstextbook(see,forexample, Ferguson1981). Wemaynowreturntoouroriginalproblem:predictingthe probabilitiesofvariousoutcomesintheselectionoftenpeople fromourvillagepopulation.Iftheprobabilitiesareworkedout for-=10,usingtheexpansiondiscussedabove,weobtainthe resultsshownintable1.2. Thereaderwillrememberthatthepointofthisdigressionwas toenableustoseejusthowrepresentativeofthepopulationa 6s-./.-.-.-/-.,- Table1. 2 MenWomenProbability 0100.001 190.010 280.044 370.117 460.205 550.246 640.205 730.117 820.044 910.010 1000.001 randomlyselectedsampleoftenpeoplemightbe.Ideally,we shouldliketheproportionofmenandwomeninthesampleto reflectthatinthepopulation:thatis,fivemenandfivewomen. Weseefromtheabovethattheprobabilityofjustthisoutcome isaboutaquarter(tobemoreprecise,0.246).Thechancesof gettingeightormoremeninthesample(or,ofcourse,eightor morewomen)are0.044+0.010+0.001,or0.055-thatis,over 1in20.Puttingthisanotherway,ifwetook20samplesoften peoplefromthevillage,wecouldexpectoneofthesetocontain eightormoremen,andonetocontaineightormorewomen. Suchasamplewouldhardlyberepresentativeofthesexdistri-butionofthevillage.Furthermore,itisnothardtoseethatany typesofunitwhichhaveasmalloverallproportioninthepopula-tionmaywellnotberepresentedinthesampleatall. Onewaytominimisethisproblemistoselectasampleby /..-.--,/-,Iftheproportionsofvarioussub-groupsinthepopulationareknown,thenrandomsamplingcan beundertakenwithineachstratum,andtheresultingsubsamples combinedtogiveanoverallsample.Forexample,inselectinga sampleoftenadultsfromourvillage,wemightchoose,byrandom methods,fivemenandfivewomen,andcombinethetwosub-samples.Suchaprocedurewouldbeanexampleof,,-/ stratifiedrandomsampling,sincetheproportionofmenand womeninthefinalsampleisthesameasinthepopulation.Asa furtherexampleofproportionalstratification,considertheprob-lemofselectingasamplethatwillbeasrepresentativeaspossible s-./.-.-.-/-.,-7 Table1.5 EarlyperiodMiddleperiodLateperiod Novels500000700000300000 Shortstories100000200000200000 oftherangeofanauthor' sliteraryproseworks.Letusassume thattheauthor' stotaloutputof2millionwordsisdividedinto genresandperiodsasintable1.3.Wemightthenchooseour samplesothat500000/2000000,or0.25ofit,camefromearly novels,700000/2000000,or0.35,frommiddlenovels,and likewise0.15fromlatenovels;0.05fromearlyshortstories,0.1 frommiddle-periodshortstories,and0.1fromlateshortstories. Withineachofthesecategories,wecouldselectoursubsample bysimplerandomsampling,orbysystematicorevenblock samplingmethods. Astratifiedframeworkhasanadditionaladvantageinthatit allowscomparisonstobemadebetweensubsamplescorresponding tothevariousgroupings.Thisispossiblewithaproportionally stratifiedsamplesuchasthosediscussedabove.Often,however, theoptimumsituationforcomparisonofsubgroupsisonein whichthesesubgroupsareofequalsize,despitetheirunequal proportionsinthepopulationasawhole.Asamplechoseninthis wayissaidtobea.,,-//,stratifiedsample.Imagine, forinstance,thatwewishtoselectasampleof120 000words fromthe2millionwordoutputofourhypotheticalauthor.Table 1.4comparesthenumberofwordstakenfromeachsubgroup under(1)proportionaland(2)disproportionalstratifiedsampling. Ifwewishtousetheresultsfromadisproportionallystratified sampletoestimatethepropertiesofthepopulationasawhole,we mustobviouslygivethevariousstratadifferentweightings.For instance,thereareonlytwo-thirdsasmanywordsfromearly novelsinthedisproportionalsampleasintheproportionalsample, sothatweshouldweightbyafactorof3/2,or1.5,thecontri-butionoftheresultsfromthissubgroup;andsoon. Weshouldalsomention-./,.-,/-,which,asthe termsuggests,consistsofsamplingbysuccessivestages,treating thesampleatonestageasthepopulationforthenext.For example,ifwewishedtoobtainasampleof100pagesfromthe 8s-./.-.-.-/-.,- Table1.4 ProportionalDisproportional Novels Early3000020000 Middle4200020000 Late1800020000 Shortstories Early600020000 Middle1200020000 Late1200020000 novelsofDickens,wemightfirstselectfivenovelsbyarandom procedure,usingthetotalsetofnovelsasthepopulation,and thenwithineachnovelselect20pages,againatrandom.Some-times,investigationswhichatfirstsightlookasiftheyarebased onsingle-stagesamplingareinfactbetterregardedasmulti-stage. Considerthesituationwherewewishtoselecttwosamplesfroma classoflanguagelearners,inordertocomparetheeffectiveness oftwodifferentteachingmethods.Thesampleschosenforcom-parisonaresamplesoftheclasstakenasapopulation;butwe shouldprobablywanttotrytogeneralisefromthatparticular classoflearnerstothewholepopulationoflearnersofsimilar characteristics,atasimilarstageofdevelopment. Thislastpointbringsustoanimportantpracticalissue.Wehave seenthattruerandomsamplingfromafinitepopulationrequires thelistingoftheentirepopulation.Itisoftenimpossible,orat leastimpracticable,toobtainalistofthewholepopulationin whichweareinterested.Insomecasesofsamplinginvolving humanpopulations,electoralandsimilarlistscanbeuseful;but evenhere,wemustbecarefulnottouse,asasamplingframe,a listthathassomein-builtbiaswhichmightprejudiceourresults. Alistoftelephonesubscribers,forexample,wouldalmostcer-tainlybiasoursampletowardsthemiddleandupperrangesofthe socialclassscale.Inmanycases,wehavetomakedowitha ' population'whichis'given'bysomepracticalcircumstance,such asthemembersofaschool,orevenaclasswithinaschool.Insuch cases,ifwewishtogeneralisebeyondthatgroupwemust,if possible,doourbesttodemonstratethatthepropertiesofthe groupdonotdifferradicallyfromthoseofthepopulationin whichwearereallyinterested.Forinstance,wemighttrytoshow .s-./.-.-.-/-.,- thatthedistributionofages,sexes,IQratingsandsocialclassesin aparticularschoolisnottoofarremovedfromthenormfor schoolchildreninthecountryasawhole.Often,however,the normitselfisunknownanddifficulttodetermine;furthermore,it maynotalwaysbeclearjustwhatfactorsarerelevant.Wemust thereforebeonourguardagainsttheunderstandabletemptation toclaimgreatergeneralisabilityforourresultsthaniswarranted bythesituation. Finallyinthissection,wemustintroducetwomoretechnical termsconnectedwithpopulationsandsamples.Thepropertiesof populationsarenormallycalled,-..,whiletheproperties ofsamplesfrompopulationsarecalled(orsometimes .-.).Imaginethatweselect1000wordsrandomlyfroma textof50 000words,andfindthattheirmean(a kindof'average') lengthis3.54letters.Thismeasureisastatisticforthesample,and isusedasanestimateofthetruepopulationparameter,which probablywillnotbeexactly3.54letters.Fortunately,asweshall see,methodsareavailableforassessingthedegreeofconfidence wemayhaveinthereliabilityofsuchestimates.Statisticsare normallysymbolisedbyRomanletters(forinstance,x,readas 'x-bar',forthemeanofasample),whileparametershavesymbols consistingofGreekletters(suchas/,' mu' ,forthemeanofa population). 1.3Thedescriptiveandinferentialfunctionsofstatistics Intheprefacetothisbook,adistinctionwasmadebetweenthe ..,..and-/..-/functionsofstatistics.Weshallnow examinethisdistinctioninrathermoredetail,bymeansofa practicalillustration. Letussupposethatwehaveselectedtwosamplesof30lan-guagelearnerseach,onetaughtbytraditionalmethods,theother bymeansofthelanguagelaboratory.Wehavegivenbothgroupsa proficiencytestinthetargetlanguage,withthefollowingresults: c/ .(languagelaboratory):marksoutof20: 15,12,11,18,15,15, 9,19,14,13,11,12,18,15,16,14,16, 17,15,17,13,14,13,15,17,19,17,18,16,14. c/ o(traditional):marksoutof20: 11,16,14,18, 6, 8, 9,14,12,12,10,15,12, 9,13,16,17,12, 8, 7,15, 5,14,13,13,12,11,13,11, 7. :s-./.-.-.-/-.,- Beforewecanmakeverymuchsenseoutofthesedata,weneedto summarisetheirimportantproperties.Wemight,forexample, constructatableshowingthenumberofstudentsineachgroup whogained0,1, 2,3,4,5marks,andsoonupto20(thatis,a /...-,./.-table).Weshouldalsowanttoknowwhat themosttypical,or'average',markisforeachgroup,andhow muchvariationthereisineachsetofmarks.Allthesearewaysof describingthedatainareadilyassimilableform,andonewhich willpermitfurtherstatisticalwork.Theseareasofdescriptive statisticsarediscussedinchapters2and3below. However,weshallalmostcertainlywanttogobeyondthemere descriptionofthedata,importantasthisis.Onepieceofinforma-tionweshallprobablyneediswhatdegreeofconfidencewemay haveinextrapolatingfromourgroupstoanyothersimilargroups taughtbysimilarmethods.Rephrasingthisintermsofsamples andpopulations,wewishtoviewourgroupsassamplesfromthe entirehypotheticalpopulationofsimilarstudentswhohavebeen orcouldbetaughtbyeachofthetwomethods.Wethuswishto usethemeasuresmadeonoursamples(samplestatistics-forin-stance,themeanmark),inordertoestimatethecorresponding parametersforthepopulationsconcerned.Theareaofestimation isdiscussedinchapter5,forwhichchapter4layssomeessential groundwork. Afurtherpieceofinformationweshall certainlywantis whether itcanbeclaimedthatoneofourgroupshasdonebetteronthe testthantheother.Wemay,perhaps,postulatethatthegroup taughtbythelanguagelaboratorymethodwillperformbetter thanthegrouptaughtbythetraditionalmethod.Hereweare concernedwiththeextremelyimportantarea of/,,/..-, Thebasisofsuchtests,anditscloserelationshipwiththedesign ofexperimentalinvestigations,formthesubjectofchapter6,and thefollowingchaptersthendiscusssomeofthemoreimportant testsavailable,theirusesandlimitations. Theareasofestimationandhypothesistestingbothinvolvethe inferringofrelationshipsfromdescriptivemeasures:together, theyconstitutetheareaofinferentialstatistics. 1.4Variables andtheirclassification Statisticsisessentiallyconcernedwith.//.quantities.Inmost investigationsofaquantitativenature,wevaryoneormoresets I s-./.-.-.-/-.,-:: Table1.5 IndependentvariableDependentvariable TypeofteachingmethodScoreonlanguagetest SentencecomplexityNo.ofsentencesrecalledcorrectlybysubjects SocialclassPercentageof-ing endingspronounced[in]by subjects TopicofnewspaperarticleSentencelength ofconditions,andmeasuretheeffectononeormorepropertiesof interesttous.Theconditionsthatweourselvesvaryarecalled -..,.-..-.//.whilethosewhoseresponsewearemeas-uringare..,.-..-.//.Someexamplesofpossiblepairings areshownintable1.5. We mayalsodistinguishbetweencontinuousandnon-continuous (ordiscrete)variables.c--...//.maytakeanyvalue withinagivenrange;forexample,thetimetakentoutterasen-tenceisacontinuousvariable.t--..or....//. ontheotherhand,cantakeonlycertainvalues;forexample,any givenwordmaybe1,2,3,4letterslong,andsoon,butitmay notbe1.5or2.61or3.14159letterslong(although,ofcourse, themeanlengthofanumberofwordsisacontinuousvariable, andmaytakevaluessuchas2.61). Afurtherimportantdimensionofvariableclassificationcon-cernsthe'levelofmeasurement'appropriatetothevariable. Somevariablesarepurelyqualitative;inrespectofsuchvariables, entitiesmaybethesameordifferent,butnot' more'or'less'. Variablesofthiskindarecalled---/.//.Asanexample, taketheoccurrenceofend-stoppedlinesinpoetry:eitheraline endsinpunctuation,andsoisend-stopped,oritdoesnot;thereis noquestionofonelinebeingmoreorlessend-stoppedthan another.Likewise(ifweassume,forthepurposesofthediscussion, thatclear-cutassignmentsofpartsofspeechcanbemade),the partofspeechofagivenwordinaparticularsentence,orinter-pretationofasentence,is anominalvariable:a wordeithercanbe classifiedasanadjectiveoritcannot. Withothervariables,itmaybepossibletorankvaluesonascale of' moreorless'.Imagine,forexample,thatweaskapanelof informantstorateasetofsentencesfor' acceptability'(definedin somepreciseway).Theratersmaywellbeabletosaythatsen-::s-./.-.-.-/-.,- tenceAismoreacceptablethansentenceB, whichis inturnmore acceptablethansentenceC,andsoon.Inthiscase,acceptability wouldbean.-/.//.Afurtherexampleofanordinal linguisticvariablemightbethepolitenessofparticularformsof requestorcommandinagivensociolinguisticsituation. Withordinalvariables,itisnotpossibletoclaimthatthereare trulyequalintervalsbetweenpointsonthescaleof' moreorless'. Forinstance,onewouldprobablynotwishtoclaimthatthe differencebetweentheacceptabilityofsentenceAandthatof sentenceBwasexactlythesameasthatbetweentheacceptability ofsentencesBandC.Withsomevariables,however,theso-called -.././/.suchstatementsarepossible.Itisnoteasyto findexamplesoftrueintervalvariablesinlinguistics,soweshall takeaneverydayphenomenoninstead.Wecansaythattempera-turesof10Cand20Cdifferbythesameamountas20Cand 30 C,or30Cand40C.Wecan,therefore,saythattemperature is anintervalvariable. Althoughwecantalkabouttheequalityoftemperatureintervals, wecannotmeaningfullyclaimthatatemperatureof20Cistwice ashotas oneof10 C,sincethezeropointontheCentigradescale isnotinanysenseanabsolutezero.Forsomevariables,however-the.//.thereisanabsolutezero,andratiosmayvalidly betaken.Consider,forexample,thetimetakentoutterasen-tence.Someonewhotakes2.48secondstoutterthesentence takesexactlytwiceaslongassomeonewhouttersitin1.24 seconds.Timeisthusa ratiovariable. Mostofthevariablesofinterestinthenaturalandphysical sciencesareoftheintervalorratiotypes,whilemanyofthosewe encounterinthestudyoflanguageareofthenominalorordinal types.Itisextremelyimportantforinvestigatorstoknowwhich typeofvariabletheyaredealingwith,becausedifferentstatistical proceduresareappropriatefordifferenttypesofvariable,aswe shallseeindetaillater.Thelevelofmeasurementofthevariable concernedinfluencesthechoiceofameasureoftypicalvalueand variability,andalsothechoiceofproceduresforhypothesis testing.Forintervalorratiovariables,theso-called,-. .canbeused;butfornominalorordinalvariables,--,-..areappropriate.Weshallhavemoretosayaboutthis matterinchapter6.Meanwhile,weshouldnotethatinvestigators ofsocialandlinguisticphenomenacommonlyassumeahigher levelofmeasurementthan,strictlyspeaking,theirdatawarrant. ::s-./.-.-.-/-.,- Datawhichoughtproperlytoberegardedasordinalareoften madetoappearinterval,bysubstitutingnumbersforranksand thenperformingarithmeticaloperationsonthenumbers.The reasonforthisisthattheparametrictestssuitableforintervaldata aremorepowerfulthannon-parametrictests.Asoft enhappensin theapplicationofstatisticalmethodstorealproblems,practical considerationsfrequentlyoutweightheconcernsofthetheoretical purist. Exercises 1Discussalternativewaysofselectingsamplesforeachofthe followinginvestigations,commentingontheadvantagesand limitationsofthemethodsyoudescribe: (i)acomparisonofwordlengthsintheplaysofShakespeare andMarlowe (ii)astudyofsomeaspectofCockneyspeech (iii)astudyofsyntacticcomplexityinthenovelsofConrad 2Classifyeachofthefollowingasanominal,ordinal,interval orratiovariable.Givereasonsforyouranswers: (i)thelengthofpausesinasampleofconversation,measured inmilliseconds (ii)therating ofwordsbyinformants, onascaleofpleasantness, rangingfrom1(veryunpleasant)to5(verypleasant) (iii)thepresenceorabsenceofafiniteverbineachclauseina particulartext (iv)thedegreeofgrammaticalityofsentences,onascalefrom0 (absolutelyungrammatical)to4(entirelygrammatical) (v)thenumberofsentencesrememberedcorrectlybyinfor-mantstenminutesafterfirsthearingthem 2Frequencydistributions 2.1Theclassificationofrawdata Aswasmentionedinsection1.3,therawdatafromaninvestiga-tionusuallyrequireclassificationbeforepatternscanreadilybe observedinthem.Letuslookagainatthesetsofscoresobtained bythetwogroupsofstudentsin thehypotheticallanguageteaching experimentdiscussedbrieflyinsection1.3,designedtotestthe effectivenessofthelanguagelaboratoryascomparedwithmore traditionalmethods.Thedataarerepeatedintable2.1. Wenoticethatthereseemtobemoresingle-figuremarksinthe groupBcolumnthaninthegroupAcolumn,andthattherangeof marksforgroupBislargerthanthatfor groupA(5-18asagainst 9-19).Otherwise,however,littlepatterncanbeseenataglance. Someofthescoresineachcolumnoccurmorethanonce.Ifwe nowcountthefrequencywithwhicheachscoreoccursinagiven column,weobtaina /...-,./.-foreachsetofscores, asshownintable2.2.Thepictureisnowclearer:weseethat15 isthemostfrequentscoreforgroupA,thefrequenciesfalling awayoneithersideofthisscore.Themostfrequentscorefor groupBis12,thefrequenciesagaintailingawayoneitherside. Thefrequencydistributionsalsoshowclearlythedifferencein variabilitybetweenthetwosetsofscores:themarksforgroupB aremorespreadthanthoseforgroupA.Thusthefrequencydistri-butionwillgiveusaroughideaaboutthe.-/.-..-,ofthe scoresandabouttheir.//,Precisemeasuresofthese propertieswillbediscussedinchapter3. 2.2Groupeddata Adistributiongivingafrequencyforeachindividualvaluetaken bythevariable,asabove,workswellwherethereisasmallnumber t...-,./.-15 Table2.1Scoresin a languagetestfortwogroups taughtbydifferentmethods Marksoutof20 GroupAGroupB (Languagelaboratory)(Traditional) (N=30)(N=30) 1511 1216 1114 1818 156 158 99 1914 1412 1312 1110 1215 1812 159 1613 1416 1617 1712 158 177 1315 145 1314 1513 1713 1912 1711 1813 1611 147 ofvalues(forinstance,thereareonly15actualvaluesofthe variableinthelanguagetestdata).Letusnowconsiderwhat happensifthevariablecantakeawiderrangeofvalues.Thedata intable2.3representthefrequencyofsentencesofparticular lengths(innumbersofwords)inthefirst100sentencesofIris Murdoch' s:/.o.//(Penguinedition,1962). t...-,./.-16 Table2. 2Frequencydistributionsfor scoreson languagetest ScoreGroupAGroupB 5/1 6/1 7//2 8//2 911//2 10/1 11//2III3 12//2HH5 13III3llll4 14an4III3 15m/6//2 16m3//2 17mi4/1 18m3/1 19n2 Suchadistributionisnot,byitself,particularlyuseful,because therearelargenumbersofvaluestakenbythevariable(sentence length),manywithverylowfrequencies.Aclearerpictureemerges ifthesentencelengthvaluesaregroupedinthemannershownin table2.4.Here,thedatahavebeenreclassifiedsothatthetotal frequencieswithinthe/-../1-5,6-10,11-15andsoon arerecorded.Althoughweobtainaclearerideaofthedistribution bygroupinginthisway,wealsolosesomeoftheoriginalinforma-tion.Weknowhowmanysentenceshavelengthsintherange1- 5 words,butwenolongerknow,fromthegroupeddata,whatpro-portionofthesehavelengthsof1,2,3,4and5words.Forthe purposeoflaterstatisticalcalculations,oneoftwoassumptions canbemade:eitherthatthefrequenciesareevenlyspreadoverthe classinterval(forexample,3.6sentencesofeachofthelengths 6,7,8,9and10fortheIrisMurdochdata);orthatthetotal frequencywithintheclassintervalisconcentratedatitsmid-point (3,8,13,andsoon,forthesentencelengthdata).Whichassump-tionwemakedependsonjustwhatwewanttodowiththedata, asweshallseelater. t...-,./.-17 Table2.3Sentencelength(words)distributionforthefirst100 sentencesofIris Murdoch's:/.o.//(withhyphenateditems treatedas singlewords) SentencelengthSentencelength I no.words)Frequency(no.words)Frequency 31 4 51 62 72 88 93 103 115 123 133 148 157 163 174 181 196 204 212 223 234 242 254 26 271 283 293 302 311 322 332 341 35 362 371 381 39 401 41 421 Table2.4Groupeddataforsentencelengthdistribution Sentencelength (no.words)Frequency 1 - 52 6 - 1 018 11-1526 16- 2018 21-2515 26-309 31-356 36-405 41-451 18t...-,./.- 2.3Histograms Anevenclearerideaofafrequencydistributioncanbeobtained byconvertingittoa/,-Fordatawhicharenotgrouped, wesimplyarrangethevaluestakenbythevariableonthehori-zontalaxis,andfrequencyvaluesontheverticalaxis,andthen drawaboxorbarovereachvaluetakenbythevariable,ata heightcorrespondingtothefrequencyfoundforthatvalue.The dataforourhypotheticallanguageteachingexperimentare presentedashistogramsinfigures2.1and2.2. Ifwearedealingwithgroupeddata,thewidthofaboxinthe histogramcorrespondstotheclassinterval,asinfigure2.3which showsthesentencelengthdistributionofthedatafrom:/.o.// Thehorizontalaxisislabelledwiththemid-pointsoftheclass intervals. 5 -12 Frequencyin :/.c/.3 4733 743127228554 r-.:..3 0002 67475316518 Usethechi-squaretesttodecidewhetherthewordlengthdis-tributionsdiffersignificantlyatthe5percentlevel. 2InthesameinvestigationofPlath' spoetry,thefollowingfre-quencieswerefound,inthefourcollectionsofpoemsstudied, foritemscontaining/// :/.c/.c-,/.r.../r-.:.. 27284626 Thetotalnumberofrunningwordsineachcollectionof poemsis::/.c/.8827;c-,/.r.7949; ../9 340;r-.:..6 610.Testwhethertheuseof ///isevenlydistributedoverthefourbooksofpoems,using a5percentsignificancelevel. :/./...124 3AgaininthePlathstudy,thefrequenciesofcertainpunctua-tionmarksinthefourcollectionswereexamined,withthe resultsshownintable9.7.Byapplyingthechi-squaretestto eachpunctuationmarkinturn,assessthesignificanceofthis aspectofthepoet' sstyle,assumingthatthepunctuationof thetexthasnotbeenalteredbyeditorialrevisions.Assume alsothattheexpectedfrequencyofagivenpunctuationmark ineachtextisproportionaltothelengthofthetext(fortext lengths,seequestion2above). Table9.7 The Colossus Crossingthe WaterArielWinterTrees Fullstop536545660524 Comma749581885608 Colon65412415 Semi-colon2519166 Exclamationmark17342954 Questionmark5167753 Dash30527751 4SentencelengthdistributionsinthePlathpoetrywerealso studied,withtheresultsshownintable9.8.Usethechi-square testtoexaminethesentencelengthdistributioninthetexts, andcommentonyourfindings. Table9.8 SentencelengthTheCrossingthe (no.words)ColossusWaterArielWinterTrees 1-59085141191 6-10159218277229 11-1511311317094 16-2066727157 21-2542504424 26-3019262314 31-3519151810 36-40219118 > 4 0299126 :/./...125 5Inquestion2ofthechapter2exercisesweregiventheinten-sitiesof100stressedand100unstressedsyllablesinasample ofspokenFrench.Usingthechi-squaretestasatestofgood-nessoffit,determinewhethereachsetofobservationsdeparts significantlyfromthenormaldistributionatthe5percent level. 6CoatesandLeech(1980),inastudyofthedistributionof variousmeaningsofthemodalverbsinBritishandAmerican English,foundthefollowingdistributionsfor' root'and ' epistemic'meaningsof-.and/..intheircorpus: BritishAmerican -.. root153150 epistemic74:47' /.. root226209 epistemic29 Calculatea valuefor x2ineachcase,andinterpretthefindings. 7Connolly(1979),studyingsyntacticchangeinMiddleEnglish, examinedthefrequenciesofcertainclausestructurepatterns inthreeearlyandthreelateMiddleEnglish(ME)texts. (i)Hisresultsonthefrequencyofcontiguityofpredicator andafollowingmajorelementofstructureindeclarative affirmative clausesareshownbelow: Non-contiguousContiguous EarlyME Text129129 Text229168 Text317158 LateME Text114159 Text229180 Text323195 Calculateax2valueforthevariationbetweenthetexts ineachgrouptakenseparately.Thenpoolthefrequencies :/./...126 forthethreetextsineachgroup,andperformasecond chi-squaretesttoassessdifferencesinthepatternof contiguitybetweenthegroups.Whatdotheresultstell you? (ii)Connollyalsostudiedthecontiguityofsubjectand predicatorinthetwogroupsoftexts,withthefollowing results: Non-contiguousContiguous EarlyME Text116124 Text227155 Text39149 LateME Text14179 Text210201 Text35203 Calculatex2valuesforthevariationwithineachgroup, andinterpretyourfindings, (iii)A'thirdstudyinvolvedthefrequencyofinversionof subjectandverbafteranintroductoryadverbialin declarativeaffirmativeclauses: No inversionInversion EarlyME Text12711 Text23416 Text33414 LateME Text110927 Text26111 Text34929 Calculatex2valuesforthevariationwithineachgroup, andinterprettheresults. 10TheF distributionanditsuses 10.1Introduction Insection5.2wediscussedthesamplingdistributionofthemeans ofsamplestakenfromthesamepopulation,andinsection7.2we extendedtheconceptofsamplingdistributiontothedifference betweenthemeansofpairsofsamplesfromapopulation.Inthe presentchapterweconsiderafurt hersamplingdistribution,con-cernedwiththevariancesofsamplesfromapopulation. Considerthecasewherewetakesamples,ofsizesA!iandN2 respectively,fromeitherasinglepopulationortwopopulations withthesamevariance.Wecalculatethevariancesfor bothsamples (sjand.andthenfindtheratioIfwedidthisforavery largenumberofpairsofsamples,theratioswouldformadistri-butionknownasthetdistribution,whosepropertiesareknown forvariousnumbersofdegreesoffreedom, thesebeing(Nx 1)and (JV2 ~" 1)forthetwosamples.Inotherwords,tablesofF,theratio ofSit oallowustofindtheprobabilityofobtainingtratios aboveanygivenvalue,fortwosamplesfromthesamepopulation, orfromtwopopulationswiththesamevariance.TableA8of appendix1givesthecriticalvaluesoftwhichareequalledor exceededwith5or1percentprobability,forvariouscombina-tionsofdegreesoffreedom,inadirectionalorone-tailedtestand inanon-directionalortwo-tailedtest. 10.2Testingforhomogeneityofvariance Thereareoccasionsonwhichwemayneedtoknowwhetheror notthevariancesoftwosamplesdiffersufficientlyforustobe abletorejectthenullhypothesisthatthevariancesofthepopula-tionsfromwhichthetwosamplesweredrawnareequal.Thisis usuallyknownastestingfor' homogeneityofvariance'.Wemay :/.t./.--...128 beinterestedinvariabilityforitsownsake(seesection3.3.1),or wemaywishtoconfirmthattheassumptionofhomogeneityof variancemadebyaparametrictestsuchasthe/-testisareason-ableone. Toperformthetest,wecalculatethevarianceforeachsample, andtaketheratioofthelargertothesmallervariance,sothatthe tvalueobtainedisneverlessthan1.Wethenlookupthecritical valueoftintableA8fortherequiredsignificancelevel.Welook alongthetopofthetableforthenumberofdegreesoffreedom (Nl1)associatedwiththelargervariance,andthenfollowthis columndownuntilwefindtherowcorrespondingtothenumber ofdegreesoffreedom(N2 1) associatedwiththesmallervariance. Theintersectionoftheappropriatecolumnandrowgivesthe requiredvalueoftWemustuse a non-directionalor two-tailedtest sinceourdecisiontoplacethebiggervarianceinthenumerator ofthetratiowaspurelyarbitrary,andwehavenotinfactpre-dictedwhichvariancewillbethegreater.Thecriticalvaluesgiven intableA8thereforecorrespondtothe10and2percentlevels inthistest.Ifthecalculatedvalueequalsorexceedsthecritical valueforaparticularsignificancelevel,wemustrejectthenull hypothesisandconcludethatthevariancesaresignificantly different.Otherwise,wemaysafelyassumethatwehavenogood evidencetoabandontheclaimofhomogeneityofvariance. Insection7.2.1.2weperformeda/-testtoassessthesignifi-canceofthedifferenceinmeansbetweentwoindependentsetsof scoresonasentencerecalltest.Wecalculatedthevariancesand foundthattheydidindeedseemquitesimilar.Letusnowshow thatwewerecorrectinassuminghomogeneityofvarianceinthis case.Thevarianceswereasfollows: =15.11for 10 16.61for

. Takingtastheratioofthelargertothesmallervariance,wehave Thereare(9 1) or8degreesoffreedomforthecalculationofthe largervariance, and(10 1) or9 degreesoffreedomforthesmaller variance.WenowlookintableA8,findingthecolumncorre-spondingtothedegreesoffreedomforthelargervariance,andthe Correctionfor page129 Paragraph2 ofSection10.3,beginning:'Inchapter7,we discussed. . . ' ,shouldbereplaced bythefollowingparagraph: Inchapter7,wediscussed.andf-testsforassessingthesignificanceofthedifference betweenthemeansfortwoindependentsamples.Sometimes,however,wemaywishto comparethemeansofmorethantwoindependentsamples.Itmightatfirst bethoughtthat wecouldsimplycompareallpossiblepairingsofsamples.Itcanbeshowntheoretically, however,thatsuchaprocedureis unwise,inthatitincreasesunacceptablytheriskofType Ierror(seepage72).Whatweneedinsteadhereistheanalysisofvariancetechnique,by meansofwhichwe cantestwhetherthereareanysignificantdifferencesbetweenthemeans overall,withouttestingeachpairseparately. :/.t./.--...129 rowcorrespondingtothedegreesoffreedomforthesmaller variance.Thecriticalvalueis3.23atthe10percentleveland5.47 atthe2percentlevel(rememberingthatwearedealingherewith anon-directionaltest).Sinceourcalculatedvalueismuchsmaller thaneitherofthesecriticalvalues,wecannotrejectthenullhypo-thesisevenatthe10percentlevel,andwecanconcludethatwe aresafeinassuminghomogeneityofvariance. 10.3Anintroductiontotheanalysisofvariance(ANOVA) technique ThemostcommonandusefulapplicationoftheF-testisinthe techniqueknownas'analysisofvariance',oftenabbreviatedto ANOVA.Thisisaverycomplexarea,andwecandonomore thangiveabriefintroductionhere. Inchapter7wediscussedz-and/-testsforassessingthesig-nificanceofthedifferencebetweenthemeansfor twoindependent samples.Sometimes,however,wemaywishtocomparethemeans ofmorethantwoindependentsamples.Wecould,ofcourse,test eachpossiblepairingbymeansofaz-or/-test,asappropriate. Thiscouldgetverytedious,however,sincewithsetsofdata thereare1)/2pairs,sothatif,forexample,wehad6sets ofresults,therewouldbe6X5/2or15individualcomparisonsto make.Bymeansoftheanalysisofvariancetechnique,wecantest whetherthereareanysignificantdifferencesbetweenthemeans overall,withouthavingtotesteachpairseparately.Ifthereare significantdifferences,wecan,ofcourse,proceedtofindout whichpairsdiffer,usinga z-or/-test. Weshallnowdiscussinverysimpletermstherationalebehind theANOVAtechnique.Supposethatwehaveanumberofsetsof scoresorfrequenciesforsomelinguisticvariable.Theoverall variabilityinthewholecollectionofobservationscanbesplitinto twoparts.First,eachgroupofobservationswillshowsomevaria-tionwithinitself,owingtosamplingvariability.Therewillalsobe variationbetweengroups.Wecancalculateavarianceestimatefor thepopulationbasedoneachofthesesourcesofvariability.Ifall theobservationscomefromthesamepopulation,orfrompopula-tionswithequalmeansandvariances,weshallexpectthatthe estimatesbasedonwithin-groupsvariabilityandbetween-groups variabilitywillberoughlyequal,and. thereforethattheirratiowill beapproximately1.Butifthesamplesarefrompopulationswith :/.t./.--...130 differentmeans,sothatthereisasourceofvariabilityotherthan justthatarisingfromsampling,thenweshallexpectthebetween-groupsvariabilitytobeconsiderablylargerthanthewithin-groups variability,sothattheratioSbetween/swithinwillbegreaterthan unity.WecanusetheF-testtodeterminewhethertheobserved departureoftheratiofromunityissignificantatanychosenlevel. Letusnowconsiderinrathermoredetailthecalculations involvedinasimpleapplicationoftheANOVAmethod.Remem-berthatwhatwearetryingtodois partitionthevariationshown inthewholecollectionofdataintotwoparts,correspondingto variabilitywithinandbetweengroups.Thevariancesassociated witheachofthetwosourcesofvariationareusuallynotthem-selvesadditive.Wecanseewhyifwerememberthattheexpression forvarianceis 2S Ot - x )2 = : sothatthenumberofdegreesoffreedom,whichwillnormallybe differentforthewithin-groupsandbetween-groupsestimates,is involved.Thesumsofsquaresofdeviationsfromthemeanare, however,additive,anditisthesethatwepartition.Ifwecallthe totalsumofsquaresssthewithin-groupssumofsquaresssr andthebetween-groupssumofsquaresss/then ss

ssrss/ Theeasiestwaytoproceedistocalculatessandss/andthen t oobtainssrbysubtraction.Therequiredseriesofstepsisthere-foreasfollows. s.,:Calculatessigivenby ss.: where x=eachobservation grandmeanofalltheobservations. Itis,ofcourse,usuallymucheasiert ousethealternativeformula whichdoesnotinvolvesubtractingthemeanfrom eachobservation: :/.t./.--...::: s.,:CalculatessThisis,infact,thesumofsquaresofthe deviationsofeachindividualgroupmeanfromthegrandmean: ss/:,.,x, -,.: Onceagain,however,thereisamoreconvenientformula: _( Z* 2 )2,( Zx3 )2|( Z*)2 OOh1'i' NiN2N3N wherethesubscripts1,2,3andsoonrefertothegroupsof observations.Thus,foreachgroupweaddupallthevalues,square theresultanddividebythenumberofvalues.Wethensumallthe resultingfigures,andfinallysubtractz.:ascalculatedforthe wholecollectionofobservations. s.,:CalculatessrThisisinfactthesurpofsquaresofthe deviationofeachobservationfromthemeanfortheparticular grouptowhichitbelongs.Itcanbecalculatedinthisway,buta moreconvenientmethodistoobtainssrbysubtraction: ssrssss/ s.,Calculatethetwovarianceestimates,or'meansquares', astheyareoftencalled.Asusual,thevarianceistherelevantsum ofsquaresdividedbythenumberofdegreesoffreedominvolved. Thenumberofdegreesoffreedomforthebetween-groupsvariance is,asexpected,onelessthanthenumberofgroups.Ifthereare/ groups,therelevantfigureisthus1).Foreachgroup,the numberofdegreesoffreedomforthecalculationofthewithin-groupsvarianceisonelessthanthenumberofobservationsinthe group.Sincethereare/groups,andobservationsintotal,the totalnumberofdegreesoffreedomforthewithin-groupsvariance is (N k).Wethereforehave 2^ b // : _&SW ./ s.,Calculatetheratioofthevarianceestimates: ovv :/.t./.--...132 s.,cLookintableA8forthecriticalvalueoftforthechosen significancelevel.Lookacrossthetopofthetabletofindthe columncorrespondingtothenumberofdegreesoffreedom associatedwiththebetween-groupsvarianceestimate,andthen tracethecolumndowntotherowcorrespondingtothenumber ofdegreesoffreedomassociatedwiththewithin-groupsestimate. Sincethealternativehypothesisisthat>Sw>thisisadirectional orone-tailedtest.Ifthecriticalvalueisequalledorexceeded, rejectthenullhypothesisthatallthegroupshaveequalmeans. TheANOVAmethodmakescertainassumptionswhichwillbe familiarfromourdiscussionofthetestinsection7.2.1.2,namely thatthepopulationsfromwhichthesamplesarederivedare normal,andthattheyareofapproximatelyequalvariance.The validityofthenormalityassumptioncanbetestedbytheapproxi-matemethodsdiscussedinsection4.3,orbythechi-squaretest forgoodnessoffitdescribedinsection9.3,providedthatthe samplesarelargeenough.Homogeneityofvariancecanbetested bythemethodgiveninsection10.2above.Fortunately,however, theANOVAtechniqueisfairlytolerantofallbutgrossdepartures fromnormalityandhomogeneityofvariance. Letusnowworkthroughanexampleoftheapplicationofthe ANOVAmethod.Intable10.1aregiventhelengths,inwords,of thelinesineachoffourpoemsbyDylanThomas:'Omakemea mask' ,'Onceitwasthecolourofsayin/' ,' Notfromthisanger' and' Thespirecranes'(c//...t.-:.::.:published byJ.M.DentandSons,1952).WeshallusetheANOVAmethod totestforsignificantdifferencesbetweenthemeanlinelengths forthefourpoemsatthe5percentlevel.Weshallneedthe followinginourfurthercalculations: Zx,=117Zx2=1 177 JVi =12 Zx 2=1182 x 3 =1092=13 Zx 3=100ZxS=752=14 Zx4=109=1091N4 =11 Zx= 4 4 4Zx2=4112N=50 s.,: ,( Zx)24442 SSt=Z x2- - -- 4112=169.28 :50 :/.t./.--...133 Table10.1LinelengthsinfourpoemsbyDylanThomas Linelengths(no.ofwords)in: Poem1Poem2Poem3Poem4 'O make me...''Onceitwas...''Notfromthis..'Thespire...' (N=12)(N=13)(N=14)(N=11) 137 99 99 1010 78 1010 1110 99 98 911 811 137 9 68 810 910 611 710 89 49 911 811 911 59 7 9 5 s.,: t.:z:.:z:.:z.:z.: OOv,"T"~r+ NiN2N3N 11721182100210924442 =+++=63.48. 1213141150 s.,: ssssss/169. 28- 63. 48=105.80. s., ss/63.48 si=-==21.16. /14 - 1 ssr105.80 :==2.30. /5 0 - 4 Correctionforpages134-5 Thelastparagraph onpage134,goingoverontopage135,shouldbechangedasfollows: Two 'furtherpiecesofworkinconnectionwiththissmallresearchproblemareleftas exercisesfor the reader.We have assumedthat the line lengths are normallydistributed,with approximatelyequal variancein thefour groups.Althoughthenumberoflinesin thepoems istoosmalltocheckeachindividuallyfornormality,itmightbeworthcheckingthatthe valuesoflinelengthforallfourpoemstakentogetherareroughlynormallydistributed. Secondly,homogeneityofvariancecanbecheckedbymeansoftheF-test,asoutlinedin section10.2.Inpractice,ofcourse,thesechecksshouldbemadebeforetheANOVAtest iscarriedout.(Readerswhoareassiduousenoughtoactonthesesuggestionsshouldfind thatthepooledsetof50linesdoesindeedhaveapproximatelynormallydistributedlengths, andthatthereis no goodevidencefor abandoningthe homogeneityofvarianceassumption.) :/.t./.--...134 Table10.2Summaryofanalysisofvarianceforlinelengthin fourpoemsbyDylanThomas SourceofSumsofDegrees ofVariance variationsquaresfreedomestimate Betweengroups63.48321.16(s^) Withingroups105.80462.30(s2J Total169.2849F=9.20 s., sg 21.16r t =9.20. s i2.30 s.,cWelookupthecriticalvalueoftforthe5percentlevel, with/ 1) =3degreesoffreedomfor thebetween-groupsvariance estimate,and/. 46degrees offreedomfor thewithin-groups estimate.Thetabledoesnotgiveusanexactvaluefor46degrees offreedom,butweshallnotb^seriouslyinerrorifwetakethe valuefor45degreesoffreedom,givingacriticalvalueofabout 2.82forthesituationweareinterestedin.Ourcalculatedvalueof 9.20ismuchlargerthanthis,sowecanconcludethatthereisa significantdifferenceinthemeans. Themainquantitiesinananalysisofvarianceareoftensum-marisedintheformofatable,suchastable10.2. Threefurtherpiecesofworkinconnectionwiththissmall researchproblemareleftasexercisesforthereader.Wehave assumedthatthelinelengthsarenormallydistributed,with approximatelyequalvarianceinthefourgroups.Althoughthe numberoflinesinthepoemsistoosmalltocheckeachindi-viduallyfornormality,itmightbeworthcheckingthatthevalues oflinelengthforallfourpoemstakentogetherareroughly normallydistributed.Secondly,homogeneityofvariancecanbe checkedbymeansoftheF-test,asoutlinedinsection10.2.In practice,ofcourse,thesechecksshouldbemadebeforethe ANOVAtestiscarriedout.Thirdly,comparisonsofallsixpossible pairsofmeanscanbeundertakenusingthe/-testfor smallsamples, t oseejustwherethesignificantdifferences inmeanslie.(Readers whoareassiduousenoughtoactonthesesuggestionsshouldfind :/.t./.--...135 thatthepooledsetof50linesdoesindeedhaveapproximately normallydistributedlengths,thatthereisnogoodevidencefor abandoningthehomogeneityofvarianceassumption,andthatthe meanforpoem3(' Notfromthisanger')differs significantlyfrom theothers,butthatpoems1,2and4donotdiffersignificantly amongthemselves.) Inthisbriefintroductiontotheanalysisofvariancetechnique, wehavedealtonlywiththesituationwhere justonefactordiffer-entiatesthegroups.Inourworkedexamplethisfactorwaswhich poemwasinvolved;itcould,however,havebeenthevariouslevels ofcomplexityofsentencesinarecalltest,variousregistersofa language,andsoon.TheANOVAtechniqueisnot,infact, restrictedtosuchone-factorinvestigations,butcanbegeneralised tocovercaseswherethereismorethanonefactorinvolved.For instance,inanexperimentonsentencerecall,wemightwishto investigatetheeffectofbothsentencecomplexityandtime elapsedbetweenhearingandrepeatingthesentences.Insucha case,weshouldwanttobeabletoseparatetheeffectsofrandom samplingvariationandtheeffectofeachvariableweareinterested in.Weshouldalsowanttoknowiftherewasanyinteraction betweenthetwovariables,i.e.sentencecomplexityandtime elapsed.MorecomplexvariantsoftheANOVAtechniqueallow investigationsofthiskind,involvingtwoorevenmorefactors, tobeanalysedstatistically.Suchcomplicationsare,however, beyondthescopeofthisbook.Forfurtherdetails,readersshould refertoastandardstatisticstextbooksuchasFerguson(1981). Exercises 1Inquestion2oftheexercisestochapter2weregiventhe lengthsof100stressedand100unstressedsyllablesina sampleofspokenFrench.Testwhetherthevariancesof stressedandunstressedsyllablelengthsdiffersignificantly atthe10percentlevel. 2Inquestion1oftheexercisestochapter7weregiventhe numberofwordtypesineachof50samplesof500running wordseach,fromtwotexts.Test,atthe10percentlevel, whetherthevariancesofthenumberoftypesper500words differsignificantlybetweenthetwotexts. 3InamoredetailedstudyofsyllablelengthinFrenchthanthat referredtoinquestion1,thelengthsweredetermined(in unitsof1/50sec)forsyllablesinfourdifferentenvironments: Correctionfor page136 Omitthelastsentenceonthe page('Ifthereare,go ontoperformpairwise/-teststoshow wherethedifferenceslie.') :/.t./.--...136 A(A =8) :unstressed,atbeginningoftonegroup,after nopause B (AT =29) :unstressed,atbeginningoftonegroup,after apause C( N=50):stressed,insideatonegroup,insideamono-syllabicword D17):stressed,insideatonegroup,atbeginningof apolysyllabicword. A:16,18,17,18,12,17,17,8 B:21,14,12,10,20,11,11,16,12,12,13,12,14,15,18, 11,10,10,16,16,17,17,14,12,13,10,15,11,11 C:21,19,23,14,14,13,16,17,18,19,24,16,12, 23, 24, 16,18,12,23,19,22,20,22,23,27,25,24, 21, 20,15, 22, 22,20, 26,19,13,17, 22,14, 23,14,17, 25, 17, 22, 20, 21, 22, 20, 12 D:15, 15, 16, 21, 18, 10, 18, 16, 15, 13, 18, 16, 16, 19, 16, 16, 16 Usetheanalysisofvariancemethodtodeterminewhether therearesignificantdifferences,atthe5percentlevel,inthe meanlengthsundertfiefourconditions. 4Aclassof10German,9Japanese^11Frenchand8Russian studentsofEnglishisgivenanexerciseontheuseofthe definiteandindefinitearticlesinEnglish.Thenumberof errorsmadebythestudentsineachnationalitygroupisas follows: c.--:,-..t.-/s.- 4635 6343 87810 21287 3926 18'58 4539 57210 332 24 1 Usetheanalysisofvariancemethodtodeterminewhether therearesignificantdifferencesinthemeansatthe5per centlevel.Ifthereare,goontoperformpairwise/-teststo showwherethedifferences lie. 11Correlation 11.1Thenatureofcorrelation Twiceinthisbook,wehavehadoccasiontomentiontheconcept ofcorrelation.Insection6.2.1,wenotedadistinctionbetween experimentalstudies,inwhichthe investigatordeliberatelymanipu-latestheindependentvariableinordertotesttheeffectonthe dependentvariable,andobservationalorcorrelationalstudies,in whichhesimplyobservestherelationship,ifany,betweentwo(or more)variables.Insection6.3.6,wediscussedtwotypesofrelation betweensamples,dependingonthedesignoftheinvestigation:in anindependentgroupsdesign,two(ormore)quitedifferentsets ofsubjectsareused;butinarepeatedmeasuresoramatchedpairs design,thesamesetofsubjects,orsubjectscloselymatchedon someparticularvariable(s),isinvolvedinthetwo(ormore) conditionsofthetest.Inthelattercase,wemustallowforthe possibilitythattheresultsofthetestforeachsubjectormatched pairofsubjectsmayreflectsomecharacteristicofthatsubjector matchedpair:inotherwords,theremaybesomecorrelation betweenthescores,whichisduetothepropertiesofthesubjects ratherthantotheindependentvariablebeinginvestigated. Correlationis,then,thatareaofstatisticswhichisconcerned withthestudyofsystematicrelationshipsbetweentwo(ormore) variables,andattemptst oanswerquestionssuchas:Dohigh valuesofvariablextendtogotogetherwithhighvaluesof variable7 ?Ordohighvaluesofxgowithlowvaluesof::Oris theresomemorecomplexrelationshipbetweenxand:or perhapsnorelationshipatall?Inthischapterweshalllookbriefly atsomeofthetechniquesusedtoanswersuchquestions.Weshall dealonlywithrelationshipsbetweentwovariables:detailsof multiplecorrelationmethodscanbefoundinmoreadvanced statisticstextbooks.v 138c./- 11.2Visualrepresentationofcorrelation:scattergrams Thefirststepinanystudyofthepossiblecorrelationbetweentwo ratioorintervallyscaledvariablesistoproducewhatiscalleda ..,-or.,-Thisisagraph,inwhichtheaxes representvaluesofthetwovariables,andeachmemberofthe sampleorpopulationbeinginvestigatedisrepresentedbyadot orcrosswhosepositionisdefinedbythevaluesofthetwo variablesforthatmember.Somepossibletypesofscattergram areshowninfigures11.1-11.6.Infigure11.1wehaveastrong positivelinearcorrelationbetweenthevariablesxand:inthat thehigherthevalueofxforagivenmemberofthesample,the higherthevalueof:tendstobe.Thepointsclusterclosely, Y V y Figure11.1Highpositivecorrelation X . Figure11.2Highnegativecorrelation c./-:: Figure11.3Lowerpositivecorrelation Figure11.4Lowernegativecorrelation #*. . Figure11.5Littleornocorrelation anditwouldbeeasytodrawastraightlinethroughthemiddle ofthem.Figure11.2showsstrongnegativecorrelation,high valuesofxbeingassociatedwithlowvaluesof:andviceversa. Figure11.3,likefigure11.1,showspositivecorrelation,buthere thecorrelationisweaker:highvaluesofxstilltendtogowith 140c./- Figure11.6Nonlinearrelationship highvaluesofY,buttherearequiteafewexceptions,sothatthe pointsdonotclustersostronglyroundastraightline.Similarly, figure11.4showsweakernegativecorrelationthanfigure11.2. Infigure11.5thereis nodiscerniblepatternofdotsatall,andthe variablesthusshowlittleornocorrelation.Infigure.11.6thereis averyclearpatternofrelationship,butitisnotlinear;rather, thereisamorecomplexcurvedpattern. Therearetworeasonswhyascattergramshouldalwaysbe drawnasthefirststepinanystudyofcorrelation.Firstly,and mostobviously,thediagramprovidesa visualindicationofwhether thereisanyappreciablecorrelation,andifsowhetheritispositive ornegative.Secondly,itisimportanttoestablishwhetheror notthepatternofrelationshipislinear,since,asweshallsee, theusualquantitativemeasuresofcorrelationassumealinear relationship. Weshallnowconstructascattergramforthedataintable11.1, whichrepresentthescoresof'20foreignlanguagelearnerson (a)alanguagecomprehensiontestand(b)atranslationtest,for theforeignlanguageintoEnglish.Wewishtoobtainsomevisual indicationofwhetherthereisanysystematicrelationshipbetween thesetsofscoresonthetwotests. Thescattergraminfigure11.7showsthatthereissomedegree ofpositivecorrelationbetweenthescoresonthetwotests.The correlationisnotextremelystrong,sincethedotsdonotcluster verycloselyroundthebeststraightline whichcouldbeconstructed torunthroughthemiddleofthem.Inthenextsection,weshall seehowameasureofthedegreeofcorrelationcanbeobtained. c./-:: Table11.1Marks oncomprehensionandtranslation testsfor20foreignlanguagelearners Scoreon Learnerno.ComprehensiontestTranslationtest 11715 21313 3128 41417 51516 689 7914 81310 91116 101413 111214 121617 13109 141816 151519 161616 171217 181612 191014 2098 11.3Thequantitativeassessmentofcorrelation: correlationcoefficients ::::s-.,.-./.-/-./-.//.- Althoughascattergramisanimportantfirststepinastudyof correlationbetweentwovariableswithanintervalorratiolevel ofmeasurement,itdoesnotgiveanyprecisequantitativeassess-mentofthedegreeofcorrelationinvolved.Tosupplementthe informationgivenbyascattergram,adescriptivestatisticknown asa./-.//.-isnormallycalculated.Theexpressions forcalculatingsuchcoefficientsaresodevisedthatavalueof+1 isobtainedforperfectpositivecorrelation,avalueof1for perfectnegativecorrelation,anda valueofzerofornocorrelation atall.Forintervalorratiovariables,theappropriatemeasureis 142c./- :Figure11.7 8 10 12 14 16Scoreon comprehensiontest Scattergramofscoresontwolanguagetests 20 theso-calledt.-,..--.-./-.//.- whichtakesintoaccounttheexactmagnitudeofeachscoreon eachvariable. Forordinalvariables,scattergramsarenotreallyappropriate, sinceanyat t empttodemonstratealinearrelationshipassumes measurementonascalewithequalintervals.Ameasureofthe correlationbetweentwoordinalvariablescan,however,be obtainedbycalculatingthes,.---/./-.//.- whichisbasedsolelyontherankingoftheobservations.For certaintypesofinvestigationinvolvingnominalvariables,athird kindofcorrelationcoefficient,the,/.//.-maybeused. Eachofthethreetypesofcorrelationcoefficientwillnowbe discussedinturn. c./-:: :::::/.t.-,..--.-./-.//.- Asstatedabove,thePearsoncorrelationcoefficientisappropriate wherebothvariablesunderconsiderationareoftheratioorinter-valtype.Itsusealsoassumesthateachsetofscoresisnormally distributed. Ifwerepresentthescoreofanymemberofthesampleonone ofthevariablesasx,andthecorrespondingscoreontheother variableas,thenthesetofx-scoreswillhaveameananda standarddeviationgivenby 2 x/ S ( x - X )2 x=sx~/ V/ whereisthenumberofpairsofobservations.Similarly,the ^-scoreswillhavea meanandstandarddeviationgivenby _2/yj X( yy)2 Anyparticularx-scorecannowbeexpressedintermsofitsdis-tanceinstandarddeviationsfromthemean(thatis,asaz-score): xx z* Similarlyforany.y-score: yy ThePearsoncorrelationcoefficientisdefinedas Z VZll / wherez..,isthesumoftheproductsofthetwoz-scores,and actsasameasureoftherelationshipbetweenthetwovariables xand,and( Nl )isthenumberofdegreesoffreedomassoci-144c./- atedwiththecoefficient.Itcanbeshownt hatthevalueofthis expressionliesbetween+1(forperfectpositivecorrelation)and 1 (forperfectnegativecorrelation),asrequired. Thevalueofthecorrelationcoefficientcould,ofcourse,be calculatedbyfindingthez-scoresoneachvariablefor eachmember ofthesample,multiplyingthem,summingtheproduct soverthe wholesample,anddividingby(Ar 1).However,bycombining theaboveequationsandrearrangingtheresult,itcanbeshown t hatisalsogivenbythefollowingexpression,whichismore convenientcomputationally: t,ss, '~y / { N^ x2- ( 2x)2} { ATy2- ( S y )2}' Letusnowcalculaterforthetwosetsoflanguagetestscores forwhichwedrewascattergraminfigure11.7.Table11.2gives eachpairofscores,togetherwiththeirsquaresandproduct ,since thesearethequantitiesweneedforthecalculationofSubsti-tutingintheequationforwehave ( 2 0X36 51)( 2 6 0X2 7 3 ) r~v T ( 2 0X35 4 0 )-2 6 02}{ ( 2 0X39 3 7 )-2 7 p } 7 30 2 0 - 7 09 8 0 ~V ? 7 08 0 0 - 6 76 0 0 )( 7 87 4 0 - 7 45 2 9 ) 20 4 020 4 0 ====0 . 5 6 . y / 32 0 0X42 1 136 7 0 . 8 6 Aswesawfromthescattergram,thetwovariablesarepositively correlated,thoughnotextremelystrongly. Thequestionwhichnowarisesisjusthowgreatthecorrelation coefficientmustbeinorderthatwemayclaimasignificantcorre-lationbetweenthevariables.Wemaysetupanullhypothesisthat thereisnocorrelationbetweenthetwovariables:thatis,thatfor thepopulationfromwhichthesamplewasdrawn,thevalueof iszero.Thequestionisnow,howlargemustthecoefficientbefor theretobe,say,a5percentchanceorlessofobtainingthe observedresultwithasampleofpairsdrawnfromapopulation inwhichthereisactuallynocorrelation?TableA9ofappendix1 givesthecriticalvaluesofforvariousnumbersofpairsofobser-vations,Forourlanguagetestscoreswehave20,andthe c./-:: Table11.2Thelanguagetestscoresandtheirsquaresand products Scoreson ComprehensionTranslation M x2 ,:xy 1715289225255 1313169169169 1281446496 1417196289238 1516225256240 89648172 91481196126 1310169100130 1116121256176 1413196169182 1214144196168 1617256289272 1091008190 1816324256288 1519225361285 1616256256256 1217144289204 1612256144192 1014100196140 98816472 2 x=260 y=273Ex2= 3540=3937l,xy=3 651 criticalvalueofatthe5percentlevelinanon-directionaltest (thatis,ifouralternativehypothesisissimplythatthetwo variablesarecorrelated)is0.444,whilethatinadirectionaltest (thatis,ifouralternativehypothesisisthatthereis positivecorre-lation)is0.378.Therefore,whicheveralternativehypothesiswe havesetup,thecorrelationcoefficientisinfactsignificantatthe 5percentlevel. :::::/.s,.---/./-.//.- Ifbothofthevariableswhosecorrelationwewishtotestare ordinal,theappropriatemeasureis,as wehaveseen,theSpearman coefficient,basedonranking.Ifonevariableisordinalandthe otherintervalorratio,thentheintervalorratiodatamaybe 146c./- treatedasordinalbyusingonlytheirrankordering(withaconse-quentlossofinformation),sothattheSpearmanmethodcanbe used. Wefirstrankthesetofscoresonthefirstvariablefrom smallest(1)tolargest(N).Similarly,werankthesetofscoreson theothervariable,againfrom1 toWherethereareties(thatis, wherethereisthesamevalueonagivenvariablefortwoormore membersofthesample),weusethemeanoftheranksthatwould havebeenoccupiedifnotiehadoccurred.(Comparethesimilar procedureintheMann-Whitneyand Wilcoxontests;seechapter8.) Foreachpairofscores,wethencalculatethedifference.. betweentheranks,andsquareit.TheSpearmancorrelationcoef-ficientp(theGreekletter' rho' )isthencalculatedas 6Xd2 t::. TableA10ofappendix1givescriticalvaluesofpforvarious numbersofpairsofobservations, Letusnowworkthroughanexampleofthe use oftheSpearman correlationcoefficient.Supposethatwehaveaskedapaneloffive ten-year-oldchildrentoratetenpassagesofEnglishaccordingto howdifficulttheyfoundthemtoread,on a scale from1(extremely easy)to10(extremelydifficult).Supposealsothatwehaveasked apaneloffiveteacherstoratethesamepassagesaccordingtohow difficulttheythinkthepassageswouldbeforten-year-oldchildren toread,againonascalefrom1to10.Wethentake,tothenearest wholenumber,themedianratingforeachpassage,byeachgroup, withtheresultsshownintable11.3.Thequestionis whetherthe twosetsofratingsaresignificantlycorrelatedat,letussay,the 5percentlevel.Weshallpredictthattheyareindeedpositively correlated,andthereforeneedadirectionaltestofsignificancefor thecorrelationcoefficient. Tocalculatepwefirstrankeachsetofratings,rememberingto allocatethemeanranktoanygroupsoftiedscores.Wethen calculatethedifferencebetweenranksforeachpassageofEnglish, andsquareit.Summationoveralltenpassagesgivest.:These operationsareshownintable11.4. Wecannowcalculatepasfollows: c./-:: Table11.3Medianratingsofreadingdifficultyforten passages,bypanelsofchildrenandteachers Medianrating Passagenumberbychildrenbyteachers 175 264 345 456 5810 677 7109 832 998 1021 Table11.4Rankingofthesetsofdifficultyratingsand calculationofh d2 Rank Median ratingMedian ratingdifference childrenRankbyteachersRank(d)d2 76.554.524 654324 4354.5- 1 . 52.25 5466- 24 881010- 24 76.577- 0 . 50.25 10109911 322200 998811 21 .1100 Xd2=20.5 TableA10showsthatthecriticalvalueofpfor TV=10 inadirec-tionaltestis0.564.Thereisthereforeasignificantcorrelation betweenthechildren'sandteachers'ratingsatthislevel(indeed, thevalueofpissignificantevenatthe1percentlevel). TheSpearmancorrelationcoefficientbecomesaninaccurate measureifthereisalargenumberoftiedranks.Underthesecir-148c./- cumstances,amoreaccuratemethodistousetheranksasvalues (withmeanranksforthetiedobservations)t ocalculateaPearson coefficient(seesection11.3.2). ::::/.,/.//.- Wherethetwovariablesweareconcernedwitharenominal,the questionofwhethertheyarecorrelatedornotresolvesitselfinto thequestion,' IfaparticularitemhaspropertyA,isthereahigh probabilitythatitwillalsohavepropertyB?'Acommonapplica-tionistothestatisticsoftestinginlanguageteachingandother fields.Herethequestionatissueis,' Ifalearnerpassesontest item1,ishealsolikelytopassonitem2?'Thegeneralsituation involvedinsuchstudiescanberepresentedasa2X2*:able. //.://.: +.o ct Here,.ocandtarethefrequenciesofobservationsinthefour cellsofthetable;forinstance,.mightbethenumberoflearners whofailtestitem1butpasstestitem2,othenumberwhopass bothtestitems,andsoon.Thecorrelationbetweenthetwo variablescanbemeasuredbythephi(0)coefficient,definedas oc.t r ,.o.ct..c.ot. Itshouldbeclearfromthisformulathatthephicoefficientis verycloselyrelatedtothechi-squarestatisticdiscussedinchapter 9.Thetwostatisticsareinfactrelatedasfollows: =V x Worx2=W2. Thesignificanceofaphicoefficientcanthereforebeassessedby calculating:andthenlookingupcriticalvaluesinthechi-c./-149 Table11.5Successandfailureontwolanguagetestitems Testitem1 Testitem2FailureSuccessTotal Success Failure 49 14 211670 30 Total6337100 squaretable(tableA7)withonedegreeoffreedom(sincewehave a2X2table:seesection9.4). Letussupposethatwewishtoknowwhethersuccessonone iteminalanguagetestissignificantlycorrelatedwithsuccessona seconditem.Imaginethatwehavetested100learners,withthe resultsshownintable11.5.Wecalculate CM LO 03 CO cd f.CO CM CO LO 00 03 r^ CM CM CM LO r-03 .o CM00CM00CMCMCNCMCM * - ' cm' * - ' CMCMCMCMCMCMCM T 00 o 00 CO CM 00 CM cd03 03 03 OcoO o p co cd co LO cd LO cmO LO co co cnLO co r-00 00 co00 LOco cmCOCMCOcmCMCMCMCMCMCMCMCMCMCMcm"CMCMCM CO 00 00LO 00 r-00CM 00CD CO O) -CO CO CO O O CO LO O 00 p O03 03 00 CD LO LO LO 00 O LO03 00 CO CM 00 cm'00CN00CM00CMCMCNCMCMCMCMCMCMCMCMCMCMCMCM CM CM LO 03 00 COCO CM CM CMCMo COCO 03 CM03 CO O00 COp CO O CMo p p 03 LO COCO LO CO CO CM 00CN00CM00CM00CMCMCMCMCMCNCM CMCMCMCMCMCMCMCM co 00 co LOCD LO00 CM 00 r-CM CN CMo CO03LO00 03 00 OCM CO CD O ^t00 p 03 cd cm O cd coLO cm00cm00cm00cm'00cm00cmcmcmCMCMCMCM CMcm'CMCN CMCMCMCM in r-LO oOCO00 00 o 00 03 CM o CM LO CM CMCM CMo O CM CM O LO00 r-CO OCM CO 03 O 03 p 't CO CM 00CM00CM00CM00CM00CM00CM00CM00cm"CMCMCNCMCMcm'CMCMcm" 00 CD 03 O CO r-CO 0300 co CMr-000000 03 CMCO 00 CM 03 CM 03LO CM CM CM ^t O 0303COCDoO CO CM 00CM00CN00CM00CM00CM00CM00CM00CM00cm"00cm"CM*CNCMCMcm' f P-; oO CD LO CO 00 LO o0303 LO LO LO CMLOOr-0000 00 00 CD CN00 CM 03 CM r-CM CM O CMCNCM00CM00CN00CM00CM00CNCOCN00CM00CM00cm'00CM00 o 05 o LO_CO 00CDCO03 CO CM OCO 03CO 00 CO CO LO CO LO CM00 LO LO CD 03CD LO CO LO LOCOr-00 CM 00 CMcm"CNCMCN00cm'COCM00CMCOCM00CM00CM00CMoo"CM00 CO i o o 03 03 03 CO CO CM 03LO r-CO O COCOCO LO CM 03o CM CD 00CM o OCO 03 CO . LO 03 O CD CO CO l o00CMCMCMCMCMCNCMCMCM00CM00CM00 CM LA 00 1 2 34 7 3 ( 37 0 1 . 3 ) 37 4 3 ( 36 6 9 . 3 ) 12 7 2 ( 11 5 7 . 9 ) 2 8 5 ( 2 5 7 . 3 ) 5 4 ( 4 1 . 2 ) 88 2 7 30 0 0 ( 2 7 7 1 . 7 ) 26 7 4 ( 2 7 4 7 . 1 ) 7 5 3 ( 8 6 7 . 1 ) 1 6 5 ( 1 9 2 . 7 ) 18 ( 3 0 . 8 ) 64 7 3 6 4 1 7 20 2 5 4 5 0 7 2 6610 154 3 7 ,( 3 4 7 3 - 3 7 0 1 . 3 )237 4 3 - 36 6 9 . 3 )2 X2=+1+ (etc. ) 3 7 0 1 . 33 6 6 9 . 3 =7 8 . 8 6 df=( 51 )( 21 )=4 . Thecriticalvalueofx2forthe5percentleveland4dfis9.49. Thereisthereforeasignificantdifferenceinthewordlength distributionsatthe5percentlevel(andindeedevenatthe 0.1percentlevel). TheCrossingWinter ColossustheWaterArielTrees:/ ///2 72 84 62 61 2 7 ( 3 4 . 3 )( 3 0 . 8 )( 3 6 . 2 )( 2 5 . 7 ) other88 0 079 2 192 9 465 8 43 25 9 9 words( 87 9 2 . 7 )( 79 1 8 . 2 )( 93 0 3 . 8 )( 65 8 4 . 3 ) 88 2 779 4 993 4 066 1 03 27 2 6 .-.....199 X2=4.48. Thecriticalvalueofx2forthe5percentlevelwith3dfis 7. 82.Wecannotrejectthenullhypothesisofevendistribution ofitemscontaining/// t.-.--/ x2 ./ c/ ./.. s,-/- /.../ Fullstop19.2037.82Yes Comma26.5037.82Yes Colon32.4037.82Yes Semi-colon7.9037.82Yes Exclamationmark39.0937.82Yes Questionmark86.2537.82Yes Dash19.1537.82Yes Examinationoftheappropriatecontingencytableswillreveal thatthe. earliestvolumeofpoetry(:/.c/..hasahigher proportionofthemore' formal'punctuationmarks(colon, semi-colon)thanexpectedonthebasisofevendistribution, whilethelasttwovolumes(../andr-.:..)havefewer ofthesepunctuationmarksthanexpected.Ontheotherhand, thelatervolumeshavemoreofthelessformalpunctuation marks(exclamationmark,questionmark,dash,comma)than expected,whereastheearliestvolumehasfewerthanexpected. c-,/.r.appearstobetransitionalintermsofits punctuation.Theseresultssupporttheauthor' sgeneralhypo-thesisthatthelanguagebecomeslessformalaswegofrom earlytolaterworks. 4x2=134.63. Thecriticalvalueof x2 for the5 percentlevel and (9 1) (4 1) or24dfis36.42.Wecanthusclaimasignificantdifferencein thedistributionofsentencelengthforthefourtexts.By consideringtheobservedandexpectedfrequenciesinthe contingencytable,itispossibletoshowthat:/.c/.has fewershortsentences,andmorelongsentences,thanexpected foranevendistribution,whiler-.:..hasmanymore shortsentences,andfewerlongones,thanexpected.c-, /.r.and../havealowproportionofveryshort 200.,,.-.: sentences( 1- 5words)andverylongsentences( >35words), andappeartobetransitionalbetweentheothertwovolumes. Overall,theresultstendtoconfirmtheconclusionsdrawn frompunctuationstudies:thereseemstobeadecreasein formalcomplexityfromearliertolaterworks,reflectinga structurallysimplerandlessformalstyle. 5Toavoidcellswithexpectedfrequenciesof o>o cm CO CNo CO 0000o ooo> o>o CN CO CN * o oooo , 'd- CN o OoOo CN LO 00o ooo 'd-TLD 00 o00 ooo CO CO CT> oo CN oCO oOO LO ' LD LD r-' CN CN CN CNoo-o> CNCNCO CT> CN 00. oo r-' CN LD* oo TCN CN CNo o>CN 00 T oo CN LO o'd- o * #CN CM CN To ooooo