统计专业英语 统计学ch02-英文版

2.1ChapterTwoGraphicalandTabularDescriptiveTechniques2.2Introduction&Re-cap…Descriptivestatisticsinvolvesarranging,summarizing,andpresentingasetofdatainsuchawaythatusefulinformationisproduced.Itsmethodsmakeuseofgraphicaltechniquesandnumericaldescriptivemeasures(suchasaverages)tosummarizeandpresentthedata.DataStatisticsInformation2.3Populations&SamplesThegraphical&tabularmethodspresentedhereapplytobothentirepopulationsandsamplesdrawnfrompopulations.PopulationSampleSubset2.4Definitions…Avariableissomecharacteristicofapopulationorsample. E.g.studentgrades. Typicallydenotedwithacapitalletter:X,Y,Z…Thevalues

ofthevariablearetherangeofpossiblevaluesforavariable. E.g.studentmarks(0..100)Dataaretheobservedvaluesofavariable. E.g.studentmarks:{67,74,71,83,93,55,48}2.5TypesofData&InformationData(atleastforpurposesofStatistics)fallintothreemaingroups:IntervalDataNominalDataOrdinalData2.6IntervalData…Interval

data

•Realnumbers,i.e.heights,weights,prices,etc. •Alsoreferredtoasquantitativeornumerical.ArithmeticoperationscanbeperformedonIntervalData,thusitsmeaningfultotalkabout2*Height,orPrice+$1,andsoon.2.7NominalData…NominalData •The

valuesofnominaldataarecategories. E.g.responsestoquestionsaboutmaritalstatus,codedas: Single=1,Married=2,Divorced=3,Widowed=4Thesedataarecategoricalinnature;arithmeticoperationsdon’tmakeanysense(e.g.doesWidowed÷2=Married?!)Nominaldataarealsocalledqualitativeorcategorical.2.8OrdinalData…Ordinal

Dataappeartobecategoricalinnature,buttheirvalueshaveanorder;arankingtothem: E.g.Collegecourseratingsystem:poor=1,fair=2,good=3,verygood=4,excellent=5

Whileitsstillnotmeaningfultodoarithmeticonthisdata(e.g.does2*fair=verygood?!),wecansaythingslike:excellent>poororfair<verygoodThatis,orderismaintainednomatterwhatnumericvaluesareassignedtoeachcategory.2.9CalculationsforTypesofDataAsmentionedabove,•

Allcalculationsarepermittedonintervaldata.•

Onlycalculationsinvolvingarankingprocessareallowedforordinaldata.•Nocalculationsareallowedfornominaldata,savecountingthenumberofobservationsineachcategory.Thislendsitselftothefollowing“hierarchyofdata”…2.10HierarchyofData…Interval Valuesarerealnumbers. Allcalculationsarevalid. Datamaybetreatedasordinalornominal.Ordinal Valuesmustrepresenttherankedorderofthedata. Calculationsbasedonanorderingprocessarevalid. Datamaybetreatedasnominalbutnotasinterval.Nominal

Valuesarethearbitrarynumbersthatrepresentcategories. Onlycalculationsbasedonthefrequenciesofoccurrencearevalid. Datamaynotbetreatedasordinalorinterval.2.11Graphical&TabularTechniquesforNominalData…Theonlyallowablecalculationonnominaldataistocountthefrequencyofeachvalueofthevariable.Wecansummarizethedatainatablethatpresentsthecategoriesandtheircountscalledafrequencydistribution.Arelativefrequencydistributionliststhecategoriesandtheproportionwithwhicheachoccurs.2.12Example2.1LightBeerPreferenceSurveyIn2006totallightbeersalesintheUnitedStateswasapproximately3milliongallonsWiththislargeamarketbreweriesoftenneedtoknowmoreaboutwhoisbuyingtheirproduct.Themarketingmanagerofamajorbrewerywantedtoanalyzethelightbeersalesamongcollegeanduniversitystudentswhododrinklightbeer.Arandomsampleof285graduatingstudentswasaskedtoreportwhichofthefollowingistheirfavoritelightbeer.2.13Example2.11.BudweiserLight2.BuschLight3.CoorsLight4.MichelobLight5.MillerLite6.NaturalLight7.OtherbrandTheresponseswererecordedusingthecodes.Constructafrequencyandrelativefrequencydistributionforthesedataandgraphicallysummarizethedatabyproducingabarchartandapiechart.

2.14Example2.1Xm02-01*

2.15FrequencyandRelativeFrequencyDistributions2.16NominalData(Frequency)BarChartsareoftenusedtodisplayfrequencies…2.17NominalData(RelativeFrequency)PieChartsshowrelativefrequencies…2.18NominalDataItallthesameinformation,(basedonthesamedata).Justdifferentpresentation.2.19Example2.2Table2.3liststhetotalenergyconsumptionoftheUnitedStatesfromallsourcesin2005.Tomakeiteasiertoseethedetailsthetablemeasurestheheatcontentinmetrictons(1,000kilograms)ofoilequivalent.Forexample,theUnitedStatesburnedanamountofcoalandcoalproductsequivalentto545,259metrictonsofoil.Useanappropriategraphicaltechniquetodepictthesefigures.2.20Table2.3 Xm02-02*Non-RenewableEnergySourcesHeatContentCoal&coalproducts 545,258 Oil 903,440 NaturalGas 517,881 Nuclear 209,890 RenewableEnergySourcesHydroelectric 18,251 SolidBiomass 52,473 Other(Liquidbiomass,geothermal, 20,533 solar,wind,andtide,wave,&Ocean) Total 2,267,7262.21Example2.2

2.22GraphicalTechniquesforIntervalDataThereareseveralgraphicalmethodsthatareusedwhenthedataareinterval(i.e.numeric,non-categorical).Themostimportantofthesegraphicalmethodsisthehistogram.Thehistogramisnotonlyapowerfulgraphicaltechniqueusedtosummarizeintervaldata,butitisalsousedtohelpexplainprobabilities.2.23Example2.4Followingderegulationoftelephoneservice,severalnewcompanieswerecreatedtocompeteinthebusinessofprovidinglong-distancetelephoneservice.Inalmostallcasesthesecompaniescompetedonpricesincetheserviceeachofferedissimilar.Pricingaserviceorproductinthefaceofstiffcompetitionisverydifficult.Factorstobeconsideredincludesupply,demand,priceelasticity,andtheactionsofcompetitors.Long-distancepackagesmayemployper-minutecharges,aflatmonthlyrate,orsomecombinationofthetwo.Determiningtheappropriateratestructureisfacilitatedbyacquiringinformationaboutthebehaviorsofcustomersandinparticularthesizeofmonthlylong-distancebills.2.24Example2.4Aspartofalargerstudy,along-distancecompanywantedtoacquireinformationaboutthemonthlybillsofnewsubscribersinthefirstmonthaftersigningwiththecompany.Thecompany’smarketingmanagerconductedasurveyof200newresidentialsubscriberswhereinthefirstmonth’sbillswererecorded.ThesedataarestoredinfileXm02-04.Thegeneralmanagerplannedtopresenthisfindingstoseniorexecutives.Whatinformationcanbeextractedfromthesedata?2.25Example2.4InExample2.1wecreatedafrequencydistributionofthe5categories.Inthisexamplewealsocreateafrequencydistributionbycountingthenumberofobservationsthatfallintoaseriesofintervals,calledclasses.I’llexplainlaterwhyIchosetheclassesIusebelow.2.26Example2.4Wehavechoseneightclassesdefinedinsuchawaythateachobservationfallsintooneandonlyoneclass.Theseclassesaredefinedasfollows:

Classes Amountsthatarelessthanorequalto15 Amountsthataremorethan15butlessthanorequalto30 Amountsthataremorethan30butlessthanorequalto45 Amountsthataremorethan45butlessthanorequalto60 Amountsthataremorethan60butlessthanorequalto75 Amountsthataremorethan75butlessthanorequalto90 Amountsthataremorethan90butlessthanorequalto105 Amountsthataremorethan105butlessthanorequalto1202.27Example2.42.28Interpret…abouthalf(71+37=108)ofthebillsare“small”,i.e.lessthan$30Thereareonlyafewtelephonebillsinthemiddlerange.(18+28+14=60)÷200=30%i.e.nearlyathirdofthephonebillsare$90ormore.2.29BuildingaHistogram…CollecttheDataCreateafrequencydistributionforthedata… How? a)Determinethenumberofclassestouse… How? Refertotable2.6:With200observations,weshouldhavebetween7&10classes…Alternative,wecoulduseSturges’formula:Numberofclassintervals=1+3.3log(n)2.30BuildingaHistogram…CollecttheDataCreateafrequencydistributionforthedata… How? a)Determinethenumberofclassestouse.[8] b)Determinehowlargetomakeeachclass… How? Lookattherangeofthedata,thatis,

Range=LargestObservation–SmallestObservation Range=$119.63–$0=$119.63

Theneachclasswidthbecomes: Range÷(#classes)=119.63÷8≈152.31BuildingaHistogram…

2.32BuildingaHistogram…

2.33ShapesofHistograms…SymmetryAhistogramissaidtobesymmetricif,whenwedrawaverticallinedownthecenterofthehistogram,thetwosidesareidenticalinshapeandsize:FrequencyVariableFrequencyVariableFrequencyVariable2.34ShapesofHistograms…SkewnessAskewedhistogramisonewithalongtailextendingtoeithertherightortheleft:FrequencyVariableFrequencyVariablePositivelySkewedNegativelySkewed2.35ShapesofHistograms…ModalityAunimodalhistogramisonewithasinglepeak,whileabimodalhistogramisonewithtwopeaks:FrequencyVariableUnimodalFrequencyVariableBimodalAmodalclassistheclasswiththelargestnumberofobservations2.36ShapesofHistograms…BellShapeAspecialtypeofsymmetric

unimodalhistogramisonethatisbellshaped:FrequencyVariableBellShapedManystatisticaltechniquesrequirethatthepopulationbebellshaped.Drawingthehistogramhelpsverifytheshapeofthepopulationinquestion.2.37HistogramComparison…Compare&contrastthefollowinghistogramsbasedondatafromEx.2.6&Ex.2.7:Thetwocourses,BusinessStatisticsandMathematicalStatisticshaveverydifferenthistograms…unimodalvs.bimodalspreadofthemarks(narrower|wider)2.38Stem&LeafDisplay…Retainsinformationaboutindividualobservationsthatwouldnormallybelostinthecreationofahistogram.Spliteachobservationintotwoparts,astemandaleaf:e.g.Observationvalue:42.19Thereareseveralwaystosplititup…Wecouldsplititatthedecimalpoint:Orsplititatthe“tens”position(whileroundingtothenearestintegerinthe“ones”position)StemLeaf4219422.39Stem&LeafDisplay…Continuethisprocessforalltheobservations.Then,usethe“stems”fortheclassesandeachleafbecomespartofthehistogram(basedonExample2.4data)asfollows…Stem Leaf

0 0000000000111112222223333345555556666666778888999999

1 000001111233333334455555667889999

2 0000111112344666778999

3 001335589

4 124445589

5 33566

6 3458

7 022224556789

8 334457889999

9 00112222233344555999

10 001344446699

11 124557889Thus,westillhaveaccesstoouroriginaldatapoint’svalue!2.40HistogramandStem&Leaf…Comparetheoverallshapesofthefigures…2.41Ogive…(pronounced“Oh-jive”)isagraphof acumulative

frequencydistribution.Wecreateanogiveinthreesteps…First,fromthefrequencydistributioncreatedearlier,calculaterelativefrequencies:RelativeFrequency=#ofobservationsinaclass Total#ofobservations2.42RelativeFrequencies…Forexample,wehad71observationsinourfirstclass(telephonebillsfrom$0.00to$15.00).Thus,therelativefrequencyforthisclassis71÷200(thetotal#ofphonebills)=0.355(or35.5%)2.43Ogive…Isagraphofacumulative

frequencydistribution.Wecreateanogiveinthreesteps…1)Calculaterelativefrequencies.2)Calculatecumulativerelativefrequenciesbyaddingthecurrentclass’relativefrequencytothepreviousclass’cumulativerelativefrequency.(Forthefirstclass,itscumulativerelativefrequencyisjustitsrelativefrequency)2.44CumulativeRelativeFrequencies…firstclass…nextclass:.355+.185=.540lastclass:.930+.070=1.00::2.45Ogive…Isagraphofacumulative

frequencydistribution.1)Calculaterelativefrequencies.2)Calculatecumulativerelativefrequencies.3)Graphthecumulativerelativefrequencies…2.46Ogive…Theogivecanbeusedtoanswerquestionslike:Whattelephonebillvalueisatthe50thpercentile?(ReferalsotoFig.2.13inyourtextbook)“around$35”2.47DescribingTimeSeriesDataObservationsmeasuredatthesamepointintimearecalledcross-sectionaldata.Observationsmeasuredatsuccessivepointsintimearecalledtime-seriesdata.Time-seriesdatagraphedonalinechart,whichplotsthevalueofthevariableontheverticalaxisagainstthetimeperiodsonthehorizontalaxis.2.48Example2.8Werecordedthemonthlyaverageretailpriceofgasolinesince1978.

Xm02-08Drawalinecharttodescribethesedataandbrieflydescribetheresults.2.49Example2.82.50Example2.9PriceofGasolinein1982-84ConstantDollarsXm02-09RemovetheeffectofinflationinExample2.8todeterminewhethergasolinepricesarehigherthantheyhavebeeninthepastafterremovingtheeffectofinflation.2.51Example2.92.52RelationshipbetweenTwoNominalVariables…Sofarwe’velookedattabularandgraphicaltechniquesforonevariable(eithernominalorintervaldata).Across-classificationtable(orcross-tabulationtable)isusedtodescribetherelationshipbetweentwonominalvariables.Across-classificationtableliststhefrequencyofeachcombinationofthevaluesofthetwovariables…2.53Example2.10InamajorNorthAmericancitytherearefourcompetingnewspapers:thePost,GlobeandMail,Sun,andStar.Tohelpdesignadvertisingcampaigns,theadvertisingmanagersofthenewspapersneedtoknowwhichsegmentsofthenewspapermarketarereadingtheirpapers.Asurveywasconductedtoanalyzetherelationshipbetweennewspapersreadandoccupation.2.54Example2.10Asampleofnewspaperreaderswasaskedtoreportwhichnewspapertheyread:GlobeandMail(1)Post(2),Star(3),Sun(4),andtoindicatewhethertheywereblue-collarworker(1),white-collarworker(2),orprofessional(3).TheresponsesarestoredinfileXm02-10.2.55Example2.10Bycountingthenumberoftimeseachofthe12combinationsoccurs,weproducedtheTable2.9. OccupationNewspaper BlueCollar WhiteCollar Professional TotalG&M 27 29 33 89Post 18 43 51 112Star 38 21 22 81Sun 37 15 20 72Total 120 108 126 354

2.56Example2.10Ifoccupationandnewspaperarerelated,thentherewillbedifferencesinthenewspapersreadamongtheoccupations.Aneasywaytoseethisistocovertthefrequenciesineachcolumntorelativefrequenciesineachcolumn.Thatis,computethecolumntotalsanddivideeachfrequencybyitscolumntotal. OccupationNewspaper BlueCollar WhiteCollar Professional G&M 27/120=.23 29/108=.27 33/126=.26 Post 18/120=.15 43/108=.40 51/126=.40 Star 38/120=.32 21/108=.19 22/126=.17 Sun 37/120=.31 15/108=.14 20/126=.16 2.57Example2.10Interpretation:Therelativefrequenciesinthecolumns2&3aresimilar,buttherearelargedifferencesbetweencolumns1and2andbetweencolumns1and3.Thistellsusthatbluecollarworkerstendtoreaddifferentnewspapersfrombothwhitecollarworkersandprofessionalsandthatwhitecollarandprofessionalsarequitesimilarintheirnewspaperchoice.dissimilarsimilar2.58GraphingtheRelationshipBetweenTwoNominalVariables…Usethedatafromthecross-classificationtabletocreatebarcharts…ProfessionalstendtoreadtheGlobe&MailmorethantwiceasoftenastheStarorSun…2.59GraphingtheRelationshipBetweenTwoIntervalVariables…Movingfromnominaldatatointervaldata,wearefrequentlyinterestedinhowtwointervalvariablesarerelated.Toexplorethisrelationship,weemployascatterdiagram,whichplotstwovariablesagainstoneanother.TheindependentvariableislabeledXandisusuallyplacedonthehorizontalaxis,whiletheother,dependentvariable,Y,ismappedtotheverticalaxis.2.60Example2.12Arealestateagentwantedtoknowtowhatextentthesellingpriceofahomeisrelatedtoitssize.Toacquirethisinformationhetookasampleof12homesthathadrecentlysold,recordingthepriceinthousandsofdollarsandthesizeinhundredsofsquarefeet.Thesedataarelistedintheaccompanyingtable.Useagraphicaltechniquetodescribetherelationshipbetweensizeandprice.Xm02-12Size 231826202214332823202718Price 3152293552612342163083062892042651952.61Example2.12Itappearsthatinfactthereisarelationship,thatis,thegreaterthehousesizethegreaterthesellingp