商业应用统计英文版ch03

1、AppliedStatisticsforBusiness,DrxxxAssociateProfessorSchoolofEconomicsandManagementxxxx,AppliedBusinessStatistics,7thed.byKenBlack,Chapter3DescriptiveStatistics,LearningObjectives,Distinguishbetweenmeasuresofcentraltendency,measuresofvariability,measuresofshape,andmeasuresofassociation.Understandthem

2、eaningsofmean,median,mode,quartile,percentile,andrange.Computemean,median,mode,percentile,quartile,range,variance,standarddeviation,andmeanabsolutedeviationonungroupeddata.Differentiatebetweensampleandpopulationvarianceandstandarddeviation.,LearningObjectives-Continued,Understandthemeaningofstandard

3、deviationasitisappliedbyusingtheempiricalruleandChebyshevstheorem(切比雪夫定理).Computethemean,median,standarddeviation,andvarianceongroupeddata.Understandboxandwhiskerplots（箱线图，或者盒须图）,skewness,andkurtosis.Computeacoefficientofcorrelationandinterpretit.,MeasuresofCentralTendency:UngroupedData,Measuresofce

4、ntraltendencyyieldinformationabout“particularplacesorlocationsinagroupofnumbers.”CommonMeasuresofLocationModeMedianMeanPercentilesQuartiles,Mode-themostfrequentlyoccurringvalueinadatasetApplicabletoalllevelsofdatameasurement(nominal,ordinal,interval,andratio)Canbeusedtodeterminewhatcategoriesoccurmo

5、stfrequentlySometimes,nomodeexists(noduplicates)BimodalInatieforthemostfrequentlyoccurringvalue,twomodesarelistedMultimodal-Datasetsthatcontainmorethantwomodes,Mode,Median,Median-middlevalueinanorderedarrayofnumbers.Halfthedataareaboveit,halfthedataarebelowitMathematically,itsthe(n+1)/2thorderedobse

6、rvationForanarraywithanoddnumberofterms,themedianisthemiddlenumbern=11=(n+1)/2th=12/2th=6thorderedobservationForanarraywithanevennumberoftermsthemedianistheaverageofthemiddletwonumbersn=10=(n+1)/2th=11/2th=5.5th=averageof5thand6thorderedobservation,ArithmeticMean,MeanistheaverageofagroupofnumbersApp

7、licableforintervalandratiodataNotapplicablefornominalorordinaldataAffectedbyeachvalueinthedataset,includingextremevaluesComputedbysummingallvaluesinthedatasetanddividingthesumbythenumberofvaluesinthedataset,ThenumberofU.S.carsinservicebytopcarrentalcompaniesinarecentyearaccordingtoAutoRentalNewsfoll

8、ows.Computethemode,themedian,andthemean.,DemonstrationProblem3.1,DemonstrationProblem3.1,SolutionsMode:9,000(twocompanieswith9,000carsinservice)Median:With13differentcompaniesinthisgroup,N=13.Themedianislocatedatthe(13+1)/2=7thposition.Becausethedataarealreadyordered,medianisthe7thterm,whichis20,000

9、.Mean:=x/N=(1,791,000/13)=137,769.23,Percentiles,Percentile-measuresofcentraltendencythatdivideagroupofdatainto100partsAtleastn%ofthedatalieatorbelowthenthpercentile,andatmost(100-n)%ofthedatalieabovethenthpercentileExample:90thpercentileindicatesthatat90%ofthedataareequaltoorlessthanit,and10%ofthed

10、atalieaboveit,CalculatingPercentiles,Tocalculatethepthpercentile,OrderthedataCalculatei=N(p/100)DeterminethepercentileIfiisawholenumber,thenusetheaverageoftheithand(i+1)thorderedobservationOtherwise,roundiuptothenexthighestwholenumber,Quartiles,Quartile-measuresofcentraltendencythatdivideagroupofdat

11、aintofoursubgroupsQ1:25%ofthedatasetisbelowthefirstquartileQ2:50%ofthedatasetisbelowthesecondquartileQ3:75%ofthedatasetisbelowthethirdquartile,Forthecarsinservicedata,n=13,soQ1:i=13(25/100)=3.25,sousethe4thorderedobservationQ1=9,000Q3:i=13(75/100)=9.75,sousethe10thorderedobservationQ3=204,000,Quarti

12、lesforDemonstrationProblem3.1,WhichMeasureDoIUse?,Whichmeasureofcentraltendencyismostappropriate?Ingeneral,themeanispreferred,sinceithasnicemathematicalproperties(inparticular,seechapter7)Themedianandquartiles,areresistanttooutliersConsiderthefollowingthreedatasets1,2,3(median=2,mean=2)1,2,6(median=

13、2,mean=3)1,2,30(median=2,mean=11)Allhavemedian=2,butthemeanissensitivetotheoutliersIngeneral,ifthereareoutliers,themedianispreferredtothemean,BoxandWhiskerPlot,WhyUseaBoxandWhiskerPlot?Boxandwhiskerplotsareveryeffectiveandeasytoread.Theysummarizedatafrommultiplesourcesanddisplaytheresultsinasinglegr

14、aph.Boxandwhiskerplotsallowforcomparisonofdatafromdifferentcategoriesforeasier,moreeffectivedecision-making.,Aboxandwhiskerplotisdevelopedfromfivestatistics.MinimumvaluethesmallestvalueinthedatasetSecondquartilethevaluebelowwhichthelower25%ofthedataarecontainedMedianvaluethemiddlenumberinarangeofnum

15、bersThirdquartilethevalueabovewhichtheupper25%ofthedataarecontainedMaximumvaluethelargestvalueinthedatasetSometypesarecalledboxandwhiskerplotswithoutliers.,MeasuresofVariability-toolsthatdescribethespreadorthedispersionofasetofdata.Providesmoremeaningfulinformationwhenusedwithmeasuresofcentraltenden

16、cyincomparisontoothergroups,MeasuresofVariability:UngroupedData,CommonMeasuresofVariabilityRangeInter-quartileRangeMeanAbsoluteDeviationVarianceandStandardDeviationCoefficientofVariation,MeasuresofSpreadorDispersion:UngroupedData,Range,ThedifferencebetweenthelargestandthesmallestvaluesinasetofdataAd

17、vantageeasytocomputeDisadvantageisaffectedbyextremevalues,InterquartileRange,InterquartileRange-rangeofvaluesbetweenthefirstandthirdquartilesRangeofthe“middlehalf”;middle50%Usefulwhenresearchersareinterestedinthemiddle50%,andnottheextremesExample:Forthecarsinservicedata,theIQRis204,0009,000=195,000,

18、Deviationsfromthemean,UsefulforintervalorratioleveldataAnexaminationofdeviationfromthemeancanrevealinformationaboutthevariabilityofthedataDeviationsareusedmostlyasatooltocomputeothermeasuresofvariabilityHowever,thesumofdeviationsfromthearithmeticmeanisalwayszero:Sum(X-)=0Therearetwowaystosolvethisco

19、nundrum,MeanAbsoluteDeviation(MAD),Onesolutionistotaketheabsolutevalueofeachdeviationaroundthemean.ThisiscalledtheMeanAbsoluteDeviationNotethatwhiletheMADisintuitivelysimple,itisrarelyusedinpractice,PopulationVariance,AnothersolutionistotaketheSumofSquaredDeviations(SSD)aboutthemeanThepopulationvari

20、anceistheaverageofthesquareddeviationsaboutthearithmeticmeanforasetofnumbers.Thepopulationvarianceisdenotedby/sigma/.,PopulationStandardDeviation,Thepopulationstandarddeviationisameasureofthespreadofadistribution.Asymmetricdistributioniscompletelydescribedbyitscenteratthemeananditsspreaddefinedbymul

21、tiplesofitsstandarddeviation.Thestandarddeviationisthesquarerootofthevariance.,WhatsthemeaningofStd.deviation?,EmpiricalruleItsusedtostatetheapproximatepercentageofvaluesthatliewithinagivennumberofstandarddeviationsfromthemeanofasetofdataifthedataarenormallydistributed.,*Empiricalrule,*Basedontheass

22、umptionthatthedataareapproximatelynormallydistributed.,ChebyshevvTheorem,Unlikeempiricalrule,Chebyshevstheoremappliestoalldistributionsregardlessoftheirshapeandthuscanbeusedwheneverthedatadistributionshapeisunknownorisnonnormal.,ChebyshevsTheoremWithinkstandarddeviationsfromthemean,k,lieatleastpropo

23、rtionofthevalues.Assumptionk1,SampleVariance,SampleVariance-averageofthesquareddeviationsfromthearithmeticmeanSampleVariancedenotedbys2,SampleStandardDeviation,SamplestandarddeviationisthesquarerootofthesamplevarianceSameunitsasoriginaldata,Theeffectivenessofdistrictattorneyscanbemeasuredbyseveralva

24、riables,includingthenumberofconvictionspermonth,thenumberofcaseshandledpermonth,andthetotalnumberofyearsofconvictionpermonth.Aresearcherusesasampleoffivedistrictattorneysinacityanddeterminesthetotalnumberofyearsofconvictionthateachattorneywonagainstdefendantsduringthepastmonth,asreportedinthefirstco

25、lumninthefollowingtabulations.Computethemeanabsolutedeviation,thevariance,andthestandarddeviationforthesefigures.,DemonstrationProblem3.6,DemonstrationProblem3.6,SolutionTheresearchercomputesthemeanabsolutedeviation,thevariance,andthestandarddeviationforthesedatainthefollowingmanner.,ZScores,Zscorer

26、epresentsthenumberofStdDevavalue(x)isaboveorbelowthemeanofasetofnumbersZscoreallowstranslationofavaluesrawdistancefromthemeanintounitsofstddevZ=(x-)/,CoefficientofVariation(CV)measuresthevolatilityofavalue(perhapsastockportfolio),relativetoitsmean.Itstheratioofthestandarddeviationtothemean,expressed

27、asapercentageUsefulwhencomparingStdDevcomputedfromdatawithdifferentmeansMeasurementofrelativedispersion,CoefficientofVariation,CoefficientofVariation,Considertwodifferentpopulations,Since15.8611.90,thefirstpopulationismorevariable,relativetoitsmean,thanthesecondpopulation,IntervalFrequency(f)Midpoin

28、t(M)f*M20-under3062515030-under40183563040-under50114549550-under60115560560-under7036519570-under8017575502150,CalculationofGroupedMean,Sometimesdataarealreadygrouped,andyouareinterestedincalculatingsummarystatistics,CumulativeClassIntervalFrequencyFrequency20-under306630-under40182440-under5011355

29、0-under60114660-under7034970-under80150N=50,MedianofGroupedData-Example,ModeofGroupedData,ClassIntervalFrequency20-under30630-under401840-under501150-under601160-under70370-under801,MidpointofthemodalclassModalclasshasthegreatestfrequency,VarianceandStandardDeviationofGroupedData,PopulationVariancea