(1.3)-BSAFC4-PPT商务统计课件

BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-1Chapter3NumericalDescriptiveMeasuresBusinessStatistics,AFirstCourse4thEditionBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-2Inthischapter,youlearn:

Todescribethepropertiesofcentraltendency,variation,andshapeinnumericaldataTocalculatedescriptivesummarymeasuresforapopulationTocalculatethecoefficientofvariationandZ-scoresToconstructandinterpretabox-and-whiskerplotTocalculatethecovarianceandthecoefficientofcorrelationLearningObjectivesBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-3ChapterTopicsMeasuresofcentraltendency,variation,andshapeMean,median,mode,geometricmeanQuartilesRange,interquartilerange,varianceandstandarddeviation,coefficientofvariation,Z-scoresSymmetricandskeweddistributionsPopulationsummarymeasuresMean,variance,andstandarddeviationTheempiricalruleandChebyshevruleBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-4ChapterTopicsFivenumbersummaryandbox-and-whiskerplotCovarianceandcoefficientofcorrelationPitfallsinnumericaldescriptivemeasuresandethicalissues(continued)BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-5SummaryMeasuresArithmeticMeanMedianModeDescribingDataNumericallyVarianceStandardDeviationCoefficientofVariationRangeInterquartileRangeGeometricMeanSkewnessCentralTendencyVariationShapeQuartilesBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-6MeasuresofCentralTendencyCentralTendencyArithmeticMeanMedianModeGeometricMeanOverviewMidpointofrankedvaluesMostfrequentlyobservedvalueBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-7ArithmeticMeanThearithmeticmean(samplemean)isthemostcommonmeasureofcentraltendencyForasampleofsizen:SamplesizeObservedvaluesBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-8ArithmeticMeanThemostcommonmeasureofcentraltendencyMean=sumofvaluesdividedbythenumberofvaluesAffectedbyextremevalues(outliers)(continued)012345678910Mean=3012345678910Mean=4BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-9MedianInanorderedarray,themedianisthe“middle”number(50%above,50%below)

Notaffectedbyextremevalues012345678910Median=3012345678910Median=3中位数BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-10FindingtheMedianThelocationofthemedian:Ifthenumberofvaluesisodd,themedianisthemiddlenumberIfthenumberofvaluesiseven,themedianistheaverageofthetwomiddlenumbersNotethatisnotthevalueofthemedian,onlythepositionofthemedianintherankeddataBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-11ModeAmeasureofcentraltendencyValuethatoccursmostoftenNotaffectedbyextremevaluesUsedforeithernumericalorcategorical(nominal)dataTheremaybenomodeTheremaybeseveralmodes01234567891011121314

Mode=90123456NoMode众数BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-12Fivehousesonahillbythebeach

ReviewExampleHousePrices:

$2,000,000

500,000

300,000

100,000

100,000BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-13ReviewExample:

SummaryStatisticsMean:($3,000,000/5) =$600,000Median:middlevalueofrankeddata

=$300,000Mode:mostfrequentvalue

=$100,000HousePrices:

$2,000,000500,000

300,000

100,000

100,000Sum$3,000,000BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-14Mean

isgenerallyused,unlessextremevalues(outliers)existThenmedian

isoftenused,sincethemedianisnotsensitivetoextremevalues.Example:Medianhomepricesmaybereportedforaregion–lesssensitivetooutliersWhichmeasureoflocation

isthe“best”?BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-15QuartilesQuartilessplittherankeddatainto4segmentswithanequalnumberofvaluespersegment25%25%25%25%Thefirstquartile,Q1,isthevalueforwhich25%oftheobservationsaresmallerand75%arelargerQ2isthesameasthemedian(50%aresmaller,50%arelarger)Only25%oftheobservationsaregreaterthanthethirdquartileQ1Q2Q3四分位数BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-16QuartileFormulasFindaquartilebydeterminingthevalueintheappropriatepositionintherankeddata,whereFirstquartileposition:

Q1=(n+1)/4

Secondquartileposition:

Q2=(n+1)/2

(themedianposition)

Thirdquartileposition:

Q3=3(n+1)/4

where

n

isthenumberofobservedvaluesBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-17(n=9)Q1isinthe

(9+1)/4=2.5positionoftherankeddata

sousethevaluehalfwaybetweenthe2ndand3rdvalues,

soQ1=12.5QuartilesSampleDatainOrderedArray:111213161617182122

Example:FindthefirstquartileQ1andQ3aremeasuresofnoncentrallocationQ2=median,ameasureofcentraltendencyBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-18(n=9)Q1isinthe

(9+1)/4=2.5positionoftherankeddata,

soQ1=12.5Q2isinthe

(9+1)/2=5thpositionoftherankeddata,

soQ2=median=16Q3isinthe

3(9+1)/4=7.5positionoftherankeddata,

soQ3=19.5QuartilesSampleDatainOrderedArray:111213161617182122

Example:(continued)BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-19GeometricMeanGeometricmeanUsedtomeasuretherateofchangeofavariableovertimeGeometricmeanrateofreturnMeasuresthestatusofaninvestmentovertimeWhereRiistherateofreturnintimeperiodiBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-20ExampleAninvestmentof$100,000declinedto$50,000attheendofyearoneandreboundedto$100,000atendofyeartwo:50%decrease100%increaseTheoveralltwo-yearreturniszero,sinceitstartedandendedatthesamelevel.BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-21ExampleUsethe1-yearreturnstocomputethearithmeticmeanandthegeometricmean:Arithmeticmeanrateofreturn:Geometricmeanrateofreturn:MisleadingresultMoreaccurateresult(continued)BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-22Samecenter,differentvariationMeasuresofVariationVariationVarianceStandardDeviationCoefficientofVariationRangeInterquartileRangeMeasuresofvariationgiveinformationonthespreadorvariabilityofthedatavalues.

BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-23RangeSimplestmeasureofvariationDifferencebetweenthelargestandthesmallestvaluesinasetofdata:Range=Xlargest–Xsmallest01234567891011121314Range=14-1=13Example:BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-24IgnoresthewayinwhichdataaredistributedSensitivetooutliers789101112Range=12-7=5789101112Range=12-7=5DisadvantagesoftheRange

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120Range=5-1=4Range=120-1=119BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-25InterquartileRangeCaneliminatesomeoutlierproblemsbyusingtheinterquartilerange

Eliminatesomehigh-andlow-valuedobservationsandcalculatetherangefromtheremainingvaluesInterquartilerange=3rdquartile–1stquartile =Q3–Q1四分位距BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-26InterquartileRangeMedian(Q2)XmaximumXminimumQ1Q3Example:25%25%25%25%1230455770Interquartilerange=57–30=27BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-27Average(approximately)ofsquareddeviationsofvaluesfromthemeanSample

variance:VarianceWhere

=meann=samplesizeXi=ithvalueofthevariableX方差BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-28StandardDeviationMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthevarianceHasthesameunitsastheoriginaldataSample

standarddeviation:标准差BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-29CalculationExample:

SampleStandardDeviationSample

Data(Xi):1012141517181824n=8Mean=X=16Ameasureofthe“average”scatteraroundthemeanBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-30MeasuringvariationSmallstandarddeviationLargestandarddeviationBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-31ComparingStandardDeviationsMean=15.5S=3.338

11121314151617181920211112131415161718192021DataBDataAMean=15.5S=0.9261112131415161718192021Mean=15.5S=4.567DataCBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-32AdvantagesofVarianceandStandardDeviationEachvalueinthedatasetisusedinthecalculationValuesfarfromthemeanaregivenextraweight

(becausedeviationsfromthemeanaresquared)BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-33CoefficientofVariationMeasuresrelativevariationAlwaysinpercentage(%)ShowsvariationrelativetomeanCanbeusedtocomparetwoormoresetsofdatameasuredindifferentunits变异系数BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-34ComparingCoefficient

ofVariationStockA:Averagepricelastyear=$50Standarddeviation=$5StockB:Averagepricelastyear=$100Standarddeviation=$5Bothstockshavethesamestandarddeviation,butstockBislessvariablerelativetoitspriceBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-35ZScoresAmeasureofdistancefromthemean(forexample,aZ-scoreof2.0meansthatavalueis2.0standarddeviationsfromthemean)Thedifferencebetweenavalueandthemean,dividedbythestandarddeviationAZscoreabove3.0orbelow-3.0isconsideredanoutlierZ分数BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-36ZScoresExample:Ifthemeanis14.0andthestandarddeviationis3.0,whatistheZscoreforthevalue18.5?Thevalue18.5is1.5standarddeviationsabovethemean(AnegativeZ-scorewouldmeanthatavalueislessthanthemean)(continued)BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-37ShapeofaDistributionDescribeshowdataaredistributedMeasuresofshapeSymmetricorskewedMean=Median

Mean<Median

Median<MeanRight-SkewedLeft-SkewedSymmetricPositiveskewNegativeskewBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-38NumericalMeasures

foraPopulationPopulationsummarymeasuresarecalledparametersThepopulationmeanisthesumofthevaluesinthepopulationdividedbythepopulationsize,Nμ=populationmeanN=populationsizeXi=ithvalueofthevariableXWhere

BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-39AverageofsquareddeviationsofvaluesfromthemeanPopulation

variance:PopulationVarianceWhere

μ=populationmeanN=populationsizeXi=ithvalueofthevariableXBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-40PopulationStandardDeviationMostcommonlyusedmeasureofvariationShowsvariationaboutthemeanIsthesquarerootofthepopulationvarianceHasthesameunitsastheoriginaldataPopulation

standarddeviation:BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-41BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-42Thepriceofrentingacarforaweek,withmanualtransmissionbutdecliningcollisiondamagewaiverin12Europeancountriesispresentedinthetable.Calculatethemeanandinterquartilerange.BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-43Ifthedatadistributionisapproximatelybell-shaped,thentheinterval:

containsabout68%ofthevaluesin thepopulationorthesampleTheEmpiricalRule68%经验法则BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-44

containsabout95%ofthevaluesin thepopulationorthesample

containsabout99.7%ofthevalues inthepopulationorthesampleTheEmpiricalRule99.7%95%BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-45Regardlessofhowthedataaredistributed,atleast(1-1/k2)x100%ofthevalueswillfallwithinkstandarddeviationsofthemean(fork>1)

Examples: (1-1/12)x100%=0%

…….....k=1(μ±1σ) (1-1/22)x100%=75%…........k=2(μ±2σ) (1-1/32)x100%=89%……….k=3(μ±3σ)ChebyshevRulewithinAtleastBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-46ExploratoryDataAnalysisBox-and-WhiskerPlot:AGraphicaldisplayofdatausing5-numbersummary:

Minimum--Q1--Median--Q3--MaximumExample:25%25%25%25%箱线图BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-47ShapeofBox-and-WhiskerPlotsTheBoxandcentrallinearecenteredbetweentheendpointsifdataaresymmetricaroundthemedianABox-and-WhiskerplotcanbeshownineitherverticalorhorizontalformatMinQ1MedianQ3MaxBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-48DistributionShapeand

Box-and-WhiskerPlotRight-SkewedLeft-SkewedSymmetricQ1Q2Q3Q1Q2Q3Q1Q2Q3PositiveskewNegativeskewBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-49Box-and-WhiskerPlotExampleBelowisaBox-and-Whiskerplotforthefollowingdata:

0222334551027Thedataarerightskewed,astheplotdepicts023527MinQ1Q2Q3MaxBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-50TheSampleCovarianceThesamplecovariancemeasuresthestrengthofthelinearrelationshipbetweentwovariables(calledbivariatedata)Thesamplecovariance:OnlyconcernedwiththestrengthoftherelationshipNocausaleffectisimplied协方差BusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-51Covariancebetweentworandomvariables:cov(X,Y)>0XandYtendtomoveinthesamedirectioncov(X,Y)<0XandYtendtomoveinoppositedirectionscov(X,Y)=0XandYareindependentInterpretingCovarianceBusinessStatistics,AFirstCourse(4e)©2006Prentice-Hall,Inc.Chap3-52CoefficientofCorrelationMeasurestherelativestrengthofthelinearrelationshipbetweentwovariables