(1.22)-PowerPoint Ch 03b商务统计课件

Chapter3,PartB

DescriptiveStatistics:NumericalMeasuresMeasuresofDistributionShape,RelativeLocation,andDetectingOutliersExploratoryDataAnalysisMeasuresofAssociationBetweenTwoVariablesTheWeightedMeanand WorkingwithGroupedDataMeasuresofDistributionShape,

RelativeLocation,andDetectingOutliersDistributionShapez-ScoresChebyshev’sTheoremEmpiricalRuleDetectingOutliersDistributionShape:SkewnessAnimportantmeasureoftheshapeofadistributioniscalledskewness.TheformulafortheskewnessofsampledataisSkewnesscanbeeasilycomputedusingstatisticalsoftware.DistributionShape:SkewnessSymmetric(notskewed)RelativeFrequency.05.10.15.20.25.30.350Skewness=0Skewnessiszero.Meanandmedianareequal.RelativeFrequency.05.10.15.20.25.30.350DistributionShape:SkewnessModeratelySkewedLeftSkewness=-.31Skewnessisnegative.Meanwillusuallybelessthanthemedian.DistributionShape:SkewnessModeratelySkewedRightRelativeFrequency.05.10.15.20.25.30.350Skewness=.31Skewnessispositive.Meanwillusuallybemorethanthemedian.DistributionShape:SkewnessHighlySkewedRightRelativeFrequency.05.10.15.20.25.30.350Skewness=1.25Skewnessispositive(oftenabove1.0).Meanwillusuallybemorethanthemedian.Seventyefficiencyapartmentswererandomlysampledinacollegetown.Themonthlyrentpricesfortheapartmentsarelistedbelowinascendingorder.

DistributionShape:SkewnessExample:ApartmentRentsRelativeFrequency.05.10.15.20.25.30.350Skewness=.92DistributionShape:SkewnessExample:ApartmentRentsThez-scoreisoftencalledthestandardizedvalue.Itdenotesthenumberofstandarddeviationsadatavaluexiisfromthemean.z-ScoresExcel’sSTANDARDIZEfunctioncanbeusedtocomputethez-score.z-ScoresAdatavaluelessthanthesamplemeanwillhaveaz-scorelessthanzero.Adatavaluegreaterthanthesamplemeanwillhaveaz-scoregreaterthanzero.Adatavalueequaltothesamplemeanwillhaveaz-scoreofzero.Anobservation’sz-scoreisameasureoftherelativelocationoftheobservationinadataset.z-ScoreofSmallestValue(425)z-ScoresStandardizedValuesforApartmentRentsExample:ApartmentRentsChebyshev’sTheoremAtleast(1-1/z2)oftheitemsinanydatasetwillbewithinzstandarddeviationsofthemean,wherezisanyvaluegreaterthan1.Chebyshev’stheoremrequiresz>1,butzneednotbeaninteger.Atleastofthedatavaluesmustbewithinofthemean.75%z=2standarddeviationsChebyshev’sTheoremAtleastofthedatavaluesmustbewithinofthemean.89%z=3standarddeviationsAtleastofthedatavaluesmustbewithinofthemean.94%z=4standarddeviationsChebyshev’sTheoremLetz=1.5with=490.80ands=54.74Atleast(1-1/(1.5)2)=1-0.44=0.56or56%oftherentvaluesmustbebetween-z(s)=490.80-1.5(54.74)=409and+z(s)=490.80+1.5(54.74)=573(Actually,86%oftherentvaluesarebetween409and573.)Example:ApartmentRentsEmpiricalRuleWhenthedataarebelievedtoapproximateabell-shapeddistribution…Theempiricalruleisbasedonthenormaldistribution,whichiscoveredinChapter6.Theempiricalrulecanbeusedtodeterminethepercentageofdatavaluesthatmustbewithinaspecifiednumberofstandarddeviationsofthemean.EmpiricalRule Fordatahavingabell-shapeddistribution:ofthevaluesofanormalrandomvariablearewithinofitsmean.68.26%+/-1standarddeviationofthevaluesofanormalrandomvariablearewithinofitsmean.95.44%+/-2standarddeviationsofthevaluesofanormalrandomvariablearewithinofitsmean.99.72%+/-3standarddeviationsEmpiricalRulexm–3sm–1sm–2sm+1sm+2sm+3sm68.26%95.44%99.72%DetectingOutliersAnoutlierisanunusuallysmallorunusuallylargevalueinadataset.Adatavaluewithaz-scorelessthan-3orgreaterthan+3mightbeconsideredanoutlier.Itmightbe:anincorrectlyrecordeddatavalueadatavaluethatwasincorrectlyincludedinthedatasetacorrectlyrecordeddatavaluethatbelongsinthedatasetDetectingOutliersThemostextremez-scoresare-1.20and2.27Using|z|>3asthecriterionforanoutlier,therearenooutliersinthisdataset.StandardizedValuesforApartmentRentsExample:ApartmentRentsExploratoryDataAnalysis

Exploratorydataanalysisproceduresenableustousesimplearithmeticandeasy-to-drawpicturestosummarizedata.Wesimplysortthedatavaluesintoascendingorderandidentifythefive-numbersummaryandthenconstructaboxplot.Five-NumberSummary1SmallestValueFirstQuartileMedianThirdQuartileLargestValue2345Five-NumberSummaryLowestValue=425FirstQuartile=445Median=475ThirdQuartile=525LargestValue=615Example:ApartmentRentsBoxPlotAboxplotisagraphicalsummaryofdatathatisbasedonafive-numbersummary.AkeytothedevelopmentofaboxplotisthecomputationofthemedianandthequartilesQ1and

Q3.Boxplotsprovideanotherwaytoidentifyoutliers.400425450475500525550575600625Aboxisdrawnwithitsendslocatedatthefirstandthirdquartiles.BoxPlotAverticallineisdrawnintheboxatthelocationofthemedian(secondquartile).Q1=445Q3=525Q2=475Example:ApartmentRentsBoxPlotLimitsarelocated(notdrawn)usingtheinterquartilerange(IQR).Dataoutsidetheselimitsareconsideredoutliers.Thelocationsofeachoutlierisshownwiththesymbol

*. continuedBoxPlotLowerLimit:Q1-1.5(IQR)=445-1.5(80)=325UpperLimit:Q3+1.5(IQR)=525+1.5(80)=645Thelowerlimitislocated1.5(IQR)belowQ1.Theupperlimitislocated1.5(IQR)aboveQ3.Therearenooutliers(valueslessthan325orgreaterthan645)intheapartmentrentdata.Example:ApartmentRentsBoxPlotWhiskers(dashedlines)aredrawnfromtheendsoftheboxtothesmallestandlargestdatavaluesinsidethelimits.400425450475500525550575600625Smallestvalueinsidelimits=425Largestvalueinsidelimits=615Example:ApartmentRentsBoxPlotAnexcellentgraphicaltechniqueformakingcomparisonsamongtwoormoregroups.MeasuresofAssociation

BetweenTwoVariablesThusfarwehaveexaminednumericalmethodsusedtosummarizethedataforonevariableatatime.Oftenamanagerordecisionmakerisinterestedintherelationshipbetweentwovariables.Twodescriptivemeasuresoftherelationshipbetweentwovariablesarecovarianceandcorrelation

coefficient.CovariancePositivevaluesindicateapositiverelationship.Negativevaluesindicateanegativerelationship.Thecovarianceisameasureofthelinearassociationbetweentwovariables.CovarianceThecovarianceiscomputedasfollows:

forsamplesforpopulationsCorrelationCoefficientJustbecausetwovariablesarehighlycorrelated,itdoesnotmeanthatonevariableisthecauseoftheother.Correlationisameasureoflinearassociationandnotnecessarilycausation.Thecorrelationcoefficientiscomputedasfollows:

forsamplesforpopulationsCorrelationCoefficientCorrelationCoefficientValuesnear+1indicateastrongpositivelinear

relationship.Valuesnear-1indicateastrongnegativelinear

relationship.Thecoefficientcantakeonvaluesbetween-1and+1.Thecloserthecorrelationistozero,theweakertherelationship.Agolferisinterestedininvestigatingtherelationship,ifany,betweendrivingdistanceand18-holescore.277.6259.5269.1267.0255.6272.9697170707169AverageDrivingDistance(yds.)Average18-HoleScoreCovarianceandCorrelationCoefficientExample:GolfingStudyCovarianceandCorrelationCoefficient277.6259.5269.1267.0255.6272.9697170707169xy10.65-7.452.150.05-11.355.95-1.01.0001.0-1.0-10.65-7.4500-11.35-5.95AverageStd.Dev.267.070.0-35.408.2192.8944TotalExample:GolfingStudySampleCovarianceSampleCorrelationCoefficientCovarianceandCorrelationCoefficientExample:GolfingStudyTheWeightedMeanand

WorkingwithGroupedDataWeightedMeanMeanforGroupedDataVarianceforGroupedDataStandardDeviationforGroupedDataWeightedMeanWhenthemeaniscomputedbygivingeachdatavalueaweightthatreflectsitsimportance,itisreferredtoasaweightedmean.Inthecomputationofagradepointaverage(GPA),theweightsarethenumberofcredithoursearnedforeachgrade.Whendatavaluesvaryinimportance,theanalystmustchoosetheweightthatbestreflectstheimportanceofeachvalue.WeightedMeanwhere:

xi

=valueofobservationi

wi=weightforobservationiGroupedDataTheweightedmeancomputationcanbeusedtoobtainapproximationsofthemean,variance,andstandarddeviationforthegroupeddata.Tocomputetheweightedmean,wetreatthe

midpointofeachclassasthoughitwerethemeanofal