商务与经济统计(原书第14版) 课件 第3章　描述统计学Ⅱ数值方法

Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann©2020CengageLearning©2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.1Chapter3-DescriptiveStatistics:NumericalMeasures3.1-MeasuresofLocation3.2-MeasuresofVariability3.3-MeasuresofDistributionShape,RelativeLocation,andDetectingOutliers3.4-Five-NumberSummariesandBoxPlots3.5-MeasuresofAssociationBetweenTwoVariables3.6-DataDashboards:AddingNumericalMeasurestoImproveEffectiveness2NumericalMeasuresIfthemeasuresarecomputedfordatafromasample,theyarecalledsamplestatistics.Ifthemeasuresarecomputedfordatafromapopulation,theyarecalledpopulationparameters.Asamplestatisticisreferredtoasthepointestimatorofthecorrespondingpopulationparameter.3MeasuresofLocationMeanMedianModeWeightedMeanGeometricMeanPercentilesQuartiles4Mean

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Seventyefficiencyapartmentswererandomlysampledinacollegetown.Themonthlyrentsfortheseapartmentsarelistedbelow.5255405505655806106755305405505705856156755305405505705906256805355455505725906256905355455505755906257005355455605756006357005355455605756006497005355455605806006507005405505655806006707155405505655806106707156Median(1of4)Themedianofadatasetisthevalueinthemiddlewhenthedataitemsarearrangedinascendingorder.Wheneveradatasethasextremevalues,themedianisthepreferredmeasureofcentrallocation.Themedianisthemeasureoflocationmostoftenreportedforannualincomeandpropertyvaluedata.Afewextremelylargeincomesorpropertyvaluescaninflatethemean.7Median(2of4)Herewehaveanoddnumberofobservations:7observations: 26,18,27,12,14,27,and19.Rewritteninascendingorder: 12,14,18,19,26,27,and27.Themedianisthemiddlevalueinthislist,sothemedian=19.8Median(3of4)Herewehaveanevennumberofobservations:8observations: 26,18,27,12,14,27,19,and30.Rewritteninascendingorder: 12,14,18,19,26,27,27,and30.Themedianistheaverageofthetwomiddlevaluesinthislist,sothemedian=(19+26)/2=22.5.9Median(4of4)Example:ApartmentRentsNoticethatthereare70valuesprovidedwhichareinascendingorder.Averagingthe35thand36thvalues:Median(575+575)/2=575.52554055056558061067553054055057058561567553054055057059062568053554555057259062569053554555057559062570053554556057560063570053554556057560064970053554556058060065070054055056558060067071554055056558061067071510ModeThemodeofadatasetisthevaluethatoccurswithgreatestfrequency.Thegreatestfrequencycanoccurattwoormoredifferentvalues.Ifthedatahaveexactlytwomodes,thedataarebimodal.Ifthedatahavemorethantwomodes,thedataaremultimodal.Themodeis550.11WeightedMean(1of3)Insomeinstances,themeaniscomputedbygivingeachobservationaweightthatreflectsitsrelativeimportance.Thechoiceofweightsdependsontheapplication.Theweightsmightbethenumberofcredithoursearnedforeachgrade,asinGPA.Inotherweightedmeancomputations,quantitiessuchaskilograms,dollars,orvolumearefrequentlyused.12WeightedMean(2of3)RonButler,ahomebuilder,islookingovertheexpensesheincurredforahousehejustbuilt.Forthepurposeofpricingfutureprojects,hewouldliketoknowtheaveragewage($/hour)hepaidtheworkersheemployed.Listedbelowarethecategoriesofworkersheemployed,alongwiththeirrespectivewageandtotalhoursworked.WorkerWage($/hr)TotalHoursCarpenter21.60520Electrician28.72230Laborer11.80410Painter19.75270Plumber24.1616013WeightedMean(3of3)Example:ConstructionWagesWorkerXsubscriptibaselineWsubscriptibaselineWsubscriptibaselineXsubscriptibaselineCarpenter21.6052011,232.0Electrician28.722306,605.6Laborer11.804104,838.0Painter19.752705,332.5Plumber24.161603,865.6EMPTYCELLEMPTYCELL1,59031,873.7FYI,theequally-weighted(simple)mean=$21.2114GeometricMean(1of2)Thegeometricmeaniscalculatedbyfindingthenthrootoftheproductofnvalues.Itisoftenusedinanalyzinggrowthratesinfinancialdata(whereusingthearithmeticmeanwillprovidemisleadingresults).Itshouldbeappliedanytimeyouwanttodeterminethemeanrateofchangeoverseveralsuccessiveperiods(beityears,quarters,weeks,...).Othercommonapplicationsinclude:changesinpopulationsofspecies,cropyields,pollutionlevels,andbirthanddeathrates.15GeometricMean(2of2)Example:RateofReturnPeriodReturninpercentageGrowthFactor1Negative6.00.9402Negative8.00.9203Negative4.00.96042.01.02055.41.054Theaveragegrowthrateperperiodis(0.97752–1)(100)=–2.248%.16Percentiles

1780thPercentileExample:ApartmentRentsThe80thpercentileisthe56thvalueplus0.8timesthedifferencebetweenthe57thand56thvalues.Sothe80thpercentile=635+0.8(649–635)=646.2.5255405505655806106755305405505705856156755305405505705906256805355455505725906256905355455505755906257005355455605756006357005355455605756006497005355455605806006507005405505655806006707155405505655806106707151880thPercentile,Part2Example:ApartmentRents“Atleast80%oftheitemstakeonavalueof646.2orless.”“Atleast20%oftheitemstakeonavalueof646.2ormore.”52554055056558061067553054055057058561567553054055057059062568053554555057259062569053554555057559062570053554556057560063570053554556057560064970053554556058060065070054055056558060067071554055056558061067071519QuartilesQuartilesarespecificpercentiles.FirstQuartile=25thPercentileSecondQuartile=50thPercentile=MedianThirdQuartile=75thPercentile20ThirdQuartile(75thPercentile)Example:ApartmentRentsThe75thpercentileisthe53rdvalueplus0.25timesthedifferencebetweenthe54thand53rdvalues.The75thpercentile=thirdquartile=625+0.25(625–625)=625.52554055056558061067553054055057058561567553054055057059062568053554555057259062569053554555057559062570053554556057560063570053554556057560064970053554556058060065070054055056558060067071554055056558061067071521MeasuresofVariabilityItisoftendesirabletoconsidermeasuresofvariability(dispersion),aswellasmeasuresoflocation.Forexample,inchoosingsupplierAorsupplierBwemightconsidernotonlytheaveragedeliverytimeforeach,butalsothevariabilityindeliverytimeforeach.Commonmeasuresofvariabilityare:RangeInterquartileRangeVarianceStandardDeviationCoefficientofVariation22RangeTherangeofadatasetisthedifferencebetweenthelargestandsmallestdatavalue.Itisthesimplestmeasureofvariability.Itisverysensitivetothesmallestandlargestdatavalues.Range=largestvalue–smallestvalue=715–525=190.23525540550565580610675530540550570585615675530540550570590625680535545550572590625690535545550575590625700535545560575600635700535545560575600649700535545560580600650700540550565580600670715540550565580610670715InterquartileRange(IQR)Theinterquartilerangeofadatasetisthedifferencebetweenthethirdquartileandthefirstquartile.Itistherangeforthemiddle50%ofthedata.Itovercomesthesensitivitytoextremedatavalues.3rdQuartile(Q3)=6251stQuartile(Q1)=545IQR=625–545=8024Variance

Thevarianceforapopulationis:25StandardDeviationThestandarddeviationofadatasetisthepositivesquarerootofthevariance.Itismeasuredinthesameunitsasthedata,makingitmoreeasilyinterpretedthanthevariance.Thestandarddeviationofasampleis: Thestandarddeviationofapopulationis: 26CoefficientofVariationThecoefficientofvariationindicateshowlargethestandarddeviationisinrelationtothemean.Thecoefficientofvariationofasampleis:Thecoefficientofvariationofapopulationis: 27SampleVariance,StandardDeviation,

andCoefficientofVariationExample:ApartmentRentsThevarianceis:Thestandarddeviationis:

Thecoefficientofvariationis:28MeasuresofDistributionShape,RelativeLocation,andDetectingOutliersDistributionShapez-ScoresChebyshev’sTheoremEmpiricalRuleDetectingOutliers29DistributionShape:Skewness(1of2)Animportantnumericalmeasureoftheshapeofadistributioniscalledskewness.TheformulafortheskewnessofsampledataisSkewnesscanbeeasilycomputedusingstatisticalsoftware.30DistributionShape:Skewness(2of2)ModeratelySkewedLeftSkewnessisnegative.Meanwillusuallybelessthanthemedian.Symmetric(notskewed)Skewnessiszero.Meanandmedianareequal.HighlySkewedRightSkewnessispositive(oftenabove1.0)Meanwillusuallybemorethanthemedian.31z-Scores(1of2)Thez-scoreisoftencalledthestandardizedvalue.Itdenotesthenumberofstandarddeviationsadatavaluex

iisfromthemean.Anobservation’sz-scoreisameasureoftherelativelocationoftheobservationinadataset.Adatavaluelessthanthesamplemeanwillhaveaz-scorelessthanzero.Adatavaluegreaterthanthesamplemeanwillhaveaz-scoregreaterthanzero.Adatavalueequaltothesamplemeanwillhaveaz-scoreofzero.32z-Scores(2of2)Example:ApartmentRentsz-ScoreofSmallestValue(525)StandardizedValuesforApartmentRents-120(circled)-1.11-1.11-1.02-1.02-1.02-1.02-1.02-0.93-0.93-0.93-0.93-0.93-0.84-0.84-0.84-0.84-0.84-0.75-0.75-0.75-0.75-0.75-0.75-0.75-0.56-0.56-0.56-0.47-0.47-0.47-0.38-0.38-0.34-0.29-0.29-0.29-0.20-0.20-0.20-0.20-0.11-0.01-0.01-0.010.170.170.170.170.350.350.440.620.620.620.811.061.081.451.451.541.541.631.811.991.991.991.992.272.2733Chebyshev’sTheorem(1of2)Atleast(1–1/z2)ofthedatavaluesmustbewithinzstandarddeviationsofthemean,wherezisanyvaluegreaterthan1.Chebyshev’stheoremrequiresz>1;butzneednotbeaninteger.Atleast75%ofthedatavaluesmustbewithinz=2standarddeviationsofthemean.Atleast89%ofthedatavaluesmustbewithinz=3standarddeviationsofthemean.Atleast94%ofthedatavaluesmustbewithinz=4standarddeviationsofthemean.34Chebyshev’sTheorem(2of2)Example:ApartmentRents35EmpiricalRule(1of2)Whenthedataarebelievedtoapproximateabell-shapeddistribution:Theempiricalrulecanbeusedtodeterminethepercentageofdatavaluesthatmustbewithinaspecifiednumberofstandarddeviationsofthemean.Theempiricalruleisbasedonthenormaldistribution,whichiscoveredinChapter6.Fordatahavingabell-shapeddistribution:Approximately68%ofthedatavalueswillbewithinonestandarddeviationofthemean.Approximately95%ofthedatavalueswillbewithintwostandarddeviationsofthemean.Almostallofthedatavalueswillbewithinthreestandarddeviationsofthemean.36EmpiricalRule(2of2)37DetectingOutliers(1of2)Anoutlierisanunusuallysmallorunusuallylargevalueinadataset.Adatavaluewithaz-scorelessthan–3orgreaterthan+3mightbeconsideredanoutlier.Itmightbe:anincorrectlyrecordeddatavalueadatavaluethatwasincorrectlyincludedinthedatasetacorrectlyrecordeddatavaluethatbelongsinthedataset38DetectingOutliers(2of2)Example:ApartmentRentsThemostextremez-scoresare-1.20and2.27.Using|z|>3asthecriterionforanoutlier,therearenooutliersinthisdataset.StandardizedValuesforApartmentRents-120(circled)-1.11-1.11-1.02-1.02-1.02-1.02-1.02-0.93-0.93-0.93-0.93-0.93-0.84-0.84-0.84-0.84-0.84-0.75-0.75-0.75-0.75-0.75-0.75-0.75-0.56-0.56-0.56-0.47-0.47-0.47-0.38-0.38-0.34-0.29-0.29-0.29-0.20-0.20-0.20-0.20-0.11-0.01-0.01-0.010.170.170.170.170.350.350.440.620.620.620.811.061.081.451.451.541.541.631.811.991.991.991.992.272.27(circled)39Five-NumberSummariesandBoxPlotsSummarystatisticsandeasy-to-drawgraphscanbeusedtoquicklysummarizelargequantitiesofdata.Twotoolsthataccomplishthisarefive-numbersummariesandboxplots.SmallestValueFirstQuartileMedianThirdQuartileLargestValue40Five-NumberSummaryExample:ApartmentRentsLowestValue=525FirstQuartile=545Median=575ThirdQuartile=625LargestValue=715525(circled)530530535535535535535540540540540545545545545545(circled)550550550550550550560560560565565570570572575(circled)575(circled)575580580580585590590590600600600600610610615625625625635649650670670675675680(circled)690700700700700715715(circled)41BoxPlot(1of3)Aboxplotisagraphicaldisplayofdatathatisbasedonafive-numbersummary.Boxplotsprovideanotherwaytoidentifyoutliers.Aboxisdrawnwithitsendslocatedatthefirstandthirdquartiles.Averticallineisdrawnintheboxatthelocationofthemedian(secondquartile).42BoxPlot(2of3)Limitsarelocated(notdrawn)usingtheinterquartilerange(IQR).Dataoutsidetheselimitsareconsideredoutliers.Thelocationofeachoutlierisshownwiththesymbol*.43BoxPlot(3of3)Example:ApartmentRentsThelowerlimitislocated1.5(IQR)belowQ1. LowerLimit:Q1–1.5(IQR)=545–1.5(80)=425Theupperlimitislocated1.5(IQR)aboveQ3. UpperLimit:Q3+1.5(IQR)=625+1.5(80)=745Therearenooutliers(valueslessthan425orgreaterthan745)intheapartmentrentdata.44MeasuresofAssociationBetweenTwoVariablesThusfarwehaveexaminednumericalmethodsusedtosummarizethedataforonevariableatatime.Oftenamanagerordecisionmakerisinterestedintherelationshipbetweentwovariables.Twodescriptivemeasuresoftherelationshipbetweentwovariablesarecovarianceandcorrelationcoefficient.45CovarianceThecovarianceisameasureofthelinearassociationbetweentwovariables.Positivevaluesindicateapositiverelationship.Negativevaluesindicateanegativerelationship.Thecovarianceiscomputedasfollows:Forsamples:Forpopulations: 46CorrelationCoefficient(1of2)Correlationisameasureoflinearassociationandnotnecessarilycausation.Justbecausetwovariablesarehighlycorrelated,itdoesnotmeanthatonevariableisthecauseoftheother.Thecorrelationcoe