Quantitative Analysis Lecture02 Numerical Descript

STA1000QuantitativeAnalysisTrimester3,2012Lecture02Weekbeginningon19thNovember2012NumericalDescriptiveMeasuresCOMMONWEALTHOFAUSTRALIACopyrightRegulations1969WARNINGThismaterialhasbeenreproducedandcommunicatedtoyoubyoronbehalfofKaplanBusinessSchoolpursuanttoPartVBoftheCopyrightAct1968(theAct).ThematerialinthiscommunicationmaybesubjecttocopyrightundertheAct.AnyfurtherreproductionorcommunicationofthismaterialbyyoumaybethesubjectofcopyrightprotectionundertheAct.Donotremovethisnotice.3Chapter4:NumericalDescriptiveTechniques…•MeasuresofcentrallocationMean,median,mode•MeasuresofvariabilityRange,standarddeviation,variance,coefficientofvariation•MeasuresofrelativestandingPercentiles,quartiles•MeasuresoflinearrelationshipCovariance,correlation,determination,leastsquaresline4ArithmeticMean(orAverage)Thisisthemostpopularandusefulmeasureofcentrallocation.SumofmeasurementsNumberofmeasurementsMean=SamplemeanPopulationmeanNxiN1iSamplesizePopulationsizenxxin1i56xxxxxx6xx654321i61iExample4.1Themeanofthesampleofsixmeasurements1,3,5,2,4,3isgivenby135433.0ExampleSupposethetelephonebillsofExample2.5representapopulationofmeasurements.Thepopulationmeanis200x...xx200x20021i2001i42.1938.4553.2143.5926ExampleWhenmanyofthemeasurementshavethesamevalue,themeasurementcanbesummarisedinafrequencytable.Supposethenumbersofchildreninasampleof20familieswererecordedasfollows:NUMBEROFCHILDREN01234NUMBEROFFAMILIES3472420families2.0204(4)2(3)7(2)4(1)3(0)20x...xx20xx2021i201i7TheArithmeticMean…•…isappropriatefordescribingmeasurementdata,e.g.heightsofpeople,marksofstudentpapers,etc.•…isseriouslyaffectedbyextremevaluescalled‘outliers’.E.g.assoonasabillionairemovesintoaneighborhood,theaveragehouseholdincomeincreasesbeyondwhatitwaspreviously!Median•Anothermostcommonlyusedmeasureofcentrallocationisthemedian.•Themedianofasetofmeasurementsisthevaluethatfallsinthemiddlewhenthemeasurementsarearrangedinorderofmagnitude.8940,40,42,43,44,45,46,123Oddnumberofobservations40,40,42,43,44,45,46Example4.2Sevenemployeesalarieswererecorded(in1000s):42,45,40,46,44,40,43.Findthemediansalary.MedianSupposethedirector’ssalaryof$123000wasaddedtothegrouprecordedbefore.Findthemediansalary.Evennumberofobservations40,40,42,43,44,45,46,123Therearetwomiddlevalues!First,sortthesalaries.Then,locatethevalueinthemiddle.First,sortthesalaries.Then,locatethevaluesinthemiddle.40,40,42,43,44,45,46,12343.5,Example4.310•Anothercommonlyusedmeasureofcentrallocationisthemode.•Themodeofasetofobservationsisthevaluethatoccursmostfrequently.•Asetofdatamayhaveonemode(ormodalclass),ortwoormoremodes.•Modeisusefulforalldatatypes,thoughmainlyusedfornominaldata.•Forlargedatasets,themodalclassismuchmorerelevantthanasingle-valuemode.Mode11•Sampleandpopulationmodesarecomputedthesameway.ThemodalclassForlargedatasetsthemodalclassismuchmorerelevantthantheasingle-valuemode.Mode12Example4.4Themanagerofamenswearstoreobservedthewaistsize(incentimeters)oftrouserssoldyesterday:77,85,90,85,82,70,85,75,85,80,77,100,85,70.Themodeofthisdatasetis85cm.Thisinformationseemsvaluable(forexample,forthedesignofanewdisplayinthestore),muchmorethan‘themedianis83.5cm’.Example4.6Astatisticianwantstoreporttheresultsofamidsemesterexam,takenby100students.ThedataappearinfileXM04-06.Findthemean,medianandmode,anddescribetheinformationtheyprovide.1314Example4.6MarksMean73.98StandardError2.1502163Median81Mode84StandardDeviation21.502163SampleVariance462.34303Kurtosis0.3936606Skewness-1.073098Range89Minimum11Maximum100Sum7398Count100Themeanprovidesinformationabouttheover-allperformanceleveloftheclass.Itcanserveasatoolformakingcomparisonswithotherclassesand/orotherexams.TheMedianindicatesthathalfoftheclassreceivedagradebelow81%,andhalfoftheclassreceivedagradeabove81%.Themodemustbeusedwhendataisnominal.Ifmarksareclassifiedbylettergrade,thefrequencyofeachgradecanbecalculated.Then,themodebecomesalogicalmeasuretocompute.Note:Ifyourdataismulti-modal,thenExcelprintsthesmallestoneorN/A.ExcelOutput15ExcelHistogramforExample4.6Frequency0102030102030405060708090100MoreBinFrequency10020330240650660570108016902810024More0ModalclassThehistogramisskewedtotheleft16RelationshipbetweenMean,MedianandMode•Ifadistributionissymmetrical,themean,medianandmodecoincide.•Thatis,ifthedistributionissymmetrical,thenMean=Median=Mode.17RelationshipbetweenMean,MedianandMode•Ifadistributionisnotsymmetrical,andskewedtotheright(positivelyskewed),thethreemeasuresdiffer.Apositivelyskeweddistribution(‘skewedtotheright’)MeanMedianMode•Ifthedistributionispositivelyskewed,thenMeanMedianMode.18•Ifadistributionisnotsymmetrical,andskewedtotheleft(negativelyskewed),thethreemeasuresdiffer.MeanMedianModeAnegativelyskeweddistribution(‘skewedtotheleft’)RelationshipbetweenMean,MedianandMode•Ifthedistributionisnegativelyskewed,thenMeanMedianMode.19•Withthreemeasuresfromwhichtochoose,whichoneshouldweuse?•Themeanisgenerallyourfirstselection.However,thereareseveralcircumstanceswhenthemedianisbetter.•Themodeisseldomthebestmeasureofcentrallocation.•Onead