Chapter2,PartB

DescriptiveStatistics:

TabularandGraphicalPresentationsCrosstabulationandScatterDiagramExploratoryDataAnalysis:Stem-and-LeafDisplayExploratoryDataAnalysisThetechniquesofexploratorydataanalysisconsistofsimplearithmeticandeasy-to-drawpicturesthatcanbeusedtosummarizedataquickly.Onesuchtechniqueisthestem-and-leafdisplay.Stem-and-LeafDisplayEachdigitonastemisaleaf.Eachlineinthedisplayisreferredtoasastem.Totherightoftheverticallinewerecordthelastdigitforeachiteminrankorder.Thefirstdigitsofeachdataitemarearrangedtotheleftofaverticalline.Itissimilartoahistogramonitsside,butithastheadvantageofshowingtheactualdatavalues.Astem-and-leafdisplayshowsboththerankorder

andshapeofthedistributionofthedata.Example:HudsonAutoRepair ThemanagerofHudsonAutowouldliketogainabetterunderstandingofthecostofpartsusedintheenginetune-upsperformedintheshop.Sheexamines50customerinvoicesfortune-ups.Thecostsofparts,roundedtothenearestdollar,arelistedonthenextslide.SampleofPartsCost($)for50Tune-ups

Example:HudsonAutoRepairStem-and-LeafDisplayStem-and-LeafDisplay567891027222256788899911223445556789990023589137778914559astemaleaf

Example:HudsonAutoRepairStretchedStem-and-LeafDisplayWheneverastemvalueisstatedtwice,thefirstvaluecorrespondstoleafvaluesof0-4,andthesecondvaluecorrespondstoleafvaluesof5-9.Ifwebelievetheoriginalstem-and-leafdisplayhascondensedthedatatoomuch,wecanstretchthe

display

verticallybyusingtwostemsforeach

leadingdigit(s).StretchedStem-and-LeafDisplay5591477789135890023555678999112234456788899922227255667788991010

Example:HudsonAutoRepairStem-and-LeafDisplayLeafUnitsWheretheleafunitisnotshown,itisassumedtoequal1.Leafunitsmaybe100,10,1,0.1,andsoon.Intheprecedingexample,theleafunitwas1.Asingledigitisusedtodefineeachleaf.Theleafunitindicateshowtomultiplythestem-and-leafnumbersinordertoapproximatetheoriginaldata.Example:LeafUnit=0.1Ifwehavedatawithvaluessuchas891011LeafUnit=0.168142078.6 11.7 9.4 9.1 10.2 11.0 8.8astem-and-leafdisplayofthesedatawillbeExample:LeafUnit=10Ifwehavedatawithvaluessuchas16171819LeafUnit=1081903171806 1717 1974 1791 1682 1910 1838astem-and-leafdisplayofthesedatawillbeThe82in1682isroundeddownto80andisrepresentedasan8.CrosstabulationsandScatterDiagrams

Crosstabulationandascatterdiagramaretwomethodsforsummarizingthedatafortwovariablessimultaneously.Oftenamanagerisinterestedintabularandgraphicalmethodsthatwillhelpunderstandthe

relationshipbetweentwovariables.Thusfarwehavefocusedonmethodsthatareusedtosummarizethedataforonevariableatatime.CrosstabulationTheleftandtopmarginlabelsdefinetheclassesforthetwovariables.Crosstabulationcanbeusedwhen:onevariableisqualitativeandtheotherisquantitative,bothvariablesarequalitative,orbothvariablesarequantitative.Acrosstabulationisatabularsummaryofdatafortwovariables.PriceRangeColonialLogSplitA-FrameTotal<$200,000>$200,0001861912554530 203515Total1001214163

HomeStyleCrosstabulationExample:FingerLakesHomesquantitativevariablecategoricalvariable

ThenumberofFingerLakeshomessoldforeachstyleandpriceforthepasttwoyearsisshownbelow.CrosstabulationInsightsGainedfromPrecedingCrosstabulationOnlythreehomesinthesampleareanA-Framestyleandpricedat$200,000ormore.Thegreatestnumberofhomes(19)inthesampleareasplit-levelstyleandpricedatlessthan$200,000.Example:FingerLakesHomesPriceRangeColonialLogSplitA-FrameTotal<$200,000>$200,0001861912554530 203515Total1001214163

HomeStyleCrosstabulationFrequencydistributionforthepricerangevariableFrequencydistributionforthehomestylevariableExample:FingerLakesHomesCrosstabulation:RoworColumnPercentagesConvertingtheentriesinthetableintorowpercentagesorcolumnpercentagescanprovideadditionalinsightabouttherelationshipbetweenthetwovariables.PriceRangeColonialLogSplitA-FrameTotal<$200,000>$200,00032.7310.9134.5521.82100100Note:rowtotalsareactually100.01duetorounding.26.6731.1135.566.67

HomeStyle(Colonialand>$200K)/(All>$200K)x100=(12/45)x100Crosstabulation:RowPercentagesExample:FingerLakesHomesPriceRangeColonialLogSplitA-Frame<$200,000>$200,00060.0030.0054.2980.0040.0070.0045.7120.00

HomeStyle100 100100100Total(Colonialand>$200K)/(AllColonial)x100=(12/30)x100Crosstabulation:ColumnPercentagesExample:FingerLakesHomesCrosstabulation:Simpson’sParadox

Insomecasestheconclusionsbaseduponanaggregatedcrosstabulationcanbecompletely

reversedifwelookattheunaggregateddata.ThereversalofconclusionsbasedonaggregateandunaggregateddataiscalledSimpson’sparadox.Wemustbecarefulindrawingconclusionsabouttherelationshipbetweenthetwovariablesintheaggregatedcrosstabulation.Dataintwoormorecrosstabulationsareoftenaggregatedtoproduceasummarycrosstabulation.Thegeneralpatternoftheplottedpointssuggests

theoverallrelationshipbetweenthevariables.Onevariableisshownonthehorizontalaxisand

theothervariableisshownontheverticalaxis.Ascatterdiagramisagraphicalpresentationoftherelationshipbetweentwoquantitativevariables.ScatterDiagramandTrendlineAtrendlineprovidesanapproximationoftherelationship.ScatterDiagramAPositiveRelationshipxyScatterDiagramANegativeRelationshipxyScatterDiagramNoApparentRelationshipxyExample:PanthersFootballTeam132131424181730x=NumberofInterceptionsy=Numberof

PointsScoredScatterDiagramThePanthersfootballteamisinterestedininvestigatingtherelationship,ifany,betweeninterceptionsmadeandpointsscored.ScatterDiagramyxNumberofInterceptionsNumberofPointsScored5101520253003512304InsightsGainedfromthePrecedingScatterDiagramTherelationshipisnotperfect;allplottedpointsinthescatterdiagramarenotonastraightline.Higherpointsscoredareassociatedwithahighernumberofinterceptions.Thescatterdiagramindicatesapositiverelationshipbetweenthenumberof