OrganizingandVisualizingVariablesChapter2ObjectivesInthischapteryoulearn:

Methodstoorganizevariables.Methodstovisualizevariables.Methodstoorganizeorvisualizemorethanonevariableatthesametime.Principlesofpropervisualizations.CategoricalDataAreOrganizedByUtilizingTablesDCOVACategoricalDataTallyingData

SummaryTable

OneCategoricalVariable

TwoCategoricalVariablesContingencyTableOrganizingCategoricalData:SummaryTableAsummarytabletalliesthefrequenciesorpercentagesofitemsinasetofcategoriessothatyoucanseedifferencesbetweencategories.

ReasonForShoppingOnline?PercentBetterPrices37%Avoidingholidaycrowdsorhassles29%Convenience18%Betterselection13%Shipsdirectly3%DCOVAMainReasonYoungAdultsShopOnlineSource:Dataextractedandadaptedfrom“MainReasonYoungAdultsShopOnline?”USAToday,December5,2012,p.1A.AContingencyTableHelpsOrganizeTwoorMoreCategoricalVariablesUsedtostudypatternsthatmayexistbetweentheresponsesoftwoormorecategoricalvariablesCrosstabulatesortalliesjointlytheresponsesofthecategoricalvariablesFortwovariablesthetalliesforonevariablearelocatedintherowsandthetalliesforthesecondvariablearelocatedinthecolumnsDCOVAContingencyTable-ExampleArandomsampleof400invoicesisdrawn.Eachinvoiceiscategorizedasasmall,medium,orlargeamount.Eachinvoiceisalsoexaminedtoidentifyifthereareanyerrors.Thisdataarethenorganizedinthecontingencytabletotheright.DCOVANoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400ContingencyTableShowingFrequencyofInvoicesCategorizedBySizeandThePresenceOfErrorsContingencyTableBasedOnPercentageOfOverallTotalNoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400DCOVANoErrorsErrorsTotalSmallAmount42.50%5.00%47.50%MediumAmount25.00%10.00%35.00%LargeAmount16.25%1.25%17.50%Total83.75%16.25%100.0%42.50%=170/40025.00%=100/40016.25%=65/40083.75%ofsampledinvoiceshavenoerrorsand47.50%ofsampledinvoicesareforsmallamounts.ContingencyTableBasedOnPercentageofRowTotalsNoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400DCOVANoErrorsErrorsTotalSmallAmount89.47%10.53%100.0%MediumAmount71.43%28.57%100.0%LargeAmount92.86%7.14%100.0%Total83.75%16.25%100.0%89.47%=170/19071.43%=100/14092.86%=65/70Mediuminvoiceshavealargerchance(28.57%)ofhavingerrorsthansmall(10.53%)orlarge(7.14%)invoices.ContingencyTableBasedOnPercentageOfColumnTotalsNoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400DCOVANoErrorsErrorsTotalSmallAmount50.75%30.77%47.50%MediumAmount29.85%61.54%35.00%LargeAmount19.40%7.69%17.50%Total100.0%100.0%100.0%50.75%=170/33530.77%=20/65Thereisa61.54%chancethatinvoiceswitherrorsareofmediumsize.TablesUsedForOrganizing

NumericalDataDCOVANumericalDataOrderedArrayCumulativeDistributionsFrequencyDistributionsOrganizingNumericalData:

OrderedArrayAnorderedarrayisasequenceofdata,inrankorder,fromthesmallestvaluetothelargestvalue.Showsrange(minimumvaluetomaximumvalue)Mayhelpidentifyoutliers(unusualobservations)AgeofSurveyedCollegeStudentsDayStudents161717181818191920202122222527323842NightStudents181819192021232832334145DCOVAOrganizingNumericalData:

FrequencyDistributionThefrequencydistributionisasummarytableinwhichthedataarearrangedintonumericallyorderedclasses.

Youmustgiveattentiontoselectingtheappropriatenumberofclassgroupingsforthetable,determiningasuitablewidthofaclassgrouping,andestablishingtheboundariesofeachclassgroupingtoavoidoverlapping.Thenumberofclassesdependsonthenumberofvaluesinthedata.Withalargernumberofvalues,typicallytherearemoreclasses.Ingeneral,afrequencydistributionshouldhaveatleast5butnomorethan15classes.Todeterminethewidthofaclassinterval,youdividetherange(Highestvalue–Lowestvalue)ofthedatabythenumberofclassgroupingsdesired.DCOVAOrganizingNumericalData:

FrequencyDistributionExampleExample:Amanufacturerofinsulationrandomlyselects20winterdaysandrecordsthedailyhightemperature24,35,17,21,24,37,26,46,58,30,32,13,12,38,41,43,44,27,53,27DCOVAOrganizingNumericalData:

FrequencyDistributionExampleSortrawdatainascendingorder:

12,13,17,21,24,24,26,27,27,30,32,35,37,38,41,43,44,46,53,58Findrange:58-12=46Selectnumberofclasses:5(usuallybetween5and15)Computeclassinterval(width):10(46/5thenroundup)Determineclassboundaries(limits):Class1:10butlessthan20Class2:20butlessthan30Class3:30butlessthan40Class4:40butlessthan50Class5:50butlessthan60Computeclassmidpoints:15,25,35,45,55Countobservations&assigntoclassesDCOVAOrganizingNumericalData:FrequencyDistributionExample

ClassMidpoints Frequency10butlessthan2015 320butlessthan3025 630butlessthan4035 540butlessthan5045 450butlessthan6055 2

Total

20Datainorderedarray:12,13,17,21,24,24,26,27,27,30,32,35,37,38,41,43,44,46,53,58DCOVAOrganizingNumericalData:Relative&PercentFrequencyDistributionExample

ClassFrequency10butlessthan203.1515%20butlessthan306.3030%30butlessthan405.2525%40butlessthan504.2020%50butlessthan602.1010%

Total

201.00100%RelativeFrequency

PercentageDCOVARelativeFrequency=Frequency/Total,e.g.0.10=2/20OrganizingNumericalData:CumulativeFrequencyDistributionExampleClass10butlessthan20 315%315%20butlessthan30 630%945%30butlessthan40 525%1470%40butlessthan50 420%1890%50butlessthan60 210%20100%Total 20100 20 100%

PercentageCumulativePercentageCumulativePercentage=CumulativeFrequency/Total*100e.g.45%=100*9/20FrequencyCumulativeFrequencyDCOVAWhyUseaFrequencyDistribution?ItcondensestherawdataintoamoreusefulformItallowsforaquickvisualinterpretationofthedataItenablesthedeterminationofthemajorcharacteristicsofthedatasetincludingwherethedataareconcentrated/clusteredDCOVAFrequencyDistributions:

SomeTipsDifferentclassboundariesmayprovidedifferentpicturesforthesamedata(especiallyforsmallerdatasets)ShiftsindataconcentrationmayshowupwhendifferentclassboundariesarechosenAsthesizeofthedatasetincreases,theimpactofalterationsintheselectionofclassboundariesisgreatlyreducedWhencomparingtwoormoregroupswithdifferentsamplesizes,youmustuseeitherarelativefrequencyorapercentagedistributionDCOVAVisualizingCategoricalDataThroughGraphicalDisplaysDCOVACategoricalDataVisualizingDataBarChartSummaryTableForOneVariableContingencyTableForTwoVariablesSideBySideBarChartPieChartParetoChartVisualizingCategoricalData:

TheBarChartThebarchartvisualizesacategoricalvariableasaseriesofbars.Thelengthofeachbarrepresentseitherthefrequencyorpercentageofvaluesforeachcategory.Eachbarisseparatedbyaspacecalledagap.

DCOVAReasonForShoppingOnline?PercentBetterPrices37%Avoidingholidaycrowdsorhassles29%Convenience18%Betterselection13%Shipsdirectly3%VisualizingCategoricalData:

ThePieChartThepiechartisacirclebrokenupintoslicesthatrepresentcategories.Thesizeofeachsliceofthepievariesaccordingtothepercentageineachcategory.

DCOVAReasonForShoppingOnline?PercentBetterPrices37%Avoidingholidaycrowdsorhassles29%Convenience18%Betterselection13%Shipsdirectly3%VisualizingCategoricalData:

TheParetoChartUsedtoportraycategoricaldataAverticalbarchart,wherecategoriesareshownindescendingorderoffrequencyAcumulativepolygonisshowninthesamegraphUsedtoseparatethe“vitalfew”fromthe“trivialmany”DCOVAVisualizingCategoricalData:

TheParetoChart(con’t)DCOVA CumulativeCause Frequency Percent PercentWarpedcardjammed 36550.41% 50.41%Cardunreadable 23432.32% 82.73%ATMmalfunctions 32 4.42% 87.15%ATMoutofcash 28 3.87% 91.02%Invalidamountrequested 23 3.18% 94.20%Wrongkeystroke 23 3.18% 97.38%Lackoffundsinaccount 19 2.62% 100.00%Total 724 100.00%Source:DataextractedfromA.Bhalla,“Don’tMisusetheParetoPrinciple,”SixSigmaForumMagazine,May2009,pp.15–18.OrderedSummaryTableForCausesOfIncompleteATMTransactionsVisualizingCategoricalData:

TheParetoChart(con’t)DCOVAThe“VitalFew”VisualizingCategoricalData:

SideBySideBarChartsThesidebysidebarchartrepresentsthedatafromacontingencytable.

DCOVAInvoiceswitherrorsaremuchmorelikelytobeofmediumsize(61.54%vs30.77%and7.69%)NoErrorsErrorsTotalSmallAmount50.75%30.77%47.50%MediumAmount29.85%61.54%35.00%LargeAmount19.40%7.69%17.50%Total100.0%100.0%100.0%VisualizingNumericalDataByUsingGraphicalDisplaysNumericalDataOrderedArrayStem-and-LeafDisplayHistogramPolygonOgiveFrequencyDistributionsandCumulativeDistributionsDCOVAStem-and-LeafDisplayAsimplewaytoseehowthedataaredistributedandwhereconcentrationsofdataexistMETHOD:Separatethesorteddataseries

intoleadingdigits(thestems)and

thetrailingdigits(the

leaves)DCOVAOrganizingNumericalData:

StemandLeafDisplayAstem-and-leafdisplayorganizesdataintogroups(calledstems)sothatthevalueswithineachgroup(theleaves)branchouttotherightoneachrow.

StemLeaf1677888992001225732842AgeofCollegeStudents DayStudents NightStudentsStemLeaf1889920138323415AgeofSurveyedCollegeStudentsDayStudents161717181818191920202122222527323842NightStudents181819192021232832334145DCOVAVisualizingNumericalData:

TheHistogramAverticalbarchartofthedatainafrequencydistributioniscalledahistogram.Inahistogramtherearenogapsbetweenadjacentbars.Theclassboundaries(orclassmidpoints)areshownonthehorizontalaxis.Theverticalaxisiseitherfrequency,relativefrequency,orpercentage.Theheightofthebarsrepresentthefrequency,relativefrequency,orpercentage.DCOVAVisualizingNumericalData:

TheHistogram

ClassFrequency10butlessthan203.151520butlessthan306.303030butlessthan405.252540butlessthan504.202050butlessthan602.1010

Total

201.00100RelativeFrequency

Percentage(Inapercentagehistogramtheverticalaxiswouldbedefinedtoshowthepercentageofobservationsperclass)DCOVAVisualizingNumericalData:

ThePolygonApercentagepolygonisformedbyhavingthemidpointofeachclassrepresentthedatainthatclassandthenconnectingthesequenceofmidpointsattheirrespectiveclasspercentages.Thecumulativepercentagepolygon,orogive,displaysthevariableofinterestalongtheXaxis,andthecumulativepercentagesalongtheYaxis.Usefulwhentherearetwoormoregroupstocompare.DCOVAVisualizingNumericalData:

ThePercentagePolygonDCOVAUsefulWhenComparingTwoorMoreGroupsVisualizingNumericalData:

ThePercentagePolygonDCOVAVisualizingTwoNumericalVariablesByUsingGraphicalDisplaysTwoNumericalVariablesScatterPlotTime-SeriesPlotDCOVAVisualizingTwoNumericalVariables:TheScatterPlotScatterplotsareusedfornumericaldataconsistingofpairedobservationstakenfromtwonumericalvariablesOnevariableismeasuredontheverticalaxisandtheothervariableismeasuredonthehorizontalaxisScatterplotsareusedtoexaminepossiblerelationshipsbetweentwonumericalvariablesDCOVAScatterPlotExampleVolumeperdayCostperday231252614029146331603816742170501885519560200DCOVAATime-SeriesPlotisusedtostudypatternsinthevaluesofanumericvariableovertimeTheTime-SeriesPlot:NumericvariableismeasuredontheverticalaxisandthetimeperiodismeasuredonthehorizontalaxisVisualizingTwoNumericalVariables:TheTimeSeriesPlotDCOVATimeSeriesPlotExampleYearNumberofFranchises1996431997541998601999732000822001952002107200399200495DCOVAAmultidimensionalcontingencytableisconstructedbytallyingtheresponsesofthreeormorecategoricalvariables.InExcelcreatingaPivotTabletoyieldaninteractivedisplayofthistype.WhileMinitabwillnotcreateaninteractivetable,ithasmanyspecializedstatistical&graphicalprocedures(notcoveredinthisbook)toanalyze&visualizemultidimensionaldata.OrganizingManyCategoricalVariables:TheMultidimensionalContingencyTableDCOVAUsingExcelPivotTablesToOrganize&VisualizeManyVariablesApivottable:SummarizesvariablesasamultidimensionalsummarytableAllowsinteractivechangingofthelevelofsummarizationandformattingofthevariablesAllowsyoutointeractively“slice”yourdatatosummarizesubsetsofdatathatmeetspecifiedcriteriaCanbeusedtodiscoverpossiblepatternsandrelationshipsinmultidimensionaldatathatsimplertablesandchartswouldfailtomakeapparent.DCOVAAMultidimensionalContingencyTableTalliesResponsesOfThreeorMoreCategoricalVariablesTwoDimensionalTableShowingTheMean10YearReturn%BrokenOutByTypeOfFund&RiskLevelDCOVAThreeDimensionalTableShowingTheMean10YearReturn%BrokenOutByTypeOfFund,MarketCap,&RiskLevelDataDiscoveryMethodsCanYieldInitialInsightsIntoDataDatadiscoveryaremethodsenabletheperformanceofpreliminaryanalysesbymanipulatinginteractivesummarizationsAreusedto:TakeacloserlookathistoricalorstatusdataReviewdataforunusualvaluesUncovernewpatternsindataDrill-downisperhapsthesimplestformofdatadiscoveryDCOVADrill-DownRevealsTheDataUnderlyingAHigher-LevelSummaryDCOVAResultsofdrillingdowntothedetailsaboutsmallmarketcapvaluefundswithlowrisk.SomeDataDiscoveryMethodsArePrimarilyVisualAtreemapissuchamethodAtreemapvisualizesthecomparisonoftwoormorevariablesusingthesizeandcolorofrectanglestorepresentvaluesWhenusedwithoneormorecategoricalvariablesitformsamultilevelhierarchyortreethatcanuncoverpatternsamongnumericalvariables.DCOVAAnExampleOfATreemapDCOVAAtreemapofthenumericalvariablesassets(size)and10-yearreturnpercentage(color)forgrowthandvaluefundsthathavesmallmarketcapitalizationsandlowriskTheChallengesinOrganizingandVisualizingVariablesWhenorganizingandvisualizingdataneedtobemindfulof:ThelimitsofothersabilitytoperceiveandcomprehendPresentationissuesthatcanundercuttheusefulnessofmethodsfromthischapter.ItiseasytocreatesummariesthatObscurethedataorCreatefalseimpressionsDCOVAAnExampleOfObscuringData,InformationOverloadDCOVAFalseImpressionsCanBeCreatedInManyWaysSelectivesummarizationPresentingonlypartofthedatacollectedImproperlyconstructedchartsPotentialpiechartissuesImproperlyscaledaxesAYaxisthatdoesnotbeginattheoriginorisabrokenaxismissingintermediatevaluesChartjunkDCOVAAnExampleofSelectiveSummarization,TheseTwoSummarizationsTellTotallyDifferentStoriesDCOVACompanyChangefromPriorYearCompanyYear1Year2Year3A+7.2%A-22.6%-33.2%+7.2%B+24.4%B-4.5%-41.9%+24.4%C+24.9%C-18.5%-31.5%+24.9%D+24.8%D-29.4%-48.1%+24.8%E+12.5%E-1.9%-25.3%+12.5%F+35.1%F-1.6%-37.8%+35.1%G+29.7%G+7.4%-13.6%+29.7%HowObviousIsItThatBothPieChartsSummarizeTheSameData?DCOVAWhyis