Chapter2,PartA

DescriptiveStatistics:

TabularandGraphicalPresentationsSummarizingCategoricalDataSummarizingQuantitativeData

Categoricaldatauselabelsornamestoidentifycategoriesoflikeitems.

Quantitativedataarenumericalvaluesthatindicatehowmuchorhowmany.SummarizingCategoricalDataFrequencyDistributionRelativeFrequencyDistributionPercentFrequencyDistributionBarChartPieChartAfrequencydistributionisatabularsummaryofdatashowingthefrequency(ornumber)ofitemsineachofseveralnon-overlappingclasses.Theobjectiveistoprovideinsightsaboutthedatathatcannotbequicklyobtainedbylookingonlyattheoriginaldata.FrequencyDistribution

GuestsstayingatMaradaInnwereaskedtoratethequalityoftheiraccommodationsasbeingexcellent,aboveaverage,average,belowaverage,orpoor.Theratingsprovidedbyasampleof20guestsare:

BelowAverageAboveAverageAboveAverageAverageAboveAverageAverageAboveAverage

AverageAboveAverageBelowAveragePoorExcellentAboveAverageAverage

AboveAverageAboveAverageBelowAveragePoorAboveAverageAverageFrequencyDistributionExample:MaradaInnFrequencyDistribution

Poor BelowAverageAverage AboveAverageExcellent

2 3 5 9

1Total 20RatingFrequencyExample:MaradaInnTherelativefrequencyofaclassisthefractionorproportionofthetotalnumberofdataitemsbelongingtotheclass.Arelativefrequencydistributionisatabularsummaryofasetofdatashowingtherelativefrequencyforeachclass.RelativeFrequencyDistributionPercentFrequencyDistributionThepercentfrequencyofaclassistherelativefrequencymultipliedby100.A

percentfrequencydistributionisatabularsummaryofasetofdatashowingthepercentfrequencyforeachclass.RelativeFrequencyand

PercentFrequencyDistributions

PoorBelowAverageAverageAboveAverageExcellent .10 .15 .25 .45

.05Total 1.0010152545

5100RelativeFrequencyPercentFrequencyRating.10(100)=101/20=.05Example:MaradaInnBarChartAbarchartisagraphicaldevicefordepictingqualitativedata.Ononeaxis(usuallythehorizontalaxis),wespecifythelabelsthatareusedforeachoftheclasses.Afrequency,relativefrequency,orpercentfrequencyscalecanbeusedfortheotheraxis(usuallytheverticalaxis).Usingabaroffixedwidthdrawnaboveeachclasslabel,weextendtheheightappropriately.Thebarsareseparatedtoemphasizethefactthateachclassisaseparatecategory.PoorBelowAverageAverageAboveAverageExcellentFrequencyRatingBarChart12345678910MaradaInnQualityRatingsParetoDiagramInqualitycontrol,barchartsareusedtoidentifythemostimportantcausesofproblems.Whenthebarsarearrangedindescendingorderofheightfromlefttoright(withthemostfrequentlyoccurringcauseappearingfirst)thebarchartiscalledaParetodiagram.Thisdiagramisnamedforitsfounder,VilfredoPareto,anItalianeconomist.PieChartThepiechartisacommonlyusedgraphicaldeviceforpresentingrelativefrequencyandpercent

frequencydistributionsforcategoricaldata.Firstdrawacircle;thenusetherelativefrequenciestosubdividethecircleintosectorsthatcorrespondtotherelativefrequencyforeachclass.Sincethereare360degreesinacircle,aclasswitharelativefrequencyof.25wouldconsume.25(360)=90degreesofthecircle.

BelowAverage15%Average25%AboveAverage45%Poor10%Excellent5%

MaradaInn

QualityRatingsPieChartInsightsGainedfromthePrecedingPieChartExample:MaradaInnOne-halfofthecustomerssurveyedgaveMaradaaqualityratingof“aboveaverage”or“excellent”(lookingattheleftsideofthepie).Thismightpleasethemanager.Foreachcustomerwhogavean“excellent”rating,thereweretwocustomerswhogavea“poor”rating(lookingatthetopofthepie).Thisshoulddispleasethemanager.SummarizingQuantitativeDataFrequencyDistributionRelativeFrequencyandPercentFrequencyDistributionsDotPlotHistogramCumulativeDistributionsOgive ThemanagerofHudsonAutowouldliketogainabetterunderstandingofthecostofpartsusedintheenginetune-upsperformedintheshop.Sheexamines50customerinvoicesfortune-ups.Thecostsofparts,roundedtothenearestdollar,arelistedonthenextslide.Example:HudsonAutoRepairFrequencyDistributionSampleofPartsCost($)for50Tune-upsFrequencyDistributionExample:HudsonAutoRepairFrequencyDistribution2.Determinethewidthofeachclass.3.Determinetheclasslimits.1.Determinethenumberofnon-overlappingclasses.Thethreestepsnecessarytodefinetheclassesforafrequencydistributionwithquantitativedataare:FrequencyDistributionGuidelinesforDeterminingtheNumberofClassesUsebetween5and20classes.Datasetswithalargernumberofelementsusuallyrequirealargernumberofclasses.Smallerdatasetsusuallyrequirefewerclasses.Thegoalistouseenoughclassestoshowthevariationinthedata,butnotsomanyclassesthatsomecontainonlyafewdataitems.FrequencyDistributionGuidelinesforDeterminingtheWidthofEachClassUseclassesofequalwidth.ApproximateClassWidth=Makingtheclassesthesamewidthreducesthechanceofinappropriateinterpretations.NoteonNumberofClassesandClassWidthInpractice,thenumberofclassesandtheappropriateclasswidtharedeterminedbytrialanderror.Onceapossiblenumberofclassesischosen,theappropriateclasswidthisfound.Theprocesscanberepeatedforadifferentnumberofclasses.FrequencyDistributionUltimately,theanalystusesjudgmenttodeterminethecombinationofthenumberofclassesandclasswidththatprovidesthebestfrequencydistributionforsummarizingthedata.FrequencyDistributionGuidelinesforDeterminingtheClassLimitsClasslimitsmustbechosensothateachdataitembelongstooneandonlyoneclass.Thelowerclasslimitidentifiesthesmallestpossibledatavalueassignedtotheclass.Theupperclasslimitidentifiesthelargestpossibledatavalueassignedtotheclass.Theappropriatevaluesfortheclasslimitsdependonthelevelofaccuracyofthedata.Anopen-endclassrequiresonlyalowerclasslimitoranupperclasslimit.FrequencyDistributionIfwechoosesixclasses:

50-59 60-6970-79 80-89 90-99 100-109

2131677

5Total50PartsCost($)FrequencyApproximateClassWidth=(109-52)/6=9.5

10Example:HudsonAutoRepairRelativeFrequencyand

PercentFrequencyDistributions

50-59 60-6970-79 80-89 90-99 100-109PartsCost($)

.04.26.32.14.14

.10Total1.00RelativeFrequency

42632141410100PercentFrequency2/50.04(100)Example:HudsonAutoRepairPercentfrequencyistherelativefrequencymultipliedby100.Only4%ofthepartscostsareinthe$50-59class.Thegreatestpercentage(32%oralmostone-third)ofthepartscostsareinthe$70-79class.30%ofthepartscostsareunder$70.10%ofthepartscostsare$100ormore.InsightsGainedfromthe%FrequencyDistribution:RelativeFrequencyand

PercentFrequencyDistributionsExample:HudsonAutoRepairDotPlotOneofthesimplestgraphicalsummariesofdataisadotplot.Ahorizontalaxisshowstherangeofdatavalues.Theneachdatavalueisrepresentedbyadotplacedabovetheaxis.DotPlot50

60

70

80

90

100

110Cost($)Tune-upPartsCostExample:HudsonAutoRepairHistogramAnothercommongraphicalpresentationofquantitativedataisahistogram.Thevariableofinterestisplacedonthehorizontalaxis.Arectangleisdrawnaboveeachclassintervalwithitsheightcorrespondingtotheinterval’sfrequency,

relativefrequency,orpercentfrequency.Unlikeabargraph,ahistogramhasnonatural

separationbetweenrectanglesofadjacentclasses.Histogram24681012141618PartsCost($)Frequency50-5960-6970-79

80-8990-99100-110Tune-upPartsCostExample:HudsonAutoRepairSymmetricHistogramsShowingSkewnessRelativeFrequency.05.10.15.20.25.30.350LefttailisthemirrorimageoftherighttailExamples:heightsandweightsofpeopleHistogramsShowingSkewnessModeratelySkewedLeftRelativeFrequency.05.10.15.20.25.30.350AlongertailtotheleftExample:examscoresModeratelyRightSkewedHistogramsShowingSkewnessRelativeFrequency.05.10.15.20.25.30.350ALongertailtotherightExample:housingvaluesHistogramsShowingSkewnessHighlySkewedRightRelativeFrequency.05.10.15.20.25.30.350AverylongtailtotherightExample:executivesalaries

Cumulativefrequencydistribution

-showsthe

numberofitemswithvalueslessthanorequaltotheupperlimitofeachclass..

Cumulativerelativefrequencydistribution–showstheproportionofitemswithvalueslessthanorequaltotheupperlimitofeachclass.CumulativeDistributions

Cumulativepercentfrequencydistribution–showsthepercentageofitemswithvalueslessthanorequaltotheupperlimitofeachclass.CumulativeDistributionsThelastentryinacumulativefrequencydistributionalwaysequalsthetotalnumberofobservations.Thelastentryinacumulativerelativefrequencydistributionalwaysequals1.00.Thelastentryinacumulativepercentfrequencydistributionalwaysequals100.CumulativeDistributionsHudsonAutoRepair

<59

<69

<79

<89

<99 <109Cost($)CumulativeFrequencyCumulativeRelativeFreq