Chapter1Statistics,Data,andStatisticalThinkingContentsTheScienceofStatisticsTypesofStatisticalApplicationsinBusinessFundamentalElementsofStatisticsProcessesTypesofDataCollectingData:SamplingandRelatedIssuesBusinessAnalytics:CriticalThinkingwithStatisticsWhereWe’reGoingIntroducethefieldofstatisticsDemonstratehowstatisticsappliestobusinessIntroducethelanguageofstatisticsandthekeyelementsofanystatisticalproblemDifferentiatebetweenpopulationandsampledataDifferentiatebetweendescriptiveandinferentialstatisticsIntroducethekeyelementsofaprocessIdentifythedifferenttypesofdataanddata-collectionmethodsDiscoverhowcriticalthinkingthroughstatisticscanhelpimproveourquantitativeliteracy1.1TheScienceofStatisticsWhatIsStatistics?CollectingDatae.g.SurveyCharacterizingDatae.g.Mean,MedianAnalyzingDatae.g.TrendsandPatternsInterpretingDatae.g.ConclusionsandDecisionsWhatIsStatistics?Statisticsisthescienceofdata.Itinvolvescollecting,classifying,summarizing,organizing,analyzing,andinterpretingnumericalinformation.1.2TypesofStatisticalApplicationsinBusinessStatistics:TwoProcessesDescribingsetsofdataDrawingconclusions(makingestimates,decisions,predictions,etc.aboutsetsofdatabasedonsampling)StatisticalMethodsStatisticalMethodsDescriptiveStatisticsInferentialStatisticsDescriptiveStatisticsDescriptivestatisticsutilizesnumericalandgraphicalmethodstoexploredata,i.e.,tolookforpatternsinadataset,tosummarizetheinformationrevealedinadataset,andtopresenttheinformationinaconvenientform.InferentialStatisticsInferentialstatisticsutilizessampledatatomakeestimates,decisions,predictions,orothergeneralizationsaboutalargersetofdata.1.3FundamentalElements

ofStatisticsFundamentalElementsExperimental(orobservational)unitObjectuponwhichwecollectdataPopulationSetofunitsweareinterestedinstudyingVariablePropertyofanindividualexperimentalunitSampleSubsetoftheunitsofapopulationP

in

Population

&

ParameterS

in

Sample

&

StatisticFundamentalElementsStatisticalInferenceEstimateorpredictionorgeneralizationaboutapopulationbasedoninformationcontainedinasampleExampleAccordingtothemostrecentNielsensurveyofcableTVnewsviewers,theaverageageofCNNviewersis60years.Supposearivalnetwork(e.g.,FOX)executivehypothesizesthattheaverageageofFOXviewersisgreaterthan60.Totestherhypothesis,shesamples200FOXviewersanddeterminestheageofeach.a.Describethepopulation.b.Describethevariableofinterest.c.Describethesample.d.Describetheinference.Example(cont)Solutiona.Describethepopulation.ThepopulationisthesetofunitsofinteresttotheTVexecutive,whichisthesetofallFOXviewers.b.Describethevariableofinterest.Theage(inyears)ofeachvieweristhevariableofinterest.Example(cont)c.Describethesample.Thesamplemustbeasubsetofthepopulation.Inthiscase,itisthe200FOXviewersselectedbytheexecutive.Example(cont)d.Describetheinference.Theinferenceofinterestinvolvesthegeneralizationoftheinformationcontainedinthesampleof200viewerstothepopulationofallFOXviewers.Inparticular,theexecutivewantstoestimatetheaverageageoftheviewersinordertodeterminewhetheritexceeds60years.Shemightaccomplishthisbycalculatingtheaverageageinthesampleandusingthesampleaveragetoestimatethepopulationaverage.FundamentalElementsMeasureofReliabilityStatement(usuallyqualified)aboutthedegreeofuncertaintyassociatedwithastatisticalinferenceFourElementsofDescriptiveStatisticalProblemsThepopulationorsampleofinterestOneormorevariables(characteristicsofthepopulationorsampleunits)thataretobeinvestigatedTables,graphs,ornumericalsummarytoolsIdentificationofpatternsinthedataFiveElementsofInferentialStatisticalProblemsThepopulationofinterestOneormorevariables(characteristicsofthepopulationunits)thataretobeinvestigatedThesampleofpopulationunitsTheinferenceaboutthepopulationbasedoninformationcontainedinthesampleAmeasureofreliabilityfortheinference1.4ProcessesProcessAprocessisaseriesofactionsoroperationsthattransformsinputstooutputs.Aprocessproducesorgeneratesoutputovertime.ProcessAprocesswhoseoperationsoractionsareunknownorunspecifiediscalledablackbox.Anysetofoutput(objectornumbers)producedbyaprocessiscalledasample.ExampleAparticularfast-foodrestaurantchainhas6,289outletswithdrive-throughwindows.Toattractmorecustomerstoitsdrive-throughservices,thecompanyisconsideringofferinga50%discounttocustomerswhowaitmorethanaspecifiednumberofminutestoreceivetheirorder.Tohelpdeterminewhatthetimelimitshouldbe,thecompanydecidedtoestimatetheaveragewaitingtimeataparticulardrive-throughwindowinDallas,Texas.For7consecutivedays,theworkertakingcustomers’ordersrecordedthetimethateveryorderwasplaced.Theworkerwhohandedtheordertothecustomerrecordedthetimeofdelivery.Inbothcases,workersusedsynchronizeddigitalclocksthatreportedthetimetothenearestsecond.Attheendofthe7-dayperiod,2,109ordershadbeentimed.Example(cont)a.DescribetheprocessofinterestattheDallasrestaurant.b.Describethevariableofinterest.c.Describethesample.d.Describetheinferenceofinterest.e.Describehowthereliabilityoftheinferencecouldbemeasured.Solutiona.Theprocessofinterestisthedrive-throughwindowataparticularfast-foodrestaurantinDallas,Texas.Itisaprocessbecauseit“produces,”or“generates,”mealsovertime—thatis,itservicescustomersovertime.Example(cont)b.Describethevariableofinterest.Thevariablethecompanymonitorediscustomerwaitingtime,thelengthoftimeacustomerwaitstoreceiveamealafterplacinganorder.Becausethestudyisfocusingonlyontheoutputoftheprocess(thetimetoproducetheoutput)andnottheinternaloperationsoftheprocess(thetasksrequiredtoproduceamealforacustomer),theprocessisbeingtreatedasablackbox.c.Describethesample.Thesamplingplanwastomonitoreveryorderoveraparticular7-dayperiod.Thesampleisthe2,109ordersthatwereprocessedduringthe7-dayperiod.Example(cont)d.Describetheinferenceofinterest.Thecompany’simmediateinterestisinlearningaboutthedrive-throughwindowinDallas.Theyplantodothisbyusingthewaitingtimesfromthesampletomakeastatisticalinferenceaboutthedrive-throughprocess.Inparticular,theymightusetheaveragewaitingtimeforthesampletoestimatetheaveragewaitingtimeattheDallasfacility.Example(cont)e.Describehowthereliabilityoftheinferencecouldbemeasured.Asforinferencesaboutpopulations,measuresofreliabilitycanbedevelopedforinferencesaboutprocesses.ThereliabilityoftheestimateoftheaveragewaitingtimefortheDallasrestaurantcouldbemeasuredbyaboundontheerrorofestimation—thatis,wemightfindthattheaveragewaitingtimeis4.2minutes,withaboundontheerrorofestimationof0.5minutes.TheimplicationwouldbethatwecouldbereasonablycertainthatthetrueaveragewaitingtimefortheDallasprocessisbetween3.7and4.7minutes.1.5TypesofDataTypesofDataQuantitativedataaremeasurementsthatarerecordedonanaturallyoccurringnumericalscale.Qualitativedataaremeasurementsthatcannotbemeasuredonanaturalnumericalscale;theycanonlybeclassifiedintooneofagroupofcategories.TypesofDataTypesofDataQuantitativeDataQualitativeDataQuantitativeDataMeasuredonanumericalscale.Thetemperature(indegreesCelsius)atwhicheachunitinasampleof20piecesofheat-resistantplasticbeginstomeltThecurrentunemploymentrate(measuredasapercentage)foreachofthe50statesThescoresofasampleof150MBAapplicantsontheGMAT,astandardizedbusinessgraduateschoolentranceexamadministerednationwideThenumberoffemaleexecutivesemployedineachofasampleof75manufacturingcompaniesQualitativeDataClassifiedintocategories.Thepoliticalpartyaffiliation(Democrat,Republican,orIndependent)inasampleof50CEOsThedefectivestatus(defectiveornot)ofeachof100computerchipsmanufacturedbyIntelThesizeofacar(subcompact,compact,midsize,orfull-size)rentedbyeachofasampleof30businesstravelersAtastetester’sranking(best,worst,etc.)offourbrandsofbarbecuesauceforapanelof10testersExampleChemicalandmanufacturingplantssometimesdischargetoxic-wastematerialssuchasDDTintonearbyriversandstreams.Thesetoxinscanadverselyaffecttheplantsandanimalsinhabitingtheriverandtheriverbank.TheU.S.ArmyCorpsofEngineersconductedastudyoffishintheTennesseeRiver(inAlabama)anditsthreetributarycreeks:FlintCreek,LimestoneCreek,andSpringCreek.Atotalof144fishwerecaptured,andthefollowingvariablesweremeasuredforeach:(continuedonnextslide)Example(cont)1.River/creekwhereeachfishwascaptured2.Species(channelcatfish,largemouthbass,orsmallmouthbuffalofish)3.Length(centimeters)4.Weight(grams)5.DDTconcentration(partspermillion)ThesedataaresavedintheDDTfile.Classifyeachofthefivevariablesmeasuredasquantitativeorqualitative.Example(cont)SolutionThevariableslength,weight,andDDTarequantitativebecauseeachismeasuredonanumericalscale:lengthincentimeters,weightingrams,andDDTinpartspermillion.Incontrast,river/creekandspeciescannotbemeasuredquantitatively:Theycanonlybeclassifiedintocategories(e.g.,channelcatfish,largemouthbass,andsmallmouthbuffalofishforspecies).Consequently,dataonriver/creekandspeciesarequalitative.1.6CollectingDataObtainingDataDatafromapublishedsourceDatafromadesignedexperimentDatafromanobservationallystudyObtainingDataPublishedsource

Book,journal,newspaper,WebsiteDesignedexperiment

ResearcherexertsstrictcontrolovertheunitsSurvey

AgroupofpeoplearesurveyedandtheirresponsesarerecordedObservationstudy

UnitsareobservedinnaturalsettingandvariablesofinterestarerecordedDesignedExperimentAdesignedexperimentisadata-collectionmethodwheretheresearcherexertsfullcontroloverthecharacteristicsoftheexperimentalunitssampled.Theseexperimentstypicallyinvolveagroupofexperimentalunitsthatareassignedthetreatmentandanuntreated(orcontrol)group.ObservationalStudyAnobservationalstudyisadata-collectionmethodwheretheexperimentalunitssampledareobservedintheirnaturalsetting.Noattemptismadetocontrolthecharacteristicsoftheexperimentalunitssampled.(Examplesincludeopinionpollsandsurveys.)SamplesArepresentativesampleexhibitscharacteristicstypicalofthosepossessedbythepopulationofinterest.Asimplerandomsampleofnexperimentalunitsisasampleselectedfromthepopulationinsuchawaythateverydifferentsampleofsizenhasanequalchanceofselection.RandomSampleAsimplerandomsampleofnexperimentalunitsisasampleselectedfromthepopulationinsuchawaythateverydifferentsampleofsizenhasanequalchanceofselection.RandomNumberGeneratorsMostresearchersrelyonrandomnumbergeneratorstoautomaticallygeneratetherandomsample.Randomnumbergeneratorsareavailableintableform,andtheyarebuiltintomoststatisticalsoftwarepackages.ExampleSupposeyouwishtoassessthefeasibilityofbuildinganewhighschool.Aspartofyourstudy,youwouldliketogaugetheopinionsofpeoplelivingclosetotheproposedbuildingsite.Theneighborhoodadjacenttothesitehas711homes.Usearandomnumbergeneratortoselectasimplerandomsampleof20householdsfromtheneighborhoodtoparticipateinthestudyExample(cont)SolutionInthisstudy,yourpopulationofinterestconsistsofthe711householdsintheadjacentneighborhood.Toensurethateverypossiblesampleof20householdsselectedfromthe711hasanequalchanceofselection(i.e.,toensureasimplerandomsample),firstassignanumberfrom1to711toeachofthehouseholdsinthepopulation.ThesenumberswereenteredintoanExcelworksheet.Now,applytherandomnumbergeneratorofExcel/XLSTAT,requestingthat20householdsbeselectedwithoutreplacement.Thefigureinyourtextonpage17showsonepossiblesetofrandomnumbersgeneratedfromXLSTAT.Youcanseethathouseholdsnumbered40,63,108,...,636arethehouseholdstobeincludedinyoursample.ImportanceofSelectionHowasampleisselectedfromapopulationisofvitalimportanceinstatisticalinferencebecausetheprobabilityofanobservedsamplewillbeusedtoinferthecharacteristicsofthesampledpopulation.RandomSamplingStratifiedrandomsampling

usedwhentheexperimentalunitsassociatedwiththepopulationcanbeseparatedintotwoormoregroupsofunits.ClustersamplingsamplenaturalgroupingofexperimentalunitsandcollectdatafromallexperimentalunitswithineachclusterRandomSamplingSystematicsampling

systematicallyselectingeverykthexperimentalunitfromalistofallexperimentalunits.Randomizedresponsesamplingusefulwhenthequestionsofapollsterarelikelytoelicitfalseanswers.NonrandomSampleErrorsSelectionbiasresultswhenasubsetoftheexperimentalunitsinthepopulationisexcludedsothattheseunitshavenochanceofbeingselectedforthesample.Nonresponsebias

resultswhentheresearchersconductingasurveyorstudyareunabletoobtaindataonallexperimentalunitsselectedforthesample.Measurementerror

referstoinaccuraciesinthevaluesofthedatarecorded.Insurveys,theerrormaybeduetoambiguousorleadingquestionsandtheinterviewer’seffectontherespondent.ExampleWhatisthemostpopulardeviceusedbyonlineshoppers?Tofindout,themobilevideoadnetworkAdColonyconducteda2019nationwidesurveyof1,000USonlineshoppersforMobileMarketer.Themostpopulardevicewasasmartphone,usedby56%oftheonlineshoppers.Otherresults:28%usedadesktoporlaptopcomputer,and16%usedatablet.a.Identifythedata-collectionmethod.b.Identifythetargetpopulation.c.Arethesampledatarepresentativeofthepopulation?Example(cont)Solutiona.Identifythedata-collectionmethod.Thedata-collectionmethodisasurvey:1,000onlineshoppersparticipatedinthestudy.b.Identifythetargetpopulation.Presumably,MobileMarketer(whocommissionedthesurvey)isinterestedinthedevicesusedbyallUSonlineshoppers.Consequently,thetargetpopulationisallconsumerswhousetheInternetforonlineshopping.Example(cont)c.Arethesampledatarepresentativeofthepopulation?Becausethe1,000respondentsclearlymakeupasubsetofthetargetpopulation,theydoformasample.WhetherornotthesampleisrepresentativeisunclearbecauseMobileMarketerprovidednodetailedinformationonhowthe1,000shopperswereselected.Iftherespondentswereobtainedusing,say,random-digittelephonedialing,thenthesampleislikelytoberepresentativebecauseitisarandomsample.Example(cont)However,ifthequestionnairewasmadeavailabletoanyonesurfingtheInternet,thentherespondentsareself-selected(i.e.,eachInternetuserwhosawthesurveychosewhetherornottorespondtoit).Suchasurveyoftensuffersfromnonresponsebias.ItispossiblethatmanyInternetuserswhochosenottorespond(orwhoneversawthequestionnaire)wouldhaveansweredthequestionsdifferently,leadingtoalower(orhigher)samplepercentage.1.7CriticalThinkingwithStatisticsStatisticalThinkingBusinessanalyticsreferstomethodologies(e.g.statisticalmethods)thatextractusefulinformationfromdatainordertomakebetterbusinessdecisions.Statisticalthinkinginvolvesapplyingrationalthoughtandthescienceofstatisticstocriticallyassessdataandinferences.Fundamentaltothethoughtprocessisthatvariationexistsinpopulationsandprocessdata.StatisticsinBusinessAnalyticsKeyIdeasTypesofStatisticalApplicationsDescriptive 1.Identifypopulationandsample(collectionofexperimentalunits) 2.Identifyvariable(s) 3.Collectdata

4.DescribedataKeyIdeasTypesofStatisticalApplicationsInferential 1.Identifypopulation(collectionofallexperimental

units) 2.Identifyvariable(s) 3.Collectsampledata(subsetofpopulation)

4.Inferenceaboutpopulationbasedonsample 5.MeasureofreliabilityforinferenceKeyIdeasTypesofData1. Quantitative(numericalinnature)2. Qualitative(categoricalinnature)KeyIdeasData-CollectionMethods1. Observational(e.g.survey)2.

Publishedsource3. DesignedexperimentKeyIdeasTypesofRandomSamples1. SimpleRandomSample2. Stratifiedrandomsample3. Clustersample4. Systematicsample5. RandomresponsesampleKeyIdeasProblemswithNonrandomSamples1. Selectionbias2.

Nonresponsebias3. MeasurementerrorChapter2MethodsforDescribingSetsofDataContentsDescribingQualitativeDataGraphicalMethodsforDescribingQuantitativeDataNumericalMeasuresofCentralTendencyNumericalMeasuresofVariabilityUsingtheMeanandStandardDeviationtoDescribeDataContents(cont)6. NumericalMeasuresofRelativeStanding7. MethodsforDetectingOutliers:BoxPlotsandz-scores8. GraphingBivariateRelationships9. TheTimeSeriesPlot10. DistortingtheTruthwithDescriptiveTechniquesWhereWe’reGoingDescribequalitativedatausinggraphsDescribedatausinggraphsDescribequantitativedatausingnumericalmeasuresDescribetherelationshipbetweentwoquantitativevariablesusinggraphsDetectingdescriptivemethodsthatdistortthetruth2.1DescribingQualitativeDataKeyTermsAclassisoneofthecategoriesintowhichqualitativedatacanbeclassified.Theclassfrequencyisthenumberofobservationsinthedatasetfallingintoaparticularclass.Theclassrelativefrequencyistheclassfrequencydividedbythetotalnumbersofobservationsinthedataset.Theclasspercentageistheclassrelativefrequencymultipliedby100.SummaryTableListscategories&numberofelementsincategoryObtainedbytallyingresponsesincategoryMayshowfrequencies(counts),%orbothRowIsCategoryTally:

||||||||

||||||||MajorCountAccounting130Economics20Management50Total200BarGraphVerticalBarsforQualitativeVariablesBarHeightShowsFrequencyor%ZeroPointPercentUsedAlsoEqualBarWidthsFrequencyPieChartShowsbreakdownoftotalquantityinto

categoriesUsefulforshowing

relativedifferencesAnglesize=(360°)(percent)Econ.10%Mgmt.25%Acct.65%Majors(360°)(10%)=36°36°ParetoDiagramLikeabargraph,butwiththecategoriesarrangedbyheightindescendingorderfromlefttoright.VerticalBarsforQualitativeVariablesBarHeightShowsFrequencyor%ZeroPointPercentUsedAlsoEqualBarWidthsFrequencyExample:50HighestPaidCEOsDataon50HighestPaidCEOsaregiveninTable2.1inthebook.Constructa

bar

graph,piechart,andPareto

diagramtodescribethedataprovidedinthefrequencysummarytablebelow.Example:BarGraphSolutionBarGraphforDegreesof50HighestPaidCEOsExample:PieChartSolutionPieChartofDegreeExample:ParetoDiagramSolutionParetoDiagramforDegreesof50HighestPaidCEOsSummaryBargraph:Thecategories(classes)ofthequalitativevariablearerepresentedbybars,wheretheheightofeachbariseithertheclassfrequency,classrelativefrequency,orclasspercentage.Piechart:Thecategories(classes)ofthequalitativevariablearerepresentedbyslicesofapie(circle).Thesizeofeachsliceisproportionaltotheclassrelativefrequency.Paretodiagram:Abargraphwiththecategories(classes)ofthequalitativevariable(i.e.,thebars)arrangedbyheightindescendingorderfromlefttoright.2.2GraphicalMethodsforDescribingQuantitativeDataDotPlotHorizontalaxisisascaleforthequantitativevariable,e.g.,percent.Thenumericalvalueofeachmeasurementislocatedonthehorizontalscalebyadot.Stem-and-LeafDisplay1.Divideeachobservationintostemvalueandleafvalues:StemsarelistedinorderinacolumnLeafvalueisplacedincorrespondingstemrowtorightofbar2.

Data:21,24,24,26,27,27,30,32,38,41262144677302841Histogram012345FrequencyRelativeFrequencyPercent0 15.5 25.5 35.5 45.5 55.5LowerBoundaryBarsTouchClassFreq.15.5–25.5325.5–35.5535.5–45.52CountNumberofHistogramClassesSomerecommendationsforselectingthenumberofintervalsinahistogramforsmallerdatasetsaregivenintheboxbelow.SummaryDotplot:Thenumericalvalueofeachquantitativemeasurementinthedatasetisrepresentedbyadotonahorizontalscale.Whendatavaluesrepeat,thedotsareplacedaboveoneanothervertically.Stem-and-leafdisplay:Thenumericalvalueofthequantitativevariableispartitionedintoa“stem”anda“l(fā)eaf.”Thepossiblestemsarelistedinorderinacolumn.Theleafforeachquantitativemeasurementinthedatasetisplacedinthecorrespondingstemrow.Leavesforobservationswiththesamestemvaluearelistedinincreasingorderhorizontally.SummaryHistogram:Thepossiblenumericalvaluesofthequantitativevariablearepartitionedintoclassintervals,whereeachintervalhasthesamewidth.Theseintervalsformthescaleofthehorizontalaxis.Thefrequencyorrelativefrequencyofobservationsineachclassintervalisdetermined.Ahorizontalbarisplacedovereachclassinterval,withheightequaltoeithertheclassfrequencyorclassrelativefrequency.2.3NumericalMeasures

ofCentralTendencyCentralTendencyThecentraltendencyofthesetofmeasurements–thatis,thetendencyofthedatatocluster,orcenter,aboutcertainnumericalvalues.VariabilityThevariabilityofthesetofmeasurements–thatis,thespreadofthedata.MeanThemeanofasetofquantitativedataisthesumofthemeasurementsdividedbythenumberofmeasurementscontainedinthedataset.Example:FindtheMeanCalculatethemeanofthefollowingsixsamplemeasurements:5,3,8,5,6.SymbolsfortheSampleandPopulationMeanInthistext,weadoptageneralpolicyofusingGreekletterstorepresentpopulationnumericaldescriptivemeasuresandRomanletterstorepresentcorrespondingdescriptivemeasuresforthesample.Thesymbolsforthemeanare Samplemean

PopulationmeanMedianThemedianofaquantitativedatasetisthemiddlenumberwhenthemeasurementsarearrangedinascending(ordescending)order.CalculatingaSampleMedian,mArrangethenmeasurementsfromsmallesttolargest.Ifnisodd,misthemiddlenumber.Ifniseven,misthemeanofthemiddletwonumbers.Example:MedianforOddNumberofMeasurementsConsiderthefollowingsampleofn=7measurements:5,7,4,5,20,6,2.RawData: 5,7,4,5,20,6,2

Ordered:

2,4,5,5,6,7,20Position: 1234567Medianofthissampleism=5.Example:MedianforEvenNumberofMeasurementsConsiderthefollowingsampleofn=6measurements:5,7,4,5,20,6.RawData: 5,7,4,5,20,6

Ordered: 4,5,5,6,7,20Position: 123

456Medianofthissampleism=5.5.ModeThemodeisthemeasurementthatoccursmostfrequentlyinthedataset.SkewedAdatasetissaidtobeskewedifonetailofthedistributionhasmoreextremeobservationsthantheothertail.Example:ModeEachof10tastetestersratedanewbrandofbarbecuesauceona10-pointscale,where1=awfuland10=excellent.Findthemodeforthe10ratingsshownbelow.87968109957Thescore9occursmostoftenandsothemodeofthetaste-ratingsis9.ShapeDescribeshowdataaredistributedMeasuresofShapeSkew=SymmetryRight-SkewedLeft-SkewedSymmetricMean

=Median

Mean

Median

Median

Mean2.4NumericalMeasures

ofVariabilityRangeTherangeofaquantitativedatasetisequaltothelargestmeasurementminusthesmallestmeasurement.Range=Largest–SmallestBothdatasetshavearangeof50.Therangeiseasytocomputeandeasytounderstand,butitisaninsensitivemeasureofdatavariation.SampleVarianceThesamplevarianceforasampleofnmeasurementsisequaltothesumofthesquareddeviationsfromthemeandividedby(n–1).Thesymbols2isusedtorepresentthesamplevariance.Samplevarianceisameasureoftheofthevariabilityofadataset.Compare:andSample1:MorevariabilityaroundthemeanSample2:LessvariabilityaroundthemeanSampleVarianceFormulaShortcutformulaforcalculatings2:SampleStandardDeviationThesamplestandarddeviation,s,isdefinedasthepositivesquarerootofthesamplevariance,s2.So,SymbolsforVarianceandStandardDeviations2=Samplevariances

=Samplestandarddeviation=Populationvariance=PopulationstandarddeviationExampleCalculatethevarianceandstandarddeviationforthesample2,3,3,3,4.SolutionFirst,findthemean:

2.5UsingtheMeanandStandardDeviationtoDescribeDataUsingtheMeanandStandardDeviationtoDescribeData:Chebyshev’sRuleChebyshev’sRuleappliestoanydataset,regardlessoftheshapeofthefrequencydistribution.InterpretingStandardDeviation:Chebyshev’sTheoremNousefulinformationAtleast3/4ofthedataAtleast8/9ofthedataInterpretingStandardDeviation:EmpiricalRuleAppliestodatasetsthataremoundshapedandsymmetricApproximately68%ofthemeasurementslieintheintervalApproximately95%ofthemeasurementslieintheintervalApproximately99.7%ofthemeasurementslieintheintervalInterpretingStandardDeviation:EmpiricalRulex–3s

x–2s

x–s

x