统计专业英语 Chap1-OverviewProbability-and-Statistics

Chapter1.OverviewandDescriptiveStatistics

WeiqiLuo(骆伟祺)SchoolofSoftwareSunYat-SenUniversityEmail：weiqi.luo@Office：#A313Textbook:

JayL.Devore,Probabilityandstatisticsforengineeringandthesciences(the6th

Edition),ChinaMachinePress,2011References:

1.MillerandFreund,“ProbabilityandStatisticsforEngineers”(the7thEdition),PublishingHouseofElectronicsIndustry,2005.2.盛骤、谢式千、潘承毅，《概率论与数理统计》第4版，高等教育出版社，2008

KaiLaiChung,“ACourseinProbabilityTheory”,(the3rdEdition),ChinaMachinePress,2010.2MATLABApowerfulsoftwarewithvarioustoolboxes,including

StatisticsToolboxImageProcessingToolboxSignalprocessingToolboxRobustControlToolboxCurveFittingToolboxFuzzyLogicToolbox

…

3PrerequisiteCoursesSE-101AdvancedMathematicsSE-103LinearAlgebraSuccessiveCoursesSE-328DigitalSignalProcessingSE-343DigitalImageProcessingSE-352InformationSecurityPatternRecognition&Machinelearningetc.4UncertaintyItcanbeassessedinformallyusingthelanguagesuchas“itisunlikely”or“probably”.

5WhatisUncertainty?Thissciencecameofgamblingin7thcenturyProbabilitymeasuresuncertaintyformally,quantitatively.It

isthemathematicallanguageofuncertainty.Statisticsshowsomeusefulinformationfromtheuncertaindata,andprovidethebasisformakingdecisionsorchoosingactions.

6WhyStudyProbability&Statistics?WeatherForecastApplications7InmedicaltreatmentApplications8e.g.RelationshipbetweensmokingandlungcancerBirthdayParadox(fromWikipedia)Applications9Benford’sLaw/FirstDigitLaw(fromWikipedia)Applications10AccountingForensicsMultimediaForensics…TimeSeriesAnalysisApplications11EconomicForecastingSalesForecastingBudgetaryAnalysisStockMarketAnalysisProcessandQualityControlInventoryStudiesetc.Moreinterestingapplicationsinreallife

/v_playlist/f1486775o1p0.htmlApplications121.1.Populations,Samples,andProcesses1.2.PictorialandTabularMethodsinDescriptiveStatistics1.3MeasuresofLocation1.4.MeasuresofVariability13Chapter1:Overview&DescriptiveStatisticsPopulationAninvestigationwilltypicallyfocusonawell-definedcollectionofobjects(units).Andapopulationisthesetofallobjectsofinterestinaparticularstudy.VariablesAnycharacteristicwhosevalue(categoricalornumerical)maychangefromoneobjecttoanotherinthepopulation.141.1.Populations,Samples,andProcessesPopulationUnit/ObjectVariables/CharacteristicsAllstudentscurrentlyintheclassStudentHeightWeightHoursofworkperweekRight/left–handedAllPrintedcircuitboardsmanufacturedduringamonthBoardTypeofdefectsNumberofdefectsLocationofdefeatsAllcampusfastfoodrestaurantsRestaurantNumberofemployeesSeatingcapacityHiring/nothiringAllbooksinlibraryBookReplacementcostFrequencyofcheckoutRepairsneeds1.1.Populations,Samples,andProcesses15ExamplesofPopulations,ObjectsandvariablesSampleAsubsetofthepopulation1.1.Populations,Samples,andProcesses16PopulationSampleAccordingtothenumberofthevariablesunderinvestigation,wehaveUnivariate:asinglevariable,e.g.thetypeoftransmission,automaticormanual,oncarsBivariate:twovariables,e.g.theheight&weightofthestudentsMultivariate:morethantwovariables,e.g.systolicbloodpressure,diastolicbloodpressureandserumcholesterollevelforeachpatient

1.1.Populations,Samples,andProcesses17DescriptivestatisticsAninvestigatorwhohascollecteddatamaywishsimplytosummarizeanddescribeimportantfeaturesofthedata.Thisentailsusingmethodsfromdescriptivestatistics

Graphicalmethods(Sec.1.2),

e.g.

Stem-and-Leafdisplay,Dotplot&histogramsNumericalsummarymeasures(Sec.1.3,1.4),

e.g.

means,standarddeviations&correlationscoefficients1.1.Populations,Samples,andProcesses18Example1.1.

Hereisdataconsistingofobservationsonx=O-ringtemperatureforeachtestfiringoractuallaunchoftheshuttlerocketengine.

1.1.Populations,Samples,andProcesses19844961408367456670698058686067727370576370785267536775617081767975765831NormalizedHistogram1.1.Populations,Samples,andProcesses202535455565758520%10%30%40%Thepercentageofthetemperatureslocatedinthebin[25,35]Inferentialstatistics

Usesampleinformationtodrawsometypeofconclusion(makeaninferenceofsomesort)aboutthepopulation.PointEstimation----Chapter6Hypothesistesting----Chapter8Estimationbyconfidenceinterval---Chapter7…1.1.Populations,Samples,andProcesses21Probability&Statistics

1.1.Populations,Samples,andProcesses22PopulationSampleDeductiveReasoning(Probability)InductiveReasoning(InferentialStatistics)Themathematicallanguageis“Probability”CollectingDataIfdataisnotproperlycollected,aninvestigatormaynotbeabletoanswerthequestionsunderconsiderationwithareasonabledegreeofconfidence.MethodsforcollectingdataRandomsampling:anyparticularsubsetofthespecifiedsizehasthesamechanceofbeingselectedStratifiedsampling:entailsseparatingthepopulationunitsintonon-overlappinggroupsandtakingasamplefromeachone.1.1.Populations,Samples,andProcesses23DescriptiveStatisticsVisualtechniques(Sec.1.2)Stem-and-LeafDisplaysDotplotsHistogramNumericalsummarymeasures(Sec.1.3&1.4)MeasuresoflocationMeasureofvariability24Notation

Samplesize:Thenumberofobservationsinasinglesamplewilloftenbedenotedbyn.

Givenadatasetconsistingofn

observationsonsomevariablex,theindividualobservationswillbedenotedbyx1,x2,x3,…,xn1.2PictorialandTabularMethodinDescriptiveStatistics25Stem-and-LeafDisplaysSupposewehaveanumericaldatasetx1,x2,x3,…,xnforwhicheachxiconsistsofatleasttwodigits.StepsforconstructingaStem-and-LeafDisplaySelectoneormoreleadingdigitsforthestemvalues.Thetrailingdigitsbecometheleaves.Listpossiblestemvaluesinaverticalcolumn.Recordtheleafforeveryobservationbesidethecorrespondingstemvalue.Indicatetheunitsforstemsandleavessomeplaceinthedisplay.1.2PictorialandTabularMethodinDescriptiveStatistics26Example:Observations:16%,33%,64%,37%,31%…

Stem-and-LeafDisplayStem|Leaf1|63|371[or3|137]6|4

1.2PictorialandTabularMethodinDescriptiveStatistics27Stem:tensdigitLeaf:onesdigitExample1.5ThefollowingFigureshowsastem-andleafdisplayof140values(colleges)ofx=thepercentageofundergraduatestudentswhoarebingedrinkers.1.2PictorialandTabularMethodinDescriptiveStatistics28041134567888921223456666777889999301122333445566667777788889999941112222233444455666666777888888999500111222233455666667777888899601111244455666778Stem:tensdigitLeaf:onesdigitAstem-and-leafdisplayconveysinformationaboutthefollowingaspectsofthedata:IdentificationofatypicalorrepresentativevalueExtentofspreadaboutthetypicalvaluePresenceofanygapsinthedataExtentofsymmetryinthedistributionofvaluesNumberandlocationofpeaksPresenceofanyoutlyingvalues1.2PictorialandTabularMethodinDescriptiveStatistics29Example1.61.2PictorialandTabularMethodinDescriptiveStatistics30|35643370|26270683|059414|90700098704513|90707350|00273604|510511405022|316968051365|8009|435464433470…904|051005011040…209Stem:ThousandsandhundredsdigitsLeaf:TensandonesdigitsStem:ThousandsdigitsLeaf:Hundreds,tensandonesdigitsExample1.7(repeatedstems)

1.2PictorialandTabularMethodinDescriptiveStatistics315H|55L|2423304H|7688964L|214214144443H|9696656Stem:tensdigitLeaf:onesdigit=5|2423305|214214144447688963|9696656Stem:tensdigitLeaf:onesdigitNote:L:theleafsare0,1,2,3or4H:theleafsare5,6,7,8or9DotplotthedatasetisreasonablysmallortherearerelativelyfewdistinctdatavaluesEachobservationisrepresentedbyadotabovethecorrespondinglocationonahorizontalmeasurementscale.Whenavalueoccursmorethanonce,thereisadotforeachoccurrence,andthesedotsarestackedvertically.Aswithastem-and-leafdisplay,adotplotgivesinformationaboutlocation,spread,extremes&gaps.1.2PictorialandTabularMethodinDescriptiveStatistics32Example1.81.2PictorialandTabularMethodinDescriptiveStatistics33304050607080844961408367456670698058686067727370576370785267536775617081767975765831HistogramTypesofvariables:Discretevariable:Avariableisdiscrete

ifitssetofpossiblevalueseitherisfiniteorelsecanbelistedinaninfinitesequence.Continuousvariable:Avariableiscontinuous

ifitspossiblevaluesconsistofanentireintervalonthenumberline.1.2PictorialandTabularMethodinDescriptiveStatistics34ConstructingaHistogramforDiscreteDataThreeSteps:Determinethefrequency(orrelativefrequency)ofeachxvalue.Markpossiblexvaluesonahorizontalscale.Drawarectanglewhoseheightisthefrequency(orrelativefrequency)ofthevalue.1.2PictorialandTabularMethodinDescriptiveStatistics35ExampleSupposethatourdatasetconsistsof200observationsonx=thenumberofmajordefectsinanewcarofacertaintype.If70ofthesexare1,thenfrequencyofthexvalue1:70relativefrequencyofthexvalue1:70/200=0.35

Note:1.2PictorialandTabularMethodinDescriptiveStatistics36Example1.91.2PictorialandTabularMethodinDescriptiveStatistics37Example1.91.2PictorialandTabularMethodinDescriptiveStatistics38010200.050.10ContinuousCase

p17.Supportthatwehave50observationsonx=fuelefficiencyofanautomobile(mpg),thesmallestofwhichis27.8andthelargestofwhichis31.41.2PictorialandTabularMethodinDescriptiveStatistics3927.528.028.529.029.530.030.531.031.5Classintervals:ContinuesDiscreteEqualorUnequalwidthConstructingaHistogramforContinuousData:Equal(orUnequal)ClassWidthsSimilartothediscretecase

Makesurethat:classwidth×rectangleheight(density)=relativefrequencyoftheclass1.2PictorialandTabularMethodinDescriptiveStatistics40TypicalHistogramShapes1.2PictorialandTabularMethodinDescriptiveStatistics41SymmetricUnimodalBimodalPositivelySkewedNegativeSkewedMultivariateDataTheabovementionedtechniqueshavebeenexclusivelyforsituationsinwhicheachobservationinadatasetiseitherasinglenumberorasinglecategory.PleaserefertoChapters11-14foranalyzingmultivariatedatasets.1.2PictorialandTabularMethodinDescriptiveStatistics42Ex.11,Ex.14,Ex.20,Ex.26Homework43TheMean

Samplemean:Thesamplemeanofobservationsx1,x2,…,xnisgivenbySamplemedian:Thesamplemediaisobtainedbyfirstorderingthenobservationsfromsmallesttolargest.Then1.3MeasuresofLocation44Example1.13(Samplemean)

x1=16.1x2=9.6x3=24.9x4=20.4x5=12.7x6=21.2x7=30.2x8=25.8x9=18.5x10=10.3x11=25.3x12=14.0x13=27.1x14=45.0x15=23.3x16=24.2x17=14.6x18=8.9x19=32.4x20=11.8x21=28.51.3MeasuresofLocation450H|96891L|27034046181H|61852L|49041233422H|585371853L|02243H|4L|4H|5010203040OutlyingvalueExample1.14(Median)

x1=15.2x2=9.3x3=7.6x4=11.9x5=10.4x6=9.7x7=20.4x8=9.4x9=11.5x10=16.2x11=9.4x12=8.3Thelistoforderedvaluedis10.411.511.9n=12iseven,thenthesamplemedianis(9.7+10.4)/2=10.05Note:thesamplemeanhereis139.3/12=MeasuresofLocation46Threedifferentsharpsforapopulationdistribution1.3MeasuresofLocation47SymmetricUnimodalPositivelySkewedNegativeSkeweduu~u~u=uu~u:Populationmeanu:Populationmedian~Why?OtherMeasuresofLocationQuartilesPercentiles

1.3MeasuresofLocation48…MedianQuartiles…1%MaybeoutlyingdataTrimmedMeansAtrimmedmeanisacompromisebetweensamplemean&samplemedian.A10%trimmedmean,forexample,wouldbecomputedbyeliminatingthesmallest10%andthelargest10%ofthesampleandthenaveragingwhatisleftover.1.4MeasuresofLocation49…10%10%SampleMeanExample1.15612623666744883898964970983100310161022102910581085108811221135119712011.4MeasuresofLocation5060080010001200x~x_Xtr(10)_RemovalRemovalNote:Trimmingproportion:5%~25%Ex.34,Ex.36,Ex.40Homework51Timeerrorforthreetypeofwatches9observationsforeachtype1.4MeasuresofVariability52*********-20-100+20+10123Q:Whichtypeisthebest?Andwhy?TheRange

Thedifferencebetweenthelargestandsmallestsamplevalues.Refertothepreviousexample,type1and2haveidenticalranges,however,thereismuchlessvariabilityinthesecondsamplethaninthefirst.Deviationsfromthemean

Measure1:x1-mean,x2-mean,…,xn-mean,thenforallcases

1.4MeasuresofVariability53Samplevariance

Thesamplevariance,denotedbys2,isgivenbyThesamplestandarddeviation,denotedbys,isthesquarerootofthevariances=sqrt(s2).

Q1:vs.

Q2:n-1vs.n

1.4MeasuresofVariability54Example1.161.4MeasuresofVariability55xixi-(xi-)20.6840.98410.96852.540.87190.76020.924-0.74410.55373.131.46192.13721.038-0.63010.39700.598-1.07011.14510.483-1.18511.40453.521.85193.42951.285-0.38310.14682.650.98190.96411.497-0.17110.0293PopulationvarianceWewilluseσ2todenotethepopulationvarianceandσtodenotethepopulationstandarddeviation.WhenthepopulationisfiniteandconsistsofNvalues,1.4MeasuresofVariability56Considerapopulationwithjust3elements{1,2,3}ThemeanofthepopulationisAndthevarianceSupposeallwecantakeisasampleof2elementstakenwithrepetitiontolearnaboutthepopulation.Wewouldlikethesampletoaccuratelyestimatethemeanandvariancevaluesofthepopulation.1.4MeasuresofVariability571.4MeasuresofVariability58PossibleSamplesofSizeTwoSamplemeanxs2

usingn=2s2

usingn–1=1{1,1}10/20/1{2,2}20/20/1{3,3}30/20/1{1,2}1.5.5/2=.25.5/1=.5(2,1)1.5.5/2=.25.5/1=.5{1,3}22/2=1.02/1=2(3,1)22/2=1.02/1=2{2,3}2.5.5/2=.25.5/1=.5(3,2)2.5.5/2=.25.5/1=.5AverageofSampleStatistics21/32/3Betterestimate!Analterexpressionforthenumeratorofs2

Ify1=x1+c,y2=x2+c,…,yn=xn+c,thensy2=sx2Ify1=cx1,y2=cx2,…..,yn=cxn,thensy2=c2sx2,sy=|c|sx,wheresx2isthesamplevarianceofthex’sandsy2isthesamplevarianceofthey’s.1.4MeasuresofVariability59BecareoftheroundingerrorswhenusingthetwodifferentexpressionsBoxplotsDescribeseveralofadataset’smostprominentfeatures:center;spread;extentandnatureofanydeparturefromsymmetry;identificationof“outliers

”,observationsthatli