HMEC Module A Lesson 04

WelcometotheHighwayMaterialsEngineeringCourse(HMEC)ModuleA,Lesson4:

CollectingData:SamplingTheory.Thislessonprovidesanunderstandingofthebasic

elementsofastatistically‐basedqualityassurance(QA)programandincludesan

introductiontoqualityassuranceaswellastechniquesforcollectingdata.

Aprinter‐friendlyversionofthelessonmaterialscanbedownloadedbyselectingthe

paperclipicon.Acopyoftheslidesandnarrationareprovidedfordownload.

If youneedtechnicalassistanceduringthetraining,pleaseselecttheHelplinkintheupper

right‐handcornerofthescreen.

Bytheendofthislesson,youwillbeableto:

•Describethebasicphasesofstatisticalanalysis;

•Definesampling;

•Explaintheimportanceofsamplingandusingallavailabledata;

•Discusssecurityanddocumentationofrandomsamplelocations;

•Explainhowasamplerelatestoapopulation;and

•Applyrandomandstratifiedrandomsamplingtechniquestoobtainvaliddata.

Duringthislesson,knowledgechecksareprovidedtotestyourunderstandingofthe

materialpresented.

Thislessonwilltakeapproximately70minutestocomplete.

During thislesson,youwillbepromptedtoreferencethelessonexercisedocument.The

documentsreferencedduringthislessonareattachedtothelessoninthepaperclipicon.

Pleasetak eamomenttoopenandprintthedocument.

Remember,specificationsguidetheacceptanceofmaterialsbyquantifyingtheriskweare

willingtoassume.Weusedatafromsamplestoindicatehowcloseto“normal”or

acceptablethematerialsare.Thereisvariabilityinallmaterials,andinsamplingandtesting

aswell.

Soaskyourselfthefollowingquestions:

•Howdoyoureducethevariabilityinthetestdata?Answer: Makesuretofollowthe

correctsamplingandtestingprocedures.

•Howdoyoumakesenseofallthosenumbersyoucollect?Answer:Onewayistoorganize

thedatacorrectlyasyouwillseeinLesson5.

Imagedescription:Manlookingupwithquestionmarks.

Therearefourphasesofstatisticalanalysis, whichinclude:

1.CollectData;

2.OrganizetheData;

3. AnalyzetheData; and

4.InterprettheData.

Selecteachphase tolearnmore.

The firstphase,tocollectdata,istheplannedprocessofobtainingarelativelysmall

numberofmeasurements(sampledata)fromafairlylargequantityofmaterial(lotor

population).Propersamplingproceduresareessentialforthecollectionofvalid,

meaningfuldata.

Thesecondphaseistoorganizethedata, whichrequiresassemblingofdataintosystematic

groupsorclassificationsfromwhichlogicalconclusionscanbedrawn.

Thethirdphase ofstatisticalanalysisistoanalyzethedata,which isanumerical

determinationofstatisticalmeasuresthatdescribetheimportantcharacteristicsofthe

data.

Thefourthandlastphaseisto interpretthedata,whichmeanstousethebasicsample

resultstoinferbroaderstatementsaboutthetotalquantityofmaterial(lotorpopulation).

Anunderstandingofbasicstatisticalandprobabilityconceptsisnecessarytoensureproper

interpretationofthesampleresultsandtounderstandhowandwhydataareoften

misinterpreted.

In thislesson,we’llonlytakealookatthefirstphase:collectdata.Specifically,how

samplingtheoryisappliedtocollectdatafororganization,analysis,andinterpretation.

Dataareoflittlevalueiftheyarecollectedinahaphazard,unplannedmanner.We’llcover

theotherphasesinthefollowinglessons.

Let’sdefinewhatasampleis.

•“Tosample,” theverb,meanstoacquireaspecimenofmaterialforthepurposeoftesting

orinspectingit.

•“Asample,”thenoun,isaportionofmaterialthatisusedforinvestigationpurposes,such

astestingorinspection.

Whenwegoouttothejobsiteandtakeaportionofmaterial,thatis“sampling”the

material.Thatportionwetookduringsamplingiscalled“thesample.”Itisthislatteruseof

thetermsamplethatisusedthroughoutthismodule.

Asamplerepresentspartofthewholeprojectandittakesmultiplesamplestoestablisha

methodtoevaluateoranalyze theacceptabilityofmaterial.

Whenwecollectdata,wetakesamples.Remembertheriskscenarioswithmarbles?We

weresamplingthebag’scontents,andthesetofthreemarbleswasthesampleofwhat

wasinthebagandwecallthat“asamplesizeofthree,”notthreesamples.

Imagedescription:Workergatheringmaterialsample.

Whydowecollectdata? Theobjectiveofstatisticalanalysis,asitrelatestoconstruction

materials,istoderiveanunderstandingofthesematerialsbyutilizing dataobtainedfroma

smallportion(asample)ofthetotalquantity(population)produced.Thisprovidesamore

rationalbasisformakingdecisions.

Thegentlemanyouseehere iscollectingconcreteslumpdatausingaslumpcone.

Imagedescription:Workergatheringmaterialsample.

Definitionsandtermsrelatedtocollectingdataareusedthroughoutthemodule. They

include:

•Sampling;

•PopulationorLot(termsareusedinterchangeably);

•Sample;and

•Data.

Selecteachtermtoseethedefinition.

Formoreinformationontransportationqualityassuranceterms,seeGlossaryof

TransportationConstructionQualityAssuranceTerms.

Hyperlinkdescription:http://onlinepubs.trb.org/onlinepubs/cir culars/ec173

Samplingistheprocessofobtainingasample.

PopulationorLotare termsthatareusedinterchangeably. Apopulationorlotisaspecific

quantityofsimilarmaterial,construction,orunitsofproduct,subjectedtoeitheran

acceptanceorprocesscontroldecision.Alot,asawhole,isassumedtobeproducedbythe

sameprocess.This istheTransportationResearchBoarddefinition.

Asampleisasetofmeasurementsorcountsthatconstituteapartorallofthepopulation.

Data isfactualinformation,suchasmeasurementsorstatistics,usedasabasisfor

reasoning,discussion,ordecisionmaking.

Typically,sampling isusedtodeterminethecharacteristics(slump orasphaltcontent,for

example)ofalargerquantityofmaterial. Thelotisusually toolargetotestorevaluatein

itsentirety,sowemustevaluateaportionofthelot, whichwecallasample, tomake

decisionsaboutthetotallot.

Imagedescription:Example ofPopulationorLot.

Imagedescription:ExampleofSampling.

Whyissamplingimportant?

•Samplingprovidesdatathatrepresentthetotalproduct.Statisticalanalysesusesampling

datatoestimatethequalityofthatproduct.

•Datacollectionallowsrationaldecisionmaking.

•Samplingisusedtodeterminethequalitycharacteristics,forexamplesomepropertyof

interest,ofalargerquantityofmaterial,orpopulation.

•Itisimpracticaltotesttheentirelot,soaportionistestedtomakedecisionsaboutthe 

totallotusingallavailabledata.

•Ifwedonotuseallofthedataprovidedbythesampling,wewill inducebiasinthe

results,whichmeansthedataarenotrepresentativeofalloftheproductbeingused.

Data collectionisdonebyobtainingasamplefromapopulation. Here,the populationisa

stripofroadwayfromwhichthesampleistak en.Thesampleprovidesinformationabouta

particularportionofthepopulation.Eachoneofthesamplesobtainedfromthestretchof

roadprovidesaseparatepieceofdata.Allofthedatacollectedduringtheprocessof

samplingfeedintotheoverallstatisticalanalysisdoneontheproduct.Inthisexample,

coresaretakenfromapavementtoestimatesomequalitycharacteristicsuchasthickness

ordensity.Sotheprocessofobtainingthedataisfromthetopdown, thatis,the

populationprovidesthesample,whichthenprovidesthedata.Buttheanalysisisfromthe

bott omup,thatis,thedataprovidesinformationaboutthesample,whichthenprovides

informationaboutthepopulation.

Imagedescription: ExampleofSampling,CollectionandData.

Thedatacollectedcaneitherbecontinuousordiscrete,dependingontheprocessusedto

collectthedata. Thetwotypesofdataare:

•Continuous; and

•Discrete.

Selecteachtermtolearnmore.

Thisisjustanintroductionintothetypeofdatathatiscollected.Thetypeofdatabecomes

importantwhenwestarttoanalyzethedataaswewillseeinLesson7.

Continuousdataresultfromameasurementprocessandareusuallytheresultofreadinga

scale(e.g.,arulerorpressuregauge).Dataofthistypeareref erredtoascontinuous

variabledatasinceallvaluesalongacontinuousscalewithinaparticularrangeare

possible.

Thepictureontheleftshowsaninspectorusingathickness gaugetomeasurethe

thicknessofa layerofasphalt,sothisiscontinuousdata.Thethicknesscouldbe1.0”,1.1”,

1.2”,etc.

Imagedescription:Aninspectorusingathicknessgaugetomeasurethethicknessofalayer

ofasphalt.

Discretedataresultfromacountingprocessorfromayesornodecisionprocess.Thistype 

ofdatacouldresultfromcountingthenumberofdowelbarsplaced,orthenumberof

reinforcingbarsinabundle.Bydefinition,then,discretedataarenotobservedona

continuousscale.Thisdistinctionisimportantwhenattemptingtoorganizeandpresentthe

datathathavebeenobtained.

Intheslide,thepictureontheright showsaninspectorcountingthepiecesofsteelsothis

isdiscretedata.Thereareeither100piecesofsteelor101pieces,forexample,not100.1

piecesofsteel.

Imagedescription:Aninspectorcountingthepiecesofsteel.

Anotheraspectofthetypeofdata, similartocontinuousanddiscrete,iswhetheritis

variabledataorattributedata.Selecteachtermtolearnmore.

Whenarecordismadeofameasuredcharacteristic,suchascompressivestrengthinkPa,

thequalityissaidtobeexpressedbyavariableformat.Sincemostconstructionmaterials

mustmeetcertainrequirements,itiscommontoexpressspecificationsinavariableformat

bygivingtheacceptableupperandlowerlimitsforameasuredvalue.

Image description:Blackboard.

Thereareinstances whereitisonlynecessarytokeeparecordofthenumberofarticles

conforming,orfailingtoconform,tothespecifiedrequirements.Thisapproachresultsin

thecollectionofattributedata.Thismightbethecasewhenitisonlynecessaryto

compareaparticularcharacteristictoagivenstandard.Iteithermeetsthestandardorit

doesnot.Twotypicalapplicationsofattributedataare:

1.Screeningteststhatcanbe"go"or"nogo.“

2.Acceptanceofindividualitemssuchaslengthsofdrainagepipe.

However,theuseofvariabledataismoreefficientthanattributedata.Itwouldtherefore

beinefficienttotrytoconv ertavariabledataspecificationintoanattributedata

specification.

Nowreturningtothediscussion ofsamples.Theirsolepurposeistobeanalyzedtoprovide

informationaboutthepropertiesofalot.

Thelargerectanglecanbe thoughtofasalot consistingofthesmallersquaresthat

representpotentialsamples. Youshouldkeepinmindthatitisalwaysthepropertiesofthe

lotthatwewishtoidentifybutthatcanbedoneonlythroughasample.Andalthoughthis

rectangleisfinite,thatis,itcontainsa certainnumber ofpotentialsamplesthatcanbe

counted,inrealityapopulationcontainsaninfinitenumberofpotentialsamples.

Imagedescription:Exampleofalot.

Obviously,todeterminethe"best"estimateofsomeproperty,suchasthickness,everybit

ofpavementinthelotshouldbetested. Thisiscalled“completeenumeration.”However,

completeenumerationisneverfeasible.Forinstance,indestructivetesting,suchastaking

cores,theentirelotwouldbedestroyed.Fornondestructivetesting,suchasnuclear

densitytests,thereisneithertimenorlaboravailableforcompleteenumeration.

Therefore,sampling istheonlypracticalsolution.

Imagedescription:Exampleofalot.

Thewaythesamplelocationsarechosenisveryimportant.

Althoughthesamplesarenumberedinthefigure,wecallthisasample sizeoffour.The

samplescollectedareusedtoprovideanestimateoftheoverallqualityofthelot.Asample

canbeselectedfromthelotofmaterial,madeupofseveralparts,andthedata fromthe

samplecanbeusedtoestimatesomepropertyinthelotinordertomakeadecision

regardingitsacceptability.

Selecttheboxtoanswerthequestion,Ifwedon’tcollectinformationontheentirelot,

howcanwebesurethatthematerialisofgoodquality?

Imagedescription:Exampleofalot.

Wecanneverbe100%sureofthequality.Wecanonlyestimatethequality.

Imagedescription:Exampleofalot.

Itisimportanttoobtainvalidsampledata.Ifsamplesofalotaregoingtobeusedtomake

determinationsaboutquality,wedon’twantthedecisionsmadeaboutthequalityofthe

materialtorunahighriskofbeingincorrect.

Therelationshipbetweenthepropertiesofthesampleandthepropertiesofthe

population isanimportantaspectofstatisticaltheor yandpracticesince"good"estimates

ofthepropertiesofapopulationrequirevalidsamples.

Obtainingvalidsamplesisnotautomatic.Thefollowingaretwopossibleproceduresfor

obtainingsamples:

1.RandomSampling;and

2.BiasedSampling.

Selecteachtermtolearnmore.

Randomsamplingisasamplingprocedurewherebyanyindividualmeasurementinthe

populationisaslikelytobeincludedasanyother.Thisisoneoftwowaystoensuresample

validity.

Imagedescription:Threeexamplesofalot.

Biasedsamplingisasamplingprocedurewherebycertainindividualmeasurementshavea

greaterchanceofbeingincludedthanothers.

Image description:Workersgatheringsamples.

Thesecondwaytoensuresamplevaliditymustalsooccurundercontrolledconditions.This

isinordertominimizetheriskofmakingawrongdecisionaboutthequalityofamaterial.

Controlledconditionsmeansthattheprocessstaysessentiallythesamethroughthelot. If

aconcretebridgedeckisbeingplacedandtheingredientsoftheconcretearechanged

duringtheplacement,controlledconditionsnolongerexist.Thismeansthelotisnolonger

thesameasitwaspreviously.

Itisrequiredthatthesetofdatato betreatedasasinglegroup thatrepresent

homogeneous data.Forexample, measurementsorcountsmadeunderthesametesting

situationandthatrepresentconstructionmaterialsthatareproducedunderessentiallythe

sameconditions. Becausesampledatasupportsomanyimportantdecisions,itisessential

thatpropersamplingproceduresareused;thatis,theagencysamplingprotocolsare

followedascloselyaspossible.Itisimportanttotak emeasurementsandperformtests

followingtheagencytestingprotocolsandunderthesameconditionsascloselyaspossible

toensuretheconsistencyofresultsisnotaffected.Itisequallyimportanttosampleina

waythatensuresthematerialsrepresenttheentirepopulationandareproducedunder

essentiallythesameconditions.

Imagedescription:CrewslayembeddedtrackfortheHiawathaLightRailindowntown

Minneapolis.

Gettingsamplesthatrepresentthepopulationistheobjectiveofanysamplingplan.But

whatsamplesizeisenough?Isonespecimenenough?Howmanyareenough?Aremore

betterthanfewer?

Itisalwaysthepropertiesofthelotorpopulationthatareneeded.Samplingistheonly

effectivemeansforestimatingtheacceptabilityofalotorpopulation.However,howmany

specimens(1,2,3)shouldcomprisethesampletoarriveatasoundestimate?

Theideapersiststhatatestonasinglesampleshowsthe"true"qualityofthematerial,

andthatifanytestresultisnotwithin somelimit,thereissomethingwrongwiththe

material,construction,sampling,ortesting.Thus,termssuchasinvestigational,check,and

refereesampleshavebeenincommonusetoeitherconfirmordocumentthese"failures."

Naturedislikesidentities;variationistherule.Therefore,anyacceptanceorprocesscontrol

samplingmustaccountforvariabilityofmaterialsorconstruction.Multiplesampling

accomplishesthisobjective.

RecalltheexampleofthemarblesusedinLesson3wherethepropertyyouwere

determiningwasthecolorofthematerial(recallthatthespecificationwas8outofthe10

marblesneededtobewhite).Itwasdiscussedthattheoriginalthreesamplestakenraised

questionsaboutthelevelofriskofdeterminingwhether8ofthe10marbleswerewhite.

Whatifonlyonemarblehadbeendrawn?Whatdoyouthinktheriskswouldbe?

Imagedescription:Amarbleboxwithanarrowpointingtoeightmarblessittingoutsideof

thebox.

Let’slookat anexampleofsampling apopulationofconcreteairtests.In thisexample,itis

clearthatasamplesizeofonewouldnotgiveenoughdatatomakeanaccuratedecision

aboutthequalityofthisproduct.

Thisfigureillustratestheresultsof30concreteaircontentteststhatwewillassume

representatotalenumeration(entirepopulation)ofallthepossibleaircontentresultsfor

alotofstructuralconcrete.Ifthespecificationlimitsforaircontentforthismixare5%+

1%,thenanyaircontentresultbetween4%and6%iswithinthespecification

requirements.Areviewofthefigureindicatesthat13ofthetests(43%thoseinred)are

outsidespecificationlimits.Conversely,57%(white)arewithinspecificationlimits.

Obviously,ifasamplesizeofonemadeuptheentiresample(oftenreferredtoasasingle

sample)youwouldnotbeabletodeterminethat43%ofthematerialwasoutsidethe

specificationlimit.

Imagedescription:Afigurethatillustratestheresultsof30concreteaircontentteststhat

wewill assumerepresentatotalenumeration(entirepopulation)ofallthepossibleair

contentresultsforalotofstructuralconcrete.

Themorespecimensthatmakeupasample,thegreaterthelikelihoodthatthesample

reflectsthetruepropertiesofthelot.

Theresultsofasampleoffiveaircontentteststakenfromthelotareshownhere.Ifa

decisionweretobe basedononlythefirsttestresult(2.1%),thenthelotwouldbe

rejected.If,however,wetakeasecondtestandtheresultis5.2%,whichiswithinthe

specificationlimits,someindecisionisintroducedsinceonetestmeetstherequirement

whiletheotherdoesnot.Increasingthesamplesizepresentsaclearerpictureregarding

howtheaircontentisvaryingwithinthelot.Thisvariationcannotbeshownonthebasisof

asampleofsizeone,butcanbeestimatedasthenumberoftests(specimens)increases

(i.e.,multiplesampling).Weconsiderthequestionofhowmanyspecimenspersample

shouldbetakeninsubsequentchapters.

Conceptually,43%ofthepopulationwasoutsidethespecificationlimits(4–6).Inthis

example,oursamplesizeoffiveshowstwooutoffivetobeoutsidethespecificationlimits

(40%).Thatisaverycloseestimate.Obviously,asamplewillnotalwaysestimatea

populationpropertythiswell,butitillustratesthepointthatmultiplesamplingimproves

ourchancesofmakingacorrectdecision.Note:don’tconfusethiswithPWL,we’ renot

thereyet.

Imagedescription:Afigureshowingtheresultsofasampleoffiveaircontentteststaken

fromthelot.

Imagedescription:Example highlightingresults.

Sowheredoweselectsamples?Theselectionofthesamplinglocationswithin thelot

mustbeentirelyrandom."Random"doesnotmean"haphazard"— itmeans thesampleis

selectedwithoutbias.

Inpracticeitmaybedifficulttotraintechnicianswhohavebeenaccustomedtoinspection

toselectsampleswithoutregardtoquality.Theirtendencyistomakesurethatdefective

materialsarerepresentedinthesample,orpossiblythatonlyacceptablematerialsare

includedinthesample,thussubconsciouslybiasingthesample.

Selecteachtypetolearnmore.

Imagedescription:Exampleofabiased lot.

Imagedescription:Exampleofarandom lot.

Biasedsamplingisanysamplingthatisnotrandom.

Biasedsamplingtechniquesinclude:

•Systematicsamplingsuchassamplingonly downthecenterline;

•Representativesampling;

•Quotasampling;

•Selectivesampling;and

•Discardingdatawithoutverifiablecause.

Imagedescription:Exampleofabiased lot.

Randomsamplingcanbegener atedbyseveralmethods.

Thethreemostcommonmethodsforgeneratingrandomnumbersare:

•Randomnumbertable;

•Calculatorwitharandomnumbergenerator.Therearemanytypesofcalculatorswith

randomnumbergenerators.(Youshouldbefamiliarwiththecalculatoryouusetoassureit

is,infact,generatingrandomnumbers);and

•Computerwitharandomnumbergenerator.

Inallcases,youshouldknowthesizeoflotandnumberofspecimens.Thisinformation

comesfromspecificationsandcontractdocuments.

Imagedescription:

Randomsamplingensuresthateachportionofalothasthesamechanceofbeingselected

forthesample.

Stratifiedrandomsamplingadditionallyinvolvestheselectionoftwoormoredefinedparts

ofagivenlot.Stratifiedsamplingisusedtoensurethatthespecimensforthesampleare

obtainedfromthroughoutthelot,andarenotconcentratedinoneportionorsectionof

thelot.Forinstance,ifrandomsamplingisuseditispossible,althoughnotlikely,thatallof

thesamples, or cores,couldbeselectedwithinthefirsthalfofthelot.Thatwouldbe

randomandwouldtellyouagreatdealaboutthefirsthalf,butnothingaboutthesecond

half.Statisticallythiswouldbecorrectbutengineering‐wisewouldnotprovidea

comfortablefeelingaboutthequalityoftheentirelot.

Thelotcanbestratifiedintoanumberofsublotsequaltothesamplesizetobeselected

fromthelot.Onecoreisthenrandomlyselectedfromwithineachsublot.Byusingthis

procedure,eachportionofthelothasthesamechanceofbeingselectedwhile,atthe

sametime,ensuringthatthesamplingisspreadoutovertheentirelotinsteadofbeing

bunchedinonearea.Thisisthetypicalprocedureused.

Imagedescription:Exampleofalot,stratifiedintoanumberofsublots.

Supposeoneistosampleanasphaltmixturefromtheroadwaytoobtaincoresfordensity

determination.Thespecificationstatesthatthelotsizeshallbe5,000linearfeet(LF)of

pavement,andthatthesamplesizeisfivecoresperlot,andthatstratifiedrandom

samplingwillbedone.Ifweassumethatthepavementwidthis12feetandthelotbegins

atstation100+00,thenwecanusearandomnumbertabletoselectthesamplinglocations

ofthesublots.

Theexampleinthenextslideillustrateshowtodeterminethelocationsforsamplingusing

arandomnumberstableforfivesublots.

Imagedescription:Exampleofarandomnumbertable.

Oneofthemostcommonmethodsfordeterminingwhenorwheretoobtainsamplesis

throughtheuseofarandomnumbertable.Arandomnumbertableisacollectionof

randomdigits.Randomnumbertablescomeinmanyforms—someareshort,someare

long,somegroupedbypairsofdigits,somewithasmanyasfivedigitspergroup.When

usingarandomnumbertable,thekeyisthatbiasmustbeavoided.

Abriefexampleofarandomnumbertableisshown.Thistableisusedinthislesson

becauseitissimple.AmorethroughprocedureistheuseofASTMD3665.Theserandom

numbersinthetableinthislessonarepresentedinpairsofdigitsand,forthemethodsthat

wewill consider,canbethoughtofastwo‐placedecimalfractions.Forexample,the

randomnumber57inthetablewouldbeusedas0.57. Samplinglocationscanbe

determinedonthebasisoftime,tonnage,volume,distance,area,etc.Theonlyother

necessaryinformationisthesizeofthelottobesampledandthenumberofsamples,or

thesample size.

Whenselectingagroupofrandomnumbers,onecanenterthetableatanypoint(but

neveratthesamepointtwice)andselecttherequiredamountofnumbers.Thenumbers

canbeselectedbycolumnsorrows,bygoingleftorright,upordown,selectingalternate

numbers,oranyotherpatterndesired,butthisdecisionshouldbemadebeforestarting

theprocessofselectingthenumbers.Otherwisethenumbersmaybebiased.

Toopentherandomnumbertableexerciseselectthepaperclipicon.

Imagedescription:Exampleofarandomnumbertable.

Therandomnumbertable canbeusedtodetermineboththetransverseandlongitudinal

locationsforthecores.Twosets(columns,rows,etc.)ofrandomnumbersareselected:

oneforthetransverseposition,theotherforthelongitudinalposition.Thespecification

statesthatasetoffiverandomnumbers(thenumberofcores)forthelongitudinal(X)

positionandfiverandomnumbersforthetransverse(Y)positionofthesamplebechosen.

Thisabbreviatedtableusesthesecondblockofnumbersfromthetableintheattached

exerciseandinitselfwaschosenrandomly.(Theblockchosen,wheretostart,andwhat

directiontouse,thatis,horizontally orverticallyshouldallbeselectedrandomly.)

Thewayqualitycontrolandacceptanceactivitiesuseoftherandomnumbertableis

explainednext.Toviewtherandomnumbertableexerciseselectthepaperclipicon.We

willcomplete thisexerciselaterinthelesson.

Imagedescription:Exampleofarandomnumbertable highlightingtheXandYpositions.

Onceyouhaveselectedthesetsofrandomnumbers,applybasicmultiplicationtoderive

longitudinalandtransversecoordinatesforeachcoreinthesublot.TheseXandYrandom

numbersaremultipliedbythesublotlengthandwidth,respectively,asshowninthe 

example:

Sublot#1(startatstation100+00forthelongitudinalreferencepointandthebottomedge

ofpavementfortheverticalreferencepoint. Theedgeofpavementtouseasthereference

pointisarbitrary,butonceselectedhastobeusedthroughoutthelot.)

•Thefirstnumberinthefirstrow, whichwasrandomlyselectedastheXcoordinateisused

forthehorizontalcoordinateinthefirst sublotandismultipliedbythesublotlength,1,000

feet. Thus,thelongitudinalcoordina teforsublot#1isX=0.74x1,000=740ft.

•Likewise,thefirstnumberinthesecondrow, whichwasrandomlyselectedastheY

coordinateisusedfortheverticalcoordinateinthefirst sublotandismultiplied bythe

sublotwidth,12feet. Thus,thevertical coordinateforsublot#1isY=0.29x12=3.5ft

measuredfromthebottom edgeofpavement.

Thissameprocedureiffollowedforsublot#2shownonthenextslide.

Imagedescription:Example ofcoresamples,highlightingsublot1.

Sublot#2(startatstation110+00)

•Thesecondnumberinthefirstrow, whichwasrandomlyselectedastheXcoordinateis

usedforthehorizontalcoordinateinthesecondsublotandismultipliedbythesublot

length,1,000feet. Thus,thelongitudinalcoordinateforsublot#2isX=0.60x1,000=600

ftandisaddedtoStation110+00 forthelongitudinalreferencepoint.

•Likewise,thesecondnumberinthesecondrow, whichwasrandomlyselectedastheY

coordinateisusedfortheverticalcoordinateinthesecondsublotandismultipliedbythe

sublotwidth,12feet.Thus,thevertical coordinateforsublot#2isY=0.21x12=2.5ft

fromthebottomedgeofthepavement.

Thissame procedureisfollowedforsublot#3,sublot#4,andsublot#5andtheresultsof

allfivelocationsareshownonthenextslide.

Imagedescription:Example ofcoresamples,highlightingsublot2.

Herearethelocationsofthefivecores.

Selecttheboxtoanswerthequestion,Doyouseeanyproblemwithanylocation?

Imagedescription:Exampleof5coresamples.

Lookatsublot#4. Thecoreisneartheedgeofthepavement. Itisoff theedgebyonly0.1

foot.Thisisoftennotpermissiblebecauseofthedifficulty ofgettingcompactionthisclose

totheedge.Butifitisnotpermittedtotak easamplethisclosetotheedgeofthe

pavement,thisrestrictionshouldbeinthesamplingprotocolsoallsamplingpersonnelwill

observeit.

Ifthissituationoccurs,itispermissiblestatisticallytotakethenextset,thatisXandY

valuestodeterminethesamplelocation.Thatwouldrequirethenextsetofrandom

numberstobeusedforsublot#5.

Imagedescription:Core4example.

Nowwe’llgiveyouachancetogener ateyourownfivelocationsusingtheexample

specificationshown.Usingtherandomnumbertableintheattachedexercise,orarandom

generatoronyourcalculator,iPhone,orcomputer,developfivesetsofrandomnumbers.

Thatis,alongitudinalandaverticalcoordinateforeach.Thenbymultiplyingtherandom

numberbythesublotdimensions,determineyourfiverandomsamplinglocations.

Take15minutestodeterminefiverandomsamplinglocationsbeforemovingonwiththis

lesson.Writetheanswersinyourex ercise.Youwillbeaskedtoshareyourresultslateron.

Toopentheexerciseselectthepaperclipicon.Usetherandomnumbertableorarandom

generatoronyourcalculator.

Imagedescription:Exampleof5coresamples.

Therandomnumberscanbecompromisediftheyareknownpriortobeingneeded,sothe

securityofthenumbersoncetheyareselectedisimportant.Itisunacceptabletopublish

therandomnumberorotherwisedeterminetherandomnumbersbeforetheyareneeded.

Theownershouldwitnesstakingthesampleandshouldtakepossessionofit.Theowneris

responsibleforthechainofcustody.Itisunacceptablefortheownertogivethesampleto

thecontractortodelivertothelabunlessthesampleissecuredwithsomesortoftamper

resistantmaterialandhasaserialnumber.

Thetonnage,orlocationandtime,forexample,determinedbytherandomnumbersneed

tobedocumented.Likewise,whentherandomnumbersandresultingtonnage,orlocation

andtimethatareused,shouldberecordedsothatverificationoftheinformationis

possible.Thismayseemtobeatrivialmatter,butinthecaseofadisputeorclaim,itcan

beoneofthedecidingpoints.Infact,acourtcase(contractorclaim)inoneStatehingedon

theagencyprovingthatthesampleswereselectedrandomly.Thecontractorarguedthat

thesamplelocationswerebiasedtoward“bad”spotsinthepavement.Theinspector’slog

bookhadtobeproducedshowingthedeterminationoftherandomlocationstorefutethis

claim.

Imagedescription:lockandcheckmark.

Matchthebasicphasesofstatisticalanalysiswiththeirdefinitions.

Phasesofstatisticalanalysis:

•Collectdata;

•Organizedata;

•Analyzedata;and

•Interpretdata.

Definitions:

a)Assemblingofdataintosystematicgroupsorclassificationsfromwhichlogical

conclusionscanbedrawn;

b)Usingthebasicsampleresultstoinferbroaderstatementsaboutthetotalquantityof

material;

c)Theplannedprocessofobtainingarelativelysmallnumberofmeasurementsfroma

fairlylargequantityofmaterial;and

d)Numericaldeterminationofstatisticalmeasuresthatdescribetheimportant

characteristicsofthedata.

Thecorrect answersare:

•CollectDataisc)Theplannedprocessofobtainingarelativelysmallnumberof

measurementsfromafairlylargequantityofmaterial;

•OrganizetheDataisa)Assemblingofdataintosystematicgroupsorclassificationsfrom

whichlogicalconclusionscanbedrawn;

•AnalyzetheDataisd)Numericaldeterminationofstatisticalmeasuresthatdescribethe

importantcharacteristicsofthedata;and

InterprettheDataisb)Usingthebasicsample resultstoinferbroaderstatementsabout

thetotalquantityofmaterial.

Selectthecorrectdefinitionofsampling:

a)Anisolatedquantityofmaterialproducedessentiallybythesameprocess;

b)Asetofmeasurementsorcountsthatconstituteapartorallofthepopulation;

c)Factualinformationusedasabasisforreasoning,discussion,ordecision making;or

d)Processofobtainingasample.

Thecorrectanswer isd)Processofobtainingasample.Sampling isaprocess.Itshouldnot

beconfusedwiththe“sample, ”whichbecomeasetonmeasurementsorcountsand

providefactualinformationfordecisionmaking.

Whyissamplingimportant?Selectallthatapply.

a)Samplingisthefirststepintheprocessofprovidingfactualinformationaboutthelot;

b)Thelotisusuallytoolargetotestinitsentirety,soaportionofthelotistestedtomake

decisionsaboutthetotallotusingallavailabledata;

c)Itensures100%ofquality;and

d)Samplingallowsrationaldecisionmaking.

Thecorrectanswersare:

a)Samplingisthefirststepintheprocessofprovidingfactualinformationaboutthelot;

b)Thelotisusuallytoolargetotestinitsentirety,soaportionofthelotistestedtomake

decisionsaboutthetotallotusingallavailabledata;and

d)Samplingallowsrationaldecisionmaking.

Samplingis thefirststepintheprocessofprovidingfactualinformationaboutthelot.

Samplingtheentirelot,thatis,“completeenumeration”isnotfeasible.Thus,itisthe

samplingthatprovidestheinformationnecessaryforrationaldecisionmaking.Ifthe

samplingisdoneincorrectly,itcanprovideinformationthatcanleadtoincorrectdecisions.

Trueorfalse?Ittakesmultiplesamplestomakeabetterdecisiontodeterminethe

acceptabilityofmaterial.

a)True;or

b)False.

Thecorrect answerisa)True.As observedinthemarbleexampleinLesson3,takingonlya

samplesizeofthreecanleadtohighriskstotheagency.Ittakesmultiplesamples,thatisa

samplesizegreaterthanone,toprovideameasureofvariabilityandmakeabetter

decisionastheacceptabilityofmaterial.

Whichofthesemethodswillgiveyouabetterevaluationoftheentirelot?Selectallthat

apply.

a)Ifalotisevaluatedusingsublotswithrandomsampling,thelotisevaluatedinamore

throughmanner;

b)Usingstratifiedrandomsamplingensureseachportionofthelothasthesamechanceof

beingselectedwhilealsoensuringthatthesamplingisspreadoutovertheentirelot;

c)Itisimportanttoevaluatethefirstpartofthelotbecausethattellswhattherestofthe

lotwillbelike;and

d)Sincethewholelotisthesameanyportionevaluatedwillbethesame.

Thecorrectanswersare:

a)Ifalotisevaluatedusingsublotswithrandomsampling,thelotisevaluatedinamore

throughmanner;and

b)Usingstratifiedrandomsamplingensureseachportionofthelothasthesamechanceof

beingselectedwhilealsoensuringthatthesamplingisspreadoutovertheentirelot.

The useofsublotswithrandomsamplingensuresthatthesamplingprocessisvalidandwill

bespreadovertheentirelot.Italsoensuresthateachportionofthelothasthesame

chanceofbeingaccepted. Samplingthefirstpartofalotdoesnotprovideanyindicationof

whattherestofthelotwill belike.Assumingthewholelotisthesameisanerroneous

assumption.

Whichofthefollowingstatementsistrue?

a)Onlytherandomnumbersshouldbedocumentedandsecured;

b)Timeandlocationoftherandomnumbersneedtobedocumentedandsecured;or

c)Therandomnumbers,time,andlocationneedtobedocumentedandsecured.

Thecorrect answerisc)Therandomnumbers,time,andlocationneedtobedocumented

andsecured.Therandomnumberscanbecompromisediftheyareknownpriortobeing

needed,sothesecurityofthenumbersoncetheyareselectedisimportant.Itisalso

importantsothatverificationoftheinformationispossible.

YouhavecompletedModuleA,Lesson4:CollectingData:SamplingTheory.Youarenow

ableto:

•Describethebasicphasesofstatisticalanalysis;

•Definesampling;

•Explaintheimportanceofsamplingandusingallavailabledata;

•Discusssecurityanddocumentationofrandomsamplelocations;

•Explainhowasamplerelatestoapopulation;and

•Applyrandomandstratifiedrandomsamplingtechniquestoobtainvaliddata.

Close thislesson,andreturn tothemodulecurriculumtoselectthenextlesson.Toclose

thiswindow,selectthe“X”intheupperright‐handcornerofyourscreen.