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Click and Learn Sampling and Normal Distribution

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OVERVIEWNormaldistribution,sometimescalledthebellcurve,isacommonwaytodescribeacontinuousdistributioninprobabilitytheoryandstatistics.Inthenaturalsciences,scientiststypicallyassumethataseriesofmeasurementsofapopulationwillbenormallydistributed,eventhoughtheactualdistributionmaybeunknown.Butevenifyouassumethatmeasurementsofapopulationshouldbenormallydistributed,asampletakenfromthatpopulationwillnotnecessarilybenormallydistributed.Whyisthat?InthisClickandLearn,youwillexplorewhatsampledistributionlookslikewhensamplesaretakenfromanidealizedpopulationofadefinedmeanandstandarddeviation.Studentswillexplorehowstandarddeviationaffectsthedistributionofmeasurementsinapopulation.Next,theywillexplorehowsamplesizeaffectsthedistributionofmeasurementsandthereforethesamplemean.Throughthisexploration,studentswilldevelopanunderstandingofhowsamplesizeaffectsthedistributionofsamplemeansdrawnfromthesamepopulationandhowthisphenomenonismodeledinanequationforcalculatingthestandarderrorofthemean.

KEYCONCEPTSANDLEARNINGOBJECTIVES

• Theappearanceofahistogramofmeasurementsinasampledependsonthepopulationfromwhichthesamplecame.

• Theappearanceofthehistogramalsodependsonthesamplesize.

• Smallsamplestakenfromanormallydistributedpopulationmaynotappeartobenormallydistributed.Largersamplesstarttoapproximateanormaldistribution.

• Whenapopulationissampledrepeatedly,ameancanbecalculatedforeachsample,toobtainmanydifferentmeans.Ifthosemeansareplottedasahistogram,theywillbeapproximatelynormallydistributed.

• Thestandarddeviationofsuchadistributionofmeansiscalledthestandarderrorofthemean.

Studentswillbeableto

• explainthatstandarddeviationisameasureofthevariationofthespreadofthedataaroundthemean.

• explainthatlargersamplesizesaredesirablewhencollectingdataaboutapopulationbecausetheyaremorelikelytoreflectthedistributionofmeasurementsinapopulation.

• calculateStandardErroroftheMean(𝑆𝐸#,,butalsocommonlyreferredtoasSE,orSEM),usingtheequation𝑆𝐸#=

'(.

• explainthat𝑆𝐸#ofthemeanisameasureofthereliabilityofthemeanofasampleasareflectionofthemeanofthepopulationfromwhichthesamplewasdrawn.

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• use𝑆𝐸#todeterminethe95%ConfidenceIntervaltoadderrorbarstoagraphandusetheseerrorbarstodetermineifthereisadifferencebetweenthepopulationsfromwhichthesamplecame.

CURRICULUMCONNECTIONS APBiology(2012-2013)SP2,SP5 NGSS(2013)SEP4

KEYTERMS

measurement,sample,population,normaldistribution,randomsampling,mean,standarddeviation,standarderrorofthemean,95%ConfidenceInterval,errorbar

TIMEREQUIREMENTSCompletingallpartsofthislessonwillrequireuptothree50-minuteclassperiods.However,someportionscanbeassignedforhomework.SUGGESTEDAUDIENCE

Part1ofthisactivityisappropriateforafirstyearandanadvanced(honors,AP,orIB)highschoolbiologycourse.Parts2and3areappropriateforanadvanced(honors,AP,orIB)highschoolorintroductorycollegebiologycourse.

PRIORKNOWLEDGE

Studentsshouldbefamiliarwith

• statisticalconceptofmeanasanaverageofasample’smeasurements.

• histogramsasadisplayofthefrequencyofmeasurementsinasample.MATERIALS

• SamplingandNormalDistributionClickandLearnathttp://www.hhmi.org/biointeractive/sampling-and-normal-distribution

• DistributionofMeansgrid(lastpageofthisdocument;thesecanbelaminatedtobereusedbymultipleclasses)

TEACHINGTIPS

• ThisactivityassumesnopriorknowledgeofStandardDeviationorStandardErroroftheMean.Therefore,itcanbeusedtointroducetheuseofstatisticstodescribeadataset.Itisimportantthatstudentscandistinguishbetweenthetermsmeasurement,sample,andpopulation.Asampleisacollectionofindividualmeasurementsdrawnfromapopulation.PriortostartingPart1,studentsshouldunderstandthatitistypicallynotpossibletomeasureeveryindividualinalargepopulation.Therefore,arandomlyselectedsampleofthepopulationismeasuredandthedataisusedtorepresentthewholepopulation.

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Click and Learn Sampling and Normal Distribution

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• Studentscanoftenrecognizethatsmallsamplesizesarenotrecommendedwhencollectingdatafromapopulation.Doingasimpledemonstrationsuchasdrawingonlyafewcoloredbeadsfromabagtodeterminethedistributionofcolorsinthebagormeasuringtheheightofonlyafewstudentstodeterminethemeanheightoftheclasscanshowstudentsthatsmallsamplesizescanoftenleadtoamisrepresentationofthepopulationknownassamplingerror.

• ThesimulationintheClickandLearnisrunbyaprogramthatcalculatesarandomsamplevaluefromanormallydistributedpopulationofinfinitesize.InPart1,thestudentcanmanipulatesamplesizeaswellaspopulationmeanandstandarddeviation.InPart2,thestudentcanmanipulatesamplesize.Dependingonthespeedofyourcomputer,resamplinginPart2cantakeafewsecondsandthecalculationsoccurringinthebackgroundarecomplicated.(Forthoseofyouwhoaremathematicallyorstatisticallyinclined,theprogramusesBox-Mullertransform.)

• AttheconclusionofPart1oftheactivity,studentsshouldbeabletoexplainwhatstandarddeviationshowsaboutthedistributionofmeasurementsinapopulation.Theywillalsobeabletoexplain,usingevidencecollectedintheactivity,whythemeansoflargersamplesizesaremorelikelytoberepresentativeofapopulation’struemean.

• AttheconclusionofPart2oftheactivity,studentswillunderstandwhytheequationfor𝑆𝐸#givesanestimateofthestandarderrorofthemeanbasedonasample’ssizeandstandarddeviation.Theywillalsobeabletousetheequationtocalculate𝑆𝐸#,95%confidenceintervals,andusethe95%CItogenerateerrorbarsonabargraph.Whilethisactivityfocusesontheeffectofsamplesize(assamplesizeincreases,𝑆𝐸#decreases),studentsshouldbeabletopredictfromtheequationthatthereisadirectrelationshipbetweenthestandarddeviationand𝑆𝐸#.

• Remindstudentsthatonpage1oftheClickandLearn“SamplingfromaNormallyDistributedPopulation,”clicking“resample”issimulatingcollectinganewrandomlyselectedsetofmeasurementsfromthepopulation.Therefore,samplemeansandstandarddeviationswilllikelybedifferentfromstudenttostudent.Thiswillnotaffectthefinaloutcomeoftheactivity.Studentsshouldalsoberemindedthatonpage2oftheClickandLearn“StandardErroroftheMean,”“resample”representsrepeatingthesamplecollection500times,andeachsampleconsistsofanumberofmeasurementsequaltothesamplesize.Thismeansthatforasamplesizeof100,thesimulationtook50,000measurements.

• InPart2,theteachermayneedtopointoutanddiscussthedifferencebetweensamplemeanandstandarddeviationandthemeanandstandarddeviationof500means.Samplemeanandstandarddeviationisdescribingthedatainthetopgraph,whilethemeanandstandarddeviationof500meansisdescribingthedatainthebottomgraph.

SUGGESTEDPROCEDUREDependingontheskilllevelofthestudentsinthecourse,thisactivitycanbedoneindependentlyorguidedbytheteacher.Theprocedurebelowisforaguidedprocess,duringwhichtheinstructorchecksforstudentunderstandingatkeypointsintheactivity.

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Introduction1.Showstudentsthegraphbelowandaskthemtointerpretit.Askthemwhattheerrorbarsmean.Whileitdependsonanindividualstudent’spriorlearning,moststudentswillnotbeabletoexplainwhattheerrorbarsmean.Ifthisisthecase,askthemtodescribetheerrorbar.Guidestudentstoobservationssuchas• Theerrorbarfordarkdoesnotoverlaptheerrorbarforlight.• Thedarkerrorbarislongerthanthelighterrorbar.• Thelengthsoftheerrorbaraboveandbelowthetopofthebarareequal.

Figure1.MeanLengthofCroftonSeedlingsafterOneWeekintheDarkorintheLight.(FromUsingBioInteractiveResourcestoTeachMathematicsandStatisticsinBiologyhttp://www.hhmi.org/biointeractive/teacher-guide-math-and-statistics)2.InstructstudentstocompletethePre-assessmentQuestion(whichcouldbecollectedonnotecardsasaformativeassessment).TheninstructstudentstoaccesstheClickandLearnathttp://www.hhmi.org/biointeractive/sampling-and-normal-distributionandcompleteitems2through5.Itisimportantatthispointtoensurethatstudentsunderstandthatanindividualmeasurementispartofasampletakenfromalargerpopulation.Pointouttostudentsthecharacteristicsofanormallydistributedpopulationbyreferencingtheredlineonthegraphandthatnumberofindividualmassmeasurementsarerepresentedbythebarsinthehistogram.Note:ThispartalongwithPart1items6and7couldbeassignedtostudentsforhomeworkpriortocompletingtherestofPart1oftheactivityinclass.PART1:SAMPLINGFROMANORMALLYDISTRIBUTEDPOPULATION1. Studentsworkthroughthetaskandcompleteitems6and7toexplorehowmodifyingthestandard

deviationaffectsthedistributionofmeasurementsinthepopulation.Itshouldbepointedouttostudentsthatchangingtheparameterschangesthesimulationprogram.Inarealdataset,thestandarddeviationisdeterminedbytheactualmeasurementsinthepopulationorsample.

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Click and Learn Sampling and Normal Distribution

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2.Havestudentsreadthesummarydescriptionofstandarddeviationanddiscussanyquestionstheyhaveaboutstandarddeviationandnormaldistribution.3.IntherestofPart1,studentsexploretheeffectofsamplesizeonthesamplemeancomparedtothetruemeanofthepopulation.Remindstudentsthattheyaresettingparametersfortheprogramrunningthesimulation(populationmean=50kgandstandarddeviation=10kg).4.Item8canbeusedasaformativeassessmenttomonitorstudentunderstandingofstandarddeviation.Acorrectstudentresponsewouldbe:“Forthispopulation,68%ofthemassesshouldbebetween40and60kg(1standarddeviation),while95%ofthemassesshouldfallbetween30and70kg(2standarddeviations).”5.Studentscompleteitems9and10.Aftercompletingthistask,studentsshouldrecognizethatasamplesizeof1000ismorelikelytogiveyouasamplemeanthatreflectsthetruemeanofthepopulationbecausethelargernumberofmeasurementswillreflectthenormaldistributionofthepopulation.Theyshouldalsorecognizethatcollectingmeasurementsfromasampleof1000individualscouldbetime-consuming,expensive,orsimplynotpractical.6.Studentscomplete“Selectingtheappropriatesamplesize”bycompletingthetaskanditems11through13.Providestudentswiththe“DistributionofMeans”grid.Thistaskcanbecompletedinpairsorasmallgroup.Thereshouldbeatleastonegraphintheclassforeachsamplesizeinthesimulation(4,9,16,25,100,400,1000).Laminatingthegridswillallowthemtobereusedbyseveralclasses.Discussitem13asawholeclass.Askstudentstojustifytheiranswertothequestionwithevidencefromthegraphs.Studentstypicallyselect100asanappropriatesamplesize.Anexampleofthedatageneratedfromthistaskisshownbelow.

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PART2:STANDARDERROROFTHEMEAN1. Items1through8canbecompletedasahomeworkassignment.Studentsusethenextpageinthe

ClickandLearntoextendtheirexplorationoftheeffectofsamplesizeonthedistributionofsamplemeans.InPart2,resamplingwillgenerateahistogramofthemeansof500samplesshowinganormaldistribution.Studentsshouldcometotheconclusionthatwhilethesamplesizedoesnotaffectthemeanof500means,itdoesaffectthestandarddeviationofthemeans.Forsmallersamplesizes,thesamplemeancouldbequitedifferentfromthepopulationmean,andthisisreflectedinthelargerstandarddeviationofthemeans.ThisshouldreinforcetheconclusiontheycametoattheendofPart1thatlargersamplesizeswillprovideabetterrepresentationofthepopulationfromwhichthesamplewasdrawn.

2. Item9introducesstudentstotheequationforstandarderrorofthemean,𝑆𝐸#=

'(.Items10and

11havestudentscalculatethe𝑆𝐸#usingtheequation.Theyarethenaskedtocomparetheempiricallymeasured𝑆𝐸#(standarddeviationof500means)tothecalculatedestimationofthe𝑆𝐸#.Studentsshouldberemindedthatthemathematicalformulafor𝑆𝐸#allowsthemtoestimatethereal𝑆𝐸#givenasmallsample,whilerepeatingsamplesmanytimesallowsthemtoempiricallymeasuretheactual𝑆𝐸#.Studentsshouldfindthat,withtheexceptionofverysmallsamplesizes,usingtheequation𝑆𝐸#=

'(isareasonablyaccuratewaytoestimatetheStandardErrorofthe

Meanfromthestandarddeviationofasampleandthesamplesize.Studentscanbenefitfromadiscussionregardinghowtheequationmodelsthedistributionofmeansthattheyobserved.Encouragestudentstodiscusswhysamplesizeisinthedenominator.Asseeninthegraphbelow,standarddeviationofthemeansdecreasesassamplesizeincreases.TheyshouldrecallfromtheirobservationsduringPart1oftheactivitythatlargersamplesizesaremorelikelytobeatruerreflectionofthepopulationfromwhichthemeasurementsaredrawnandthatverysmallsamplesizesoftenresultininaccuraterepresentationsofthepopulation.Itishelpfultoreferstudentsbacktothedistributionof10meanstheyplottedinPart1oftheactivity.Emphasizethatthe𝑆𝐸#equationallowsonetoestimatethespreadofthemeansthatwouldbeexpectedfrommanysamplesdrawnfromthesamepopulationfromthestandarddeviationandsamplesizeofasingleobservedsample.

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Click and Learn Sampling and Normal Distribution

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Theeffectofsamplestandarddeviationonthe𝑆𝐸#isnotexploredinthesimulation.Thiswasdonetoavoidthemisconceptionthatsamplesizeaffectssamplestandarddeviation.Itstillmaybehelpfultodiscusstheeffectstandarddeviationofthesamplehasonthestandarderrorofthemean.Thestandarddeviationofthesampleisinthenumeratoroftheequationbecauseamorevariedpopulation(largersamplestandarddeviation)willincreasethelikelihoodthatthesamplemeasurementswillnotbeagoodrepresentationofthepopulationfromwhichtheyaretaken.3. Havestudentsreadthesummaryandthencompleteitems12and13tolearnhowtousethe

standarderrorofthemeantogenerate95%ConfidenceIntervalerrorbars.Readingthesummarywillshowstudentshowtointerprettheseerrorbars.

PART3:APPLYWHATYOUHAVELEARNEDThedatapresentedisauthenticdatacollectedbystudentsconductinganexperimenttotesttheeffectofpectinaseandcellulaseonturningapplesauceintoapplejuice.Thispartoftheactivitycanbegivenasanassessmenttodeterminestudents’levelofunderstandingoftheconceptofstandarderrorofthemeanandhowtousethestatistictoanalyzetheexperimentaldata.RECOMMENDEDFOLLOW-UPACTIVITIESEvolutioninAction:DataAnalysis(http://www.hhmi.org/biointeractive/evolution-action-data-analysis)RosemaryandPeterGranthaveprovidedmorphologicalmeasurements,includingwinglength,bodymass,andbeakdepth,takenfromasampleof100mediumgroundfinches(Geospizafortis)livingontheislandofDaphneMajorintheGalápagosarchipelago.ThecompletedatasetisavailableintheaccompanyingExcelspreadsheet.Inoneactivity,entitled“EvolutioninAction:GraphingandStatistics,”studentsareguidedthroughtheanalysisofthissampleoftheGrants’databyconstructingandinterpretinggraphs,andcalculatingandinterpretingdescriptivestatistics.Thesecondactivity,“EvolutioninAction:StatisticalAnalysis,”providesanexampleofhowthedatasetcanbeanalyzedusingstatisticaltests,inparticulartheStudent’st-testforindependentsamples,tohelpdrawconclusionsabouttheroleofnaturalselectiononmorphologicaltraitsbasedonmeasurements.LizardEvolutionVirtualLab(http://www.hhmi.org/biointeractive/lizard-evolution-virtual-lab)IntheLizardEvolutionVirtualLab,studentsexploretheevolutionoftheanolelizardsintheCaribbeanbycollectingandanalyzingtheirowndata.Thevirtuallabincludesfourmodulesthatinvestigatedifferentconceptsinevolutionarybiology,includingadaptation,convergentevolution,phylogeneticanalysis,reproductiveisolation,andspeciation.Eachmoduleinvolvesdatacollection,calculations,analysis,andansweringquestions.AUTHORValerieMay,WoodstockAcademy,WoodstockCT

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Click and Learn Sampling and Normal Distribution

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