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Sampling and Normal Distribution
Educator Materials
OVERVIEW
Normaldistribution,sometimescalledthebellcurve,isacommonwaytodescribeacontinuous
distributioninprobabilitytheoryandstatistics.Inthenaturalsciences,scientiststypicallyassumethata
seriesofmeasurementsofapopulationwillbenormallydistributed,eventhoughtheactualdistribution
maybeunknown.Butevenifyouassumethatmeasurementsofapopulationshouldbenormally
distributed,asampletakenfromthatpopulationwillnotnecessarilybenormallydistributed.Whyis
that?
InthisClickandLearn,youwillexplorewhatsampledistributionlookslikewhensamplesaretakenfrom
anidealizedpopulationofadefinedmeanandstandarddeviation.Studentswillexplorehowstandard
deviationaffectsthedistributionofmeasurementsinapopulation.Next,theywillexplorehowsample
sizeaffectsthedistributionofmeasurementsandthereforethesamplemean.Throughthisexploration,
studentswilldevelopanunderstandingofhowsamplesizeaffectsthedistributionofsamplemeans
drawnfromthesamepopulationandhowthisphenomenonismodeledinanequationforcalculating
thestandarderrorofthemean.
KEYCONCEPTSANDLEARNINGOBJECTIVES
•
Theappearanceofahistogramofmeasurementsinasampledependsonthepopulationfrom
whichthesamplecame.
•
Theappearanceofthehistogramalsodependsonthesamplesize.
•
Smallsamplestakenfromanormallydistributedpopulationmaynotappeartobenormally
distributed.Largersamplesstarttoapproximateanormaldistribution.
•
Whenapopulationissampledrepeatedly,ameancanbecalculatedforeachsample,toobtain
manydifferentmeans.Ifthosemeansareplottedasahistogram,theywillbeapproximately
normallydistributed.
•
Thestandarddeviationofsuchadistributionofmeansiscalledthestandarderrorofthemean.
Studentswillbeableto
•
explainthatstandarddeviationisameasureofthevariationofthespreadofthedataaround
themean.
•
explainthatlargersamplesizesaredesirablewhencollectingdataaboutapopulationbecause
theyaremorelikelytoreflectthedistributionofmeasurementsinapopulation.
•
calculateStandardErroroftheMean(SE)usingtheequationSE𝑥 = •
explainthatSEofthemeanisameasureofthereliabilityofthemeanofasampleasareflection
ofthemeanofthepopulationfromwhichthesamplewasdrawn.
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Sampling and Normal Distribution
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Educator Materials
useSEtodeterminethe95%ConfidenceIntervaltoadderrorbarstoagraphandusethese
errorbarstodetermineifthereisadifferencebetweenthepopulationsfromwhichthesample
came.
CURRICULUMCONNECTIONS
APBiology(2012-2013)SP2,SP5
KEYTERMS
NGSS(2013)SEP4
measurement,sample,population,normaldistribution,randomsampling,mean,standarddeviation,
standarderrorofthemean,95%ConfidenceInterval,errorbar
TIMEREQUIREMENTS
Completingallpartsofthislessonwillrequireuptothree50-minuteclassperiods.However,some
portionscanbeassignedforhomework.
SUGGESTEDAUDIENCE
Part1ofthisactivityisappropriateforafirstyearandanadvanced(honors,AP,orIB)highschool
biologycourse.Parts2and3areappropriateforanadvanced(honors,AP,orIB)highschoolor
introductorycollegebiologycourse.
PRIORKNOWLEDGE
Studentsshouldbefamiliarwith
•
statisticalconceptofmeanasanaverageofasample’smeasurements.
•
histogramsasadisplayofthefrequencyofmeasurementsinasample.
MATERIALS
•
SamplingandNormalDistributionClickandLearnat
http://www.hhmi.org/biointeractive/sampling-and-normal-distribution
•
DistributionofMeansgrid(lastpageofthisdocument;thesecanbelaminatedtobereusedby
multipleclasses)
TEACHINGTIPS
•
ThisactivityassumesnopriorknowledgeofStandardDeviationorStandardErroroftheMean.
Therefore,itcanbeusedtointroducetheuseofstatisticstodescribeadataset.Itisimportant
thatstudentscandistinguishbetweenthetermsmeasurement,sample,andpopulation.A
sampleisacollectionofindividualmeasurementsdrawnfromapopulation.Priortostarting
Part1,studentsshouldunderstandthatitistypicallynotpossibletomeasureeveryindividualin
alargepopulation.Therefore,arandomlyselectedsampleofthepopulationismeasuredand
thedataisusedtorepresentthewholepopulation.
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Sampling and Normal Distribution
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•
Studentscanoftenrecognizethatsmallsamplesizesarenotrecommendedwhencollecting
datafromapopulation.Doingasimpledemonstrationsuchasdrawingonlyafewcoloredbeads
fromabagtodeterminethedistributionofcolorsinthebagormeasuringtheheightofonlya
fewstudentstodeterminethemeanheightoftheclasscanshowstudentsthatsmallsample
sizescanoftenleadtoamisrepresentationofthepopulationknownassamplingerror.
•
ThesimulationintheClickandLearnisrunbyaprogramthatcalculatesarandomsamplevalue
fromanormallydistributedpopulationofinfinitesize.InPart1,thestudentcanmanipulate
samplesizeaswellaspopulationmeanandstandarddeviation.InPart2,thestudentcan
manipulatesamplesize.Dependingonthespeedofyourcomputer,resamplinginPart2can
takeafewsecondsandthecalculationsoccurringinthebackgroundarecomplicated.(Forthose
ofyouwhoaremathematicallyorstatisticallyinclined,theprogramusesBox-Mullertransform.)
•
AttheconclusionofPart1oftheactivity,studentsshouldbeabletoexplainwhatstandard
deviationshowsaboutthedistributionofmeasurementsinapopulation.Theywillalsobeable
toexplain,usingevidencecollectedintheactivity,whythemeansoflargersamplesizesare
morelikelytoberepresentativeofapopulation’struemean.
•
AttheconclusionofPart2oftheactivity,studentswillunderstandwhytheequationforSEM
givesanestimateofthestandarderrorofthemeanbasedonasample’ssizeandstandard
deviation.TheywillalsobeabletousetheequationtocalculateSEM,95%confidenceintervals,
andusethe95%CItogenerateerrorbarsonabargraph.Whilethisactivityfocusesonthe
effectofsamplesize(assamplesizeincreases,SEMdecreases),studentsshouldbeableto
predictfromtheequationthatthereisadirectrelationshipbetweenthestandarddeviationand
SEM.
•
Remindstudentsthatonpage1oftheClickandLearn“SamplingfromaNormallyDistributed
Population,”clicking“resample”issimulatingcollectinganewrandomlyselectedsetof
measurementsfromthepopulation.Therefore,samplemeansandstandarddeviationswill
likelybedifferentfromstudenttostudent.Thiswillnotaffectthefinaloutcomeoftheactivity.
Studentsshouldalsoberemindedthatonpage2oftheClickandLearn“StandardErrorofthe
Mean,”“resample”representsrepeatingthesamplecollection500times,andeachsample
consistsofanumberofmeasurementsequaltothesamplesize.Thismeansthatforasample
sizeof100,thesimulationtook50,000measurements.
•
InPart2,theteachermayneedtopointoutanddiscussthedifferencebetweensamplemean
andstandarddeviationandthemeanandstandarddeviationof500means.Samplemeanand
standarddeviationisdescribingthedatainthetopgraph,whilethemeanandstandard
deviationof500meansisdescribingthedatainthebottomgraph.
SUGGESTEDPROCEDURE
Dependingontheskilllevelofthestudentsinthecourse,thisactivitycanbedoneindependentlyor
guidedbytheteacher.Theprocedurebelowisforaguidedprocess,duringwhichtheinstructor
checksforstudentunderstandingatkeypointsintheactivity.
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Introduction
1.Showstudentsthegraphbelowandaskthemtointerpretit.Askthemwhattheerrorbarsmean.
Whileitdependsonanindividualstudent’spriorlearning,moststudentswillnotbeabletoexplain
whattheerrorbarsmean.Ifthisisthecase,askthemtodescribetheerrorbar.Guidestudentsto
observationssuchas
• Theerrorbarfordarkdoesnotoverlaptheerrorbarforlight.
• Thedarkerrorbarislongerthanthelighterrorbar.
• Thelengthsoftheerrorbaraboveandbelowthetopofthebarareequal.
Figure1.MeanLengthofCroftonSeedlingsafterOneWeekintheDarkorintheLight.(FromUsingBioInteractive
ResourcestoTeachMathematicsandStatisticsinBiologyhttp://www.hhmi.org/biointeractive/teacher-guidemath-and-statistics)
2.InstructstudentstocompletethePre-assessmentQuestion(whichcouldbecollectedonnotecards
asaformativeassessment).TheninstructstudentstoaccesstheClickandLearnat
http://www.hhmi.org/biointeractive/sampling-and-normal-distributionandcompleteitems2through5.
Itisimportantatthispointtoensurethatstudentsunderstandthatanindividualmeasurementispart
ofasampletakenfromalargerpopulation.Pointouttostudentsthecharacteristicsofanormally
distributedpopulationbyreferencingtheredlineonthegraphandthatnumberofindividualmass
measurementsarerepresentedbythebarsinthehistogram.Note:ThispartalongwithPart1items6
and7couldbeassignedtostudentsforhomeworkpriortocompletingtherestofPart1oftheactivityin
class.
PART1:SAMPLINGFROMANORMALLYDISTRIBUTEDPOPULATION
1. Studentsworkthroughthetaskandcompleteitems6and7toexplorehowmodifyingthestandard
deviationaffectsthedistributionofmeasurementsinthepopulation.Itshouldbepointedoutto
studentsthatchangingtheparameterschangesthesimulationprogram.Inarealdataset,the
standarddeviationisdeterminedbytheactualmeasurementsinthepopulationorsample.
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2.Havestudentsreadthesummarydescriptionofstandarddeviationanddiscussanyquestionsthey
haveaboutstandarddeviationandnormaldistribution.
3.IntherestofPart1,studentsexploretheeffectofsamplesizeonthesamplemeancomparedtothe
truemeanofthepopulation.Remindstudentsthattheyaresettingparametersfortheprogramrunning
thesimulation(populationmean=50kgandstandarddeviation=10kg).
4.Item8canbeusedasaformativeassessmenttomonitorstudentunderstandingofstandard
deviation.Acorrectstudentresponsewouldbe:“Forthispopulation,68%ofthemassesshouldbe
between40and60kg(1standarddeviation),while95%ofthemassesshouldfallbetween30and70kg
(2standarddeviations).”
5.Studentscompleteitems9and10.Aftercompletingthistask,studentsshouldrecognizethata
samplesizeof1000ismorelikelytogiveyouasamplemeanthatreflectsthetruemeanofthe
populationbecausethelargernumberofmeasurementswillreflectthenormaldistributionofthe
population.Theyshouldalsorecognizethatcollectingmeasurementsfromasampleof1000individuals
couldbetime-consuming,expensive,orsimplynotpractical.
6.Studentscomplete“Selectingtheappropriatesamplesize”bycompletingthetaskanditems11
through13.Providestudentswiththe“DistributionofMeans”grid.Thistaskcanbecompletedinpairs
orasmallgroup.Thereshouldbeatleastonegraphintheclassforeachsamplesizeinthesimulation
(4,9,16,25,100,400,1000).Laminatingthegridswillallowthemtobereusedbyseveralclasses.
Discussitem13asawholeclass.Askstudentstojustifytheiranswertothequestionwithevidencefrom
thegraphs.Studentstypicallyselect100asanappropriatesamplesize.Anexampleofthedata
generatedfromthistaskisshownbelow.
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PART2:STANDARDERROROFTHEMEAN
1. Items1through8canbecompletedasahomeworkassignment.Studentsusethenextpageinthe
ClickandLearntoextendtheirexplorationoftheeffectofsamplesizeonthedistributionofsample
means.InPart2,resamplingwillgenerateahistogramofthemeansof500samplesshowinga
normaldistribution.Studentsshouldcometotheconclusionthatwhilethesamplesizedoesnot
affectthemeanof500means,itdoesaffectthestandarddeviationofthemeans.Forsmaller
samplesizes,thesamplemeancouldbequitedifferentfromthepopulationmean,andthisis
reflectedinthelargerstandarddeviationofthemeans.Thisshouldreinforcetheconclusionthey
cametoattheendofPart1thatlargersamplesizeswillprovideabetterrepresentationofthe
populationfromwhichthesamplewasdrawn.
$
2. Item9introducesstudentstotheequationforstandarderrorofthemean,SE𝑥 = .Items10and
%
11havestudentscalculatetheSEusingtheequation.Theyarethenaskedtocomparethe
empiricallymeasuredSEM(standarddeviationof500means)tothecalculatedestimationofthe
SEM.StudentsshouldberemindedthatthemathematicalformulaforSEMallowsthemtoestimate
therealSEMgivenasmallsample,whilerepeatingsamplesmanytimesallowsthemtoempirically
measuretheactualSEM.Studentsshouldfindthat,withtheexceptionofverysmallsamplesizes,
$
usingtheequationSE𝑥 = isareasonablyaccuratewaytoestimatetheStandardErrorofthe
%
Meanfromthestandarddeviationofasampleandthesamplesize.Studentscanbenefitfroma
discussionregardinghowtheequationmodelsthedistributionofmeansthattheyobserved.
Encouragestudentstodiscusswhysamplesizeisinthedenominator.Asseeninthegraphbelow,
standarddeviationofthemeansdecreasesassamplesizeincreases.Theyshouldrecallfromtheir
observationsduringPart1oftheactivitythatlargersamplesizesaremorelikelytobeatruer
reflectionofthepopulationfromwhichthemeasurementsaredrawnandthatverysmallsample
sizesoftenresultininaccuraterepresentationsofthepopulation.
Itishelpfultoreferstudentsbacktothedistributionof10meanstheyplottedinPart1ofthe
activity.EmphasizethattheSEequationallowsonetoestimatethespreadofthemeansthatwould
beexpectedfrommanysamplesdrawnfromthesamepopulationfromthestandarddeviationand
samplesizeofasingleobservedsample.
Standard Deviation of Mean
6
5
4
3
2
1
0
0
100
200
300
400
500
Sample Size (n)
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TheeffectofsamplestandarddeviationontheSEMisnotexploredinthesimulation.Thiswasdoneto
avoidthemisconceptionthatsamplesizeaffectssamplestandarddeviation.Itstillmaybehelpfulto
discusstheeffectstandarddeviationofthesamplehasonthestandarderrorofthemean.Thestandard
deviationofthesampleisinthenumeratoroftheequationbecauseamorevariedpopulation(larger
samplestandarddeviation)willincreasethelikelihoodthatthesamplemeasurementswillnotbeagood
representationofthepopulationfromwhichtheyaretaken.
3. Havestudentsreadthesummaryandthencompleteitems12and13tolearnhowtousethe
standarderrorofthemeantogenerate95%ConfidenceIntervalerrorbars.Readingthesummary
willshowstudentshowtointerprettheseerrorbars.
PART3:APPLYWHATYOUHAVELEARNED
Thedatapresentedisauthenticdatacollectedbystudentsconductinganexperimenttotesttheeffect
ofpectinaseandcellulaseonturningapplesauceintoapplejuice.
Thispartoftheactivitycanbegivenasanassessmenttodeterminestudents’levelofunderstandingof
theconceptofstandarderrorofthemeanandhowtousethestatistictoanalyzetheexperimentaldata.
RECOMMENDEDFOLLOW-UPACTIVITIES
EvolutioninAction:DataAnalysis(http://www.hhmi.org/biointeractive/evolution-action-data-analysis)
RosemaryandPeterGranthaveprovidedmorphologicalmeasurements,includingwinglength,body
mass,andbeakdepth,takenfromasampleof100mediumgroundfinches(Geospizafortis)livingonthe
islandofDaphneMajorintheGalápagosarchipelago.Thecompletedatasetisavailableinthe
accompanyingExcelspreadsheet.
Inoneactivity,entitled“EvolutioninAction:GraphingandStatistics,”studentsareguidedthroughthe
analysisofthissampleoftheGrants’databyconstructingandinterpretinggraphs,andcalculatingand
interpretingdescriptivestatistics.Thesecondactivity,“EvolutioninAction:StatisticalAnalysis,”provides
anexampleofhowthedatasetcanbeanalyzedusingstatisticaltests,inparticulartheStudent’st-test
forindependentsamples,tohelpdrawconclusionsabouttheroleofnaturalselectiononmorphological
traitsbasedonmeasurements.
LizardEvolutionVirtualLab(http://www.hhmi.org/biointeractive/lizard-evolution-virtual-lab)
IntheLizardEvolutionVirtualLab,studentsexploretheevolutionoftheanolelizardsintheCaribbean
bycollectingandanalyzingtheirowndata.
Thevirtuallabincludesfourmodulesthatinvestigatedifferentconceptsinevolutionarybiology,
includingadaptation,convergentevolution,phylogeneticanalysis,reproductiveisolation,and
speciation.Eachmoduleinvolvesdatacollection,calculations,analysis,andansweringquestions.
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