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www.BioInteractive.org Educator Materials Click and Learn Sampling and Normal Distribution Revised October 2017 Page 1 of 8 OVERVIEW Normal distribution, sometimes called the bell curve, is a common way to describe a continuous distribution in probability theory and statistics. In the natural sciences, scientists typically assume that a series of measurements of a population will be normally distributed, even though the actual distribution may be unknown. But even if you assume that measurements of a population should be normally distributed, a sample taken from that population will not necessarily be normally distributed. Why is that? In this Click and Learn, you will explore what sample distribution looks like when samples are taken from an idealized population of a defined mean and standard deviation. Students will explore how standard deviation affects the distribution of measurements in a population. Next, they will explore how sample size affects the distribution of measurements and therefore the sample mean. Through this exploration, students will develop an understanding of how sample size affects the distribution of sample means drawn from the same population and how this phenomenon is modeled in an equation for calculating the standard error of the mean. KEY CONCEPTS AND LEARNING OBJECTIVES • The appearance of a histogram of measurements in a sample depends on the population from which the sample came. • The appearance of the histogram also depends on the sample size. • Small samples taken from a normally distributed population may not appear to be normally distributed. Larger samples start to approximate a normal distribution. • When a population is sampled repeatedly, a mean can be calculated for each sample, to obtain many different means. If those means are plotted as a histogram, they will be approximately normally distributed. • The standard deviation of such a distribution of means is called the standard error of the mean. Students will be able to • explain that standard deviation is a measure of the variation of the spread of the data around the mean. • explain that larger sample sizes are desirable when collecting data about a population because they are more likely to reflect the distribution of measurements in a population. • calculate Standard Error of the Mean ( #, , but also commonly referred to as SE, or SEM), using the equation # = ’ ( . • explain that # of the mean is a measure of the reliability of the mean of a sample as a reflection of the mean of the population from which the sample was drawn.

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Page 1: Click and Learn Sampling and Normal Distribution Educator ... · Normal distribution, sometimes called the bell curve, is a common way to describe a continuous distribution in probability

www.BioInteractive.org

Educator Materials

Click and Learn Sampling and Normal Distribution

RevisedOctober2017Page1of8

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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Educator Materials

Click and Learn Sampling and Normal Distribution

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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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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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Educator Materials

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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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Educator Materials

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RevisedOctober2017Page7of8

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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RevisedOctober2017Page8of8


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