心理統(tǒng)計學(xué)英文版課件L2 Frequency Distributions and Probability

1FrequencyDistributionsandProbability2FrequencyDistributionsAftercollectingdata,thefirsttaskforaresearcheristoorganizeandsimplifythedatasothatitispossibletogetageneraloverviewoftheresults.Thisisthegoalofdescriptivestatisticaltechniques.Onemethodforsimplifyingandorganizingdataistoconstructafrequencydistribution.3FrequencyDistributions(cont.)Afrequencydistributionisanorganizedtabulationshowingexactlyhowmanyindividualsarelocatedineachcategoryonthescaleofmeasurement.Afrequencydistributionpresentsanorganizedpictureoftheentiresetofscores,anditshowswhereeachindividualislocatedrelativetoothersinthedistribution.4FrequencyDistributionTablesAfrequencydistributiontableconsistsofatleasttwocolumns-onelistingcategoriesonthescaleofmeasurement(X)andanotherforfrequency(f).IntheXcolumn,valuesarelistedfromthehighesttolowest,withoutskippingany.Forthefrequencycolumn,talliesaredeterminedforeachvalue(howofteneachXvalueoccursinthedataset).ThesetalliesarethefrequenciesforeachXvalue.ThesumofthefrequenciesshouldequalN.5FrequencyDistributionTables(cont.)Athirdcolumncanbeusedfortheproportion(p)foreachcategory:p=f/N.Thesumofthepcolumnshouldequal1.00.AfourthcolumncandisplaythepercentageofthedistributioncorrespondingtoeachXvalue.Thepercentageisfoundbymultiplyingpby100.Thesumofthepercentagecolumnis100%.6RegularFrequencyDistributionWhenafrequencydistributiontablelistsalloftheindividualcategories(Xvalues)itiscalledaregularfrequencydistribution.7GroupedFrequencyDistributionSometimes,however,asetofscorescoversawiderangeofvalues.Inthesesituations,alistofalltheXvalueswouldbequitelong-toolongtobea“simple”presentationofthedata.Toremedythissituation,agroupedfrequencydistributiontableisused.8GroupedFrequencyDistribution(cont.)Inagroupedtable,theXcolumnlistsgroupsofscores,calledclassintervals,ratherthanindividualvalues.Theseintervalsallhavethesamewidth,usuallyasimplenumbersuchas2,5,10,andsoon.Eachintervalbeginswithavaluethatisamultipleoftheintervalwidth.Theintervalwidthisselectedsothatthetablewillhaveapproximatelytenintervals.9FrequencyDistributionGraphsInafrequencydistributiongraph,thescorecategories(Xvalues)arelistedontheXaxisandthefrequenciesarelistedontheYaxis.Whenthescorecategoriesconsistofnumericalscoresfromanintervalorratioscale,thegraphshouldbeeitherahistogramorapolygon.10HistogramsInahistogram,abariscenteredaboveeachscore(orclassinterval)sothattheheightofthebarcorrespondstothefrequencyandthewidthextendstothereallimits,sothatadjacentbarstouch.12PolygonsInapolygon,adotiscenteredaboveeachscoresothattheheightofthedotcorrespondstothefrequency.Thedotsarethenconnectedbystraightlines.Anadditionallineisdrawnateachendtobringthegraphbacktoazerofrequency.14BargraphsWhenthescorecategories(Xvalues)aremeasurementsfromanominaloranordinalscale,thegraphshouldbeabargraph.Abargraphisjustlikeahistogramexceptthatgapsorspacesareleftbetweenadjacentbars.16RelativefrequencyManypopulationsaresolargethatitisimpossibletoknowtheexactnumberofindividuals(frequency)foranyspecificcategory.Inthesesituations,populationdistributionscanbeshownusingrelativefrequencyinsteadoftheabsolutenumberofindividualsforeachcategory.18SmoothcurveIfthescoresinthepopulationaremeasuredonanintervalorratioscale,itiscustomarytopresentthedistributionasasmoothcurveratherthanajaggedhistogramorpolygon.Thesmoothcurveemphasizesthefactthatthedistributionisnotshowingtheexactfrequencyforeachcategory.20FrequencydistributiongraphsFrequencydistributiongraphsareusefulbecausetheyshowtheentiresetofscores.Ataglance,youcandeterminethehighestscore,thelowestscore,andwherethescoresarecentered.Thegraphalsoshowswhetherthescoresareclusteredtogetherorscatteredoverawiderange.21ShapeAgraphshowstheshapeofthedistribution.Adistributionissymmetricaliftheleftsideofthegraphis(roughly)amirrorimageoftherightside.Oneexampleofasymmetricaldistributionisthebell-shapednormaldistribution.Ontheotherhand,distributionsareskewedwhenscorespileupononesideofthedistribution,leavinga"tail"ofafewextremevaluesontheotherside.22PositivelyandNegatively

SkewedDistributionsInapositivelyskeweddistribution,thescorestendtopileupontheleftsideofthedistributionwiththetailtaperingofftotheright.Inanegativelyskeweddistribution,thescorestendtopileupontherightsideandthetailpointstotheleft.24Percentiles,PercentileRanks,

andInterpolationTherelativelocationofindividualscoreswithinadistributioncanbedescribedbypercentilesandpercentileranks.ThepercentilerankforaparticularXvalueisthepercentageofindividualswithscoresequaltoorlessthanthatXvalue.WhenanXvalueisdescribedbyitsrank,itiscalledapercentile.25Percentiles,PercentileRanks,

andInterpolation(cont.)Tofindpercentilesandpercentileranks,twonewcolumnsareplacedinthefrequencydistributiontable:Oneisforcumulativefrequency(cf)andtheotherisforcumulativepercentage(c%).Eachcumulativepercentageidentifiesthepercentilerankfortheupperreallimitofthecorrespondingscoreorclassinterval.Whenscoresorpercentagesdonotcorrespondtoupperreallimitsorcumulativepercentages,youmustuseinterpolationtodeterminethecorrespondingranksandpercentiles.Interpolationisamathematicalprocessbasedontheassumptionthatthescoresandthepercentageschangeinaregular,linearfashionasyoumovethroughanintervalfromoneendtotheother.26InterpolationWhenscoresorpercentagesdonotcorrespondtoupperreallimitsorcumulativepercentages,youmustuseinterpolationtodeterminethecorrespondingranksandpercentiles.Interpolationisamathematicalprocessbasedontheassumptionthatthescoresandthepercentageschangeinaregular,linearfashionasyoumovethroughanintervalfromoneendtotheother.28Stem-and-LeafDisplaysAstem-and-leafdisplayprovidesaveryefficientmethodforobtaininganddisplayingafrequencydistribution.Eachscoreisdividedintoastemconsistingofthefirstdigitordigits,andaleafconsistingofthefinaldigit.Finally,yougothroughthelistofscores,oneatatime,andwritetheleafforeachscorebesideitsstem.Theresultingdisplayprovidesanorganizedpictureoftheentiredistribution.Thenumberofleafsbesideeachstemcorrespondstothefrequency,andtheindividualleafsidentifytheindividualscores.30ProbabilityProbabilityisamethodformeasuringandquantifyingthelikelihoodofobtainingaspecificsamplefromaspecificpopulation.Wedefineprobabilityasafractionoraproportion.Theprobabilityofanyspecificoutcomeisdeterminedbyaratiocomparingthefrequencyofoccurrenceforthatoutcomerelativetothetotalnumberofpossibleoutcomes. 32Probability(cont.)Wheneverthescoresinapopulationarevariableitisimpossibletopredictwithperfectaccuracyexactlywhichscoreorscoreswillbeobtainedwhenyoutakeasamplefromthepopulation.Inthissituation,researchersrelyonprobabilitytodeterminetherelativelikelihoodforspecificsamples.Thus,althoughyoumaynotbeabletopredictexactlywhichvalue(s)willbeobtainedforasample,itispossibletodeterminewhichoutcomeshavehighprobabilityandwhichhavelowprobability.33Probability(cont.)Probabilityisdeterminedbyafractionorproportion.Whenapopulationofscoresisrepresentedbyafrequencydistribution,probabilitiescanbedefinedbyproportionsofthedistribution.Ingraphs,probabilitycanbedefinedasaproportionofareaunderthecurve.34ProbabilityandtheNormalDistributionIfaverticallineisdrawnthroughanormaldistribution,severalthingsoccur. 1.Theexactlocationofthelinecanbe specifiedbyaz-score. 2.Thelinedividesthedistribution into twosections.Thelargersectionis calledthebodyandthesmallersection iscalledthetail.35ProbabilityandtheNormalDistribution(cont.)Theunitnormaltablelistsseveraldifferentproportionscorrespondingtoeachz-scorelocation.ColumnAofthetablelistsz-scorevalues.Foreachz-scorelocation,columnsBandClisttheproportionsinthebodyandtail,respectively.Finally,columnDliststheproportionbetweenthemeanandthez-scorelocation.Becauseprobabilityisequivalenttoproportion,thetablevaluescanalsobeusedtodetermineprobabilities.37ProbabilityandtheNormalDistribution(cont.)Tofindtheprobabilitycorrespondingtoaparticularscore(Xvalue),youfirsttransformthescoreintoaz-score,thenlookupthez-scoreinthetableandreadacrosstherowtofindtheappropriateproportion/probability.Tofindthescore(Xvalue)correspondingtoaparticularproportion,youfirstlookuptheproportioninthetable,readacrosstherowtofindthecorrespondingz-score,andthentransformthez-scoreintoanXvalue.38PercentilesandPercentileRanksThepercentilerankforaspecificXvalueisthepercentageofindividualswithscoresatorbelowthatvalue.Whenascoreisreferredtobyitsrank,thescoreiscalledapercentile.Thepercentilerankforascoreinanormaldistributionissimplytheproportiontotheleftofthescore.40ProbabilityandtheBinomialDistributionBinomialdistributionsareformedbyaseriesofobservations(forexample,100cointosses)forwhichthereareexactlytwopossibleoutcomes(headsandtails).ThetwooutcomesareidentifiedasAandB,withprobabilitiesofp(A)=pandp(B)=q.ThedistributionshowstheprobabilityforeachvalueofX,whereXisthenumberofoccurrencesofAinaseriesofnobservations.41ProbabilityandtheBinomialDistribution(cont.)Whenpnandqnarebothgreaterthan10,thebinomialdistributioniscloselyapproximatedbyanormaldistributi