Lec Notes 110 Stats Sample · Mean of the distribution of differences between means..... 144

Lecture1–Displayingdata..........................................................................................................................................................12

Statistics........................................................................................................................................................................13

Statisticalmethod...............................................................................................................................................................13

Variables.......................................................................................................................................................................13

Value.............................................................................................................................................................................15

Score.............................................................................................................................................................................15

TypeofResearch...........................................................................................................................................................15

LevelofMeasurement...................................................................................................................................................15

Numeric/Quantitativevariables.........................................................................................................................................15

Ordinal/Rank-ordervariables(inorderonly).....................................................................................................................................15

Equalintervalvariables......................................................................................................................................................................17

Categorical/Nominalvariables...........................................................................................................................................17

Frequencytable............................................................................................................................................................17

Makingafrequencytable...................................................................................................................................................17

Groupedfrequencytables..................................................................................................................................................19

Histograms....................................................................................................................................................................19

Frequencypolygons......................................................................................................................................................20

Shapesofdistributions..................................................................................................................................................20

Numberofpeaks...........................................................................................................................................................20

Isitroughlysymmetrical?.............................................................................................................................................22

Kurtosis.........................................................................................................................................................................22

Discretevariable...........................................................................................................................................................22

Continuousvariable......................................................................................................................................................22

Flooreffect....................................................................................................................................................................22

Ceilingeffect.................................................................................................................................................................22

Normalcurve................................................................................................................................................................22

Lecture2–Centraltendencyandvariability................................................................................................................................24

Centraltendency...........................................................................................................................................................25

Mean...................................................................................................................................................................................25

Importantconcepts............................................................................................................................................................................25

Calculatingthemean .......................................................................................................................................................27

Mode..................................................................................................................................................................................27

Median................................................................................................................................................................................27

Whichcentraltendencymeasure..................................................................................................................................28

Variability.....................................................................................................................................................................30

Measuresofvariability.......................................................................................................................................................30

Range..................................................................................................................................................................................30

Interquartilerange(IQR)....................................................................................................................................................30

Variance.............................................................................................................................................................................30

Calculatingthevariance ........................................................................................................................30

Example:Numberoftherapysessions..............................................................................................................................31

Importantfeaturesofthevariance...................................................................................................................................31

SumofSquares(SS)=å(X-M)2.........................................................................................................................................32

Thestandarddeviation(Measuresofvariability) ..............................................................................32

SDformula ...............................................................................................32

Example:Numberoftherapysessions...............................................................................................................................32

Outlier...........................................................................................................................................................................32

Computationalformula.................................................................................................................................................32

Definitionalformula......................................................................................................................................................33

Lecture3–Standardisedscores:Zscores....................................................................................................................................34

Someexamplestoconsider...........................................................................................................................................35

Zscores.........................................................................................................................................................................35

DistributionofZscores.......................................................................................................................................................35

CalculatingaZscorefromarawscore ..........................................................................................................36

Example..............................................................................................................................................................................36

InterpretingZscores...........................................................................................................................................................36

NXM =

Example..............................................................................................................................................................................36

Example..............................................................................................................................................................................37

ImplicationsofZscores......................................................................................................................................................37

Example:Comparingscoresfromdifferentdistributions...................................................................................................37

Therelativeachievementof3friends................................................................................................................................39

CalculatingarawscorefromaZscore(fromZscoretorawscore) MZSDX += )x( ....................................................39

Example:IQdata................................................................................................................................................................39

ImportantfeaturesofZscores...........................................................................................................................................41

Whenthedistributionisnormal,Zscorestellusevenmore.............................................................................................41

Thebasisofpercentagesonanormaldistribution............................................................................................................41

Percentile............................................................................................................................................................................41

Lecture4–Correlation................................................................................................................................................................43

TypesofVariablesinResearch......................................................................................................................................44

DependentVariable(DV)....................................................................................................................................................44

IndependentVariable(IV)..................................................................................................................................................44

Examples.............................................................................................................................................................................44

MajorTypesofResearchDesign....................................................................................................................................44

Descriptioninanobservationalstudyoftwocontinuousvariables...............................................................................46

Graphingpairsofvariables:Scatterplot.............................................................................................................................46

Drawingascatterplot.........................................................................................................................................................................46

ConstructingaScatterplot.................................................................................................................................................................46

Patternsoflinearrelationship...........................................................................................................................................................47

Patternsofrelationship.....................................................................................................................................................................49

Quantifyingtherelationship:Correlation......................................................................................................................51

CalculatingthecorrelationcoefficientrNZZ

r Yxå= ......................................................................................................51

Crossproducts(SZXZY)........................................................................................................................................................................52

Makingsenseofr:proportionatereductioninerrororCoefficientofdetermination:r2tellsustheproportionofvariability.......56

Lecture5–Inferentialstatistics...................................................................................................................................................58

IntroductiontoInferentialStatistics..............................................................................................................................59

Thenormalcurve..........................................................................................................................................................59

Background.........................................................................................................................................................................59

Thenormaldistribution:areasunderthenormalcurve....................................................................................................59

1SDandthenormaldistribution........................................................................................................................................59

2SDandthenormaldistribution........................................................................................................................................60

Findingpercentagesusinganormalcurvetable................................................................................................................60

Tipsforusinganormalcurvetable....................................................................................................................................62

IQscoresexample1...........................................................................................................................................................................62

IQscoresexample2...........................................................................................................................................................................64

Findingrawscoresfrompercentages.................................................................................................................................64

Probability....................................................................................................................................................................66

Calculatingprobability ...................................................................................66

Expectedrelativefrequency...............................................................................................................................................67

Probabilityandexpectation................................................................................................................................................67

Zscoresandprobability......................................................................................................................................................67

Samplesandpopulations..............................................................................................................................................67

Methodsofsampling..........................................................................................................................................................69

Populationparametersandsamplestatistics....................................................................................................................69

Lecture6–Hypothesistesting.....................................................................................................................................................70

Errorsinhypothesistesting...........................................................................................................................................71

Example:brainaffectedbyradiation.............................................................................................................................71

Twopossibilities.................................................................................................................................................................71

Statisticalsignificance:The‘magical’p<.05..................................................................................................................73

Interpretationissues.....................................................................................................................................................73

Hypothesistesting.........................................................................................................................................................73

Theprocessofhypothesistesting..................................................................................................................................73

Step1:Formulatingresearchandnullhypotheses............................................................................................................73

Step2:Identifyingthecomparisondistribution.................................................................................................................74

Step3:Determiningthecut-offscore................................................................................................................................74

Step4:Wheredoesyoursamplescoresitonthecomparisondistribution?.....................................................................76

Step5:Decisiontime:Shouldthenullhypothesisberejected?........................................................................................76

Theimplicationsofyourdecision..................................................................................................................................76

One-tailedandtwo-tailedhypothesistests...................................................................................................................76

Directionalhypotheses..................................................................................................................................................76

Two-tailedtests............................................................................................................................................................78

Cut-offpointsfortwo-tailedtests.................................................................................................................................78

Thenormalcurve:One-andtwo-tailedtests.................................................................................................................78

DeterminingCut-offPointswithTwo-TailedTests.........................................................................................................78

Comparisonofoneandtwo-tailedtests........................................................................................................................80

Summarysofar…..........................................................................................................................................................80

Anexample...................................................................................................................................................................82

Errorsinhypothesistesting:Terminology......................................................................................................................82

Errorswhenresultissignificant:Type1error................................................................................................................83

Errorswhenresultisnotsignificant:Type2error..........................................................................................................84

Errorsinhypothesistesting...........................................................................................................................................84

Errorsinhypothesistesting:Table.................................................................................................................................84

Correctdecision...........................................................................................................................................................................85


TypeIerrors:whenH0isactuallytrue...........................................................................................................................85


TypeIIerrors:whenH1isactuallytrueandH0isfalse...................................................................................................85


RelationshipbetweenTypeIandTypeIIerrors.............................................................................................................87

Power...........................................................................................................................................................................87

JuryTrialExampleofErrors...........................................................................................................................................88

Lecture7–Thedistributionofmeans..........................................................................................................................................89

Distributionofmeans:Thelogic....................................................................................................................................90

Hypothesistestingwithsamples...................................................................................................................................90

Samplesfrompopulations.............................................................................................................................................90

Samplingvariability.......................................................................................................................................................92

Minimisingerror...........................................................................................................................................................92

Sowhatdistributiondoweneed?.................................................................................................................................92

Distributionsofmeans..................................................................................................................................................92

Whydoesthisdistributionnormalise?..........................................................................................................................94

Characteristicsofthedistributionofmeans:#1.............................................................................................................94


.........................................................................................................................................................96

Measuringvariabilityinsamplemeans..............................................................................................................................96

StandardErroroftheMean................................................................................................................................................98

IncreaseN,decreaseError.................................................................................................................................................98


Threetypesofdistributions:Populations.....................................................................................................................100

Threetypesofdistributions:Samples...........................................................................................................................100

Threetypesofdistributions:Distributionsofmeans....................................................................................................100

Threetypesofdistributions..........................................................................................................................................100

ComparisonofThreeTypesofDistributions.................................................................................................................101

Hypothesistestingwithsamples..................................................................................................................................101

Hypothesistestingagainstaknownpopulation............................................................................................................101

Example1....................................................................................................................................................................101

BacktoourNuclearPowerPlantTown.........................................................................................................................103

Step4:Wheredoesyoursamplemeansit?.....................................................................................................................103

(thisscreenwillbeintheexam).......................................................................................................................................105

Estimationandconfidenceintervals.............................................................................................................................105

Ourexample.....................................................................................................................................................................107

95%confidenceintervalsofsample.................................................................................................................................107

Usingconfidenceintervalstotesthypotheses.................................................................................................................107

OurClassExample............................................................................................................................................................109

Howconfidentarewe......................................................................................................................................................110

DidwemakeanError?.....................................................................................................................................................110

Lecture8–ttests:singlesampleanddependentmeans...........................................................................................................112

Example#1:“Stopstress”.............................................................................................................................................113

Ztestsàttests:ageneralintroduction.......................................................................................................................113

Estimatingthepopulationstandarddeviationfromthesampledata.............................................................................115

WhyN-1?Themysteryof“DegreesofFreedom”............................................................................................................115

Estimatingthestandarddeviationofthecomparisondistribution..................................................................................115

Zformulaàtformula(one-sampletests).......................................................................................................................115

ShortCuttogetSM..........................................................................................................................................................117

Theonesamplettest...................................................................................................................................................117

Thecomparisondistribution............................................................................................................................................117

Thetdistributionvs.normaldistribution.........................................................................................................................119

Thetdistributionvs.normaldistributionrecut-offscores..............................................................................................119

Tipsforusingthettable(A-2,p.675)..............................................................................................................................119

WorkingthroughExample#1:“Stopstress”..................................................................................................................122

1.Statingthehypotheses.................................................................................................................................................122

2.Determiningthecharacteristicsofthecomparisondistribution..................................................................................122

3.DeterminethecriticalvaluetorejectH0......................................................................................................................123

4.Determinethetvaluei.e.,determineyoursample’sscoreonthecomparisondistribution(thetdistribution)........123

5.Comparethescorestomakeadecision.......................................................................................................................123

Anotherwaytouseournewtdistribution...................................................................................................................123

Thettestfordependentmeans(repeatedmeasures)....................................................................................................124

Differencescores..............................................................................................................................................................124

Singlesampletodependentmeasuresttest................................................................................................................125

1.Statingthehypotheses.................................................................................................................................................125

2.Determiningthecharacteristicsofthecomparisondistribution..................................................................................125

3.DeterminethecriticalvaluetorejectH0......................................................................................................................127

4.Determinethetvaluei.e.,determineyoursample’sscoreonthecomparisondistribution(thetdistribution)........127

5.Comparethescorestomakeadecision.......................................................................................................................128

ConfidenceIntervalsaroundtheMean.........................................................................................................................128

UsingConfidenceIntervalstoTestHypothesisofMeanDifference................................................................................129

APAStyleWrite-Up......................................................................................................................................................129

FullAPAWrite-Up.............................................................................................................................................................129

Assumptionsofthettest.............................................................................................................................................129

Situationswhereweuseattestfordependentmeans................................................................................................131

Example#3:Neighbourhoodattachment.....................................................................................................................131

Step1................................................................................................................................................................................131

Step2................................................................................................................................................................................131

Step3................................................................................................................................................................................133

Step4................................................................................................................................................................................133

Step5................................................................................................................................................................................133

Example#4:NeighbourhoodAttachment;RepeatedMeasuresDesign........................................................................133

Lecture9–ttestforindependentmeans..................................................................................................................................136

Thettestforindependentmeans................................................................................................................................137

Thelogicunderlyingtheindependentmeansttest......................................................................................................137

WorkingourwaytoSdifferenceDistributionofsamplemeans.............................................................................................137

Distributionofdifferencesbetweenmeans.....................................................................................................................138

Identifyingthedistribution...............................................................................................................................................140

z-testsvst-test..................................................................................................................................................................140

VarianceofComparisonDistribution...............................................................................................................................140

Identifyingthedistribution...............................................................................................................................................142

KeyDistributionsinHypothesisTesting...........................................................................................................................142

ComparisonDistributions.................................................................................................................................................142

Stepsintheprocessofcalculatingindependentgroupsttest.......................................................................................144

Example:Dyslexiaandcolouroverlays.........................................................................................................................144

Meanofthedistributionofdifferencesbetweenmeans.................................................................................................144

Estimatedpopulationvariancefrombothsamples..........................................................................................................146

Thepooledestimateofthepopulationvariance.............................................................................................................146

Weightingvarianceestimatesaccordingtodf.................................................................................................................146

Calculatingthevariancesofthetwodistributionsofmeans...........................................................................................148

Thedistributionofthedifferencesbetweenthemeans..................................................................................................148

Equalsamplesize..............................................................................................................................................................148

Theshapeofthedistributionofthedifferencesbetweenmeans...................................................................................148

Calculatingthetscorecorrespondingtoyoursamples...................................................................................................148

StepsforatTestforIndependentMeans........................................................................................................................149

DyslexiaandcolouroverlaysexampleStep1:Statehypotheses....................................................................................................149

Step2:Determinecharacteristicsofthecomparisondistribution..................................................................................................151

Step3:Determinethecut-offscore................................................................................................................................................151

Step4:Calculatethetscore(determinesamplescoreoncomparisondistribution)......................................................................152

Step5:DecisionregardingH0...........................................................................................................................................................152

APAstylewrite-up.......................................................................................................................................................152

Assumptionsofthettestforindependentmeans........................................................................................................153

Effectsizeinttests.......................................................................................................................................................154

Cohen’sd..........................................................................................................................................................................154

EtaSquaredη2..................................................................................................................................................................154

EasytoCalculate...............................................................................................................................................................154

EffectSizeandPower.......................................................................................................................................................156

Lecture10–Chi-squaretests.....................................................................................................................................................158

Statisticaloptions…......................................................................................................................................................159

Example:Attachmentstyles#1....................................................................................................................................159

Observedandexpectedfrequencies:Whatwehavevs.whatweexpect.......................................................................161

DeterminingExpectedFrequencies:Whenallcategoriesareequal................................................................................161

Chi-square(c2)testforgoodnessoffit.........................................................................................................................161

Expectedandobservedfrequencies.................................................................................................................................161

Calculatingthec2statistics...............................................................................................................................................163

Example:Attachmentstyles#1........................................................................................................................................163

Testingsignificance:c2distributions................................................................................................................................164

Example:Attachmentstyles#1........................................................................................................................................164

Reviewofstepsforcalculatingthechi-squarestatistic.................................................................................................164

Example:Attachmentstyles#2....................................................................................................................................164

c2distributions.............................................................................................................................................................166

Heavymetalpollutionandmentalhealthexample:.....................................................................................................166

Chi-square(c2)testforindependence..........................................................................................................................168

H0:independent(unrelated).............................................................................................................................................170

Example.......................................................................................................................................................................170

Contingencytable.............................................................................................................................................................170

Calculatingtheexpectedfrequencies..............................................................................................................................172

Calculatingthec2statistics...............................................................................................................................................172

Decision............................................................................................................................................................................174

Genderandreportedchildabuseexample:..................................................................................................................174

Assumptionsofc2tests................................................................................................................................................176

Effectsizeinc2tests(strengthofrelationshipinc2testsofindependence)...................................................................176

Chi-SquareTestsinResearchArticles...............................................................................................................................176

Lecture11–IntroductiontoQualitativeResearch.....................................................................................................................177

RelevanceofQualitativeResearch...............................................................................................................................178

FeaturesofQualitativeResearch..................................................................................................................................178

ParadigmsinSocialResearch........................................................................................................................................180

Importantconcept.......................................................................................................................................................180

PositivistParadigm...........................................................................................................................................................180

SocialConstructionistParadigm.......................................................................................................................................180

ParadigmsinSocialResearch...........................................................................................................................................182

Quantitativevs.QualitativeResearch..............................................................................................................................182

DeductiveReasoning........................................................................................................................................................................182

InductiveReasoning.........................................................................................................................................................................183

BeyondParadigmWars................................................................................................................................................183

ProcessofQualitativeResearch...................................................................................................................................185

TheoryinQualitativeResearch.....................................................................................................................................185

MoreaboutTheory...........................................................................................................................................................187

PrinciplesofResearchEthics........................................................................................................................................187

EthicsofQualitativeResearch..........................................................................................................................................189

HowtoActEthically..........................................................................................................................................................189

ChecklistforTakingEthicalIssuesintoAccount...............................................................................................................189

Summary......................................................................................................................................................................190

Lecture1–Displayingdata

• Variables

• Frequencytables

• Groupedfrequencytables

• Histograms

• Frequencypolygons

• Shapesofdistributions

Statistics Determiningiftrueornot.

Statisticalmethod Descriptive

- Information/dataissummarisedsoastobemoreeasilyunderstood

- describingdata:e.g.whatdoesthesampleof2000represent

Inferential

- Inferringsomething

- usedtodrawconclusionsaboutregularitiesinthedata

- Applyingtothepopulation.Whatpeopleingeneralmaylooklikefromthedatacollected?

- Probability

- Statisticallysignificance

Variables acharacteristicthatcanhavedifferentvalues

(e.g.,age,religion,reactiontime,anxietylevel)

somethingwhichisabletovaryortakedifferentvaluesisavariable

- acrosspeople:gender,height,weight

- withinpeople:height,weight,jobsatisfaction

workwithpsychologicalmaterials

- oftenusescoresonparticulartestsasvariables

- e.g.,extroversion-introversionscore

IndependentVariable(IV)

- variablecanchange

- notdependentonothervariable,worksindependent

- cause

Dependentvariable(DV)

- affectsbychangesintheIV

DVdependsonIV

Value Apossiblenumberorcategorythatascorecanhave(e.g.,1,2,3orfemale)

Justanumberorcategory.

Numberavariablecantake,e.g.0-10

Score Particularperson’svalueonavariable(e.g.,3,6orBuddhist)

TypeofResearch Observational/Naturalisticresearch

- can’ttalkaboutcause/effect

- cantalkaboutrelationship

- notacontrolenvironment

Experimental/Controlresearch

- controlenvironment

- isolateallothervariable

- manipulateIV

Levelof

Measurement

(Kindsofvariables)

Typesofunderlyingnumericalinformationprovidedbyameasure,suchasequal-interval,rank-order,and

nominal(categorical)

Numeric/Quantitative

variables

- variableswhosevaluesarenumbers(asopposedtoanominalvariable)

- generallyusenumberstodenotedifferentvaluesofavariable,e.g.68kg

- 2typesofnumericvariables

o Magnitude

o Equalityofintervals:hasmagnitudeandequalintervals

Ordinal/Rank-order

variables(inorder

only)

• numericvariableinwhichthevaluesareranked,suchasclassstandingorplacefinishedina

race.

• Numericvariableinwhichvaluescorrespondtotherelativepositionofthingsmeasured

• differenceinmagnitudeimplied,Nosetmagnitudebetweenthe2

• notequalintervalsbetweenranks

• grouphasorder,e.g.race,1st2nd3rd,stillacategory1st(10seconds)2nd(11secs)3rd(14secs),

magnitude

• ranks:e.g.,placeinclass,orderinahorserace

• e.g.GPAbetweenbeing2ndand3rdintheclasscouldbedifferentto8thand9th

Equalinterval

variables

• variableinwhichthenumbersstandforapproximatelyequalamountsofwhatisbeing

measured

• Numericvariableinwhichdifferencesbetweenvaluescorrespondtodifferencesinthe

underlyingthingbeingmeasured

• hasmagnitude

• differenceinmagnitudeimplied

• equalintervalsareassumed

• e.g.,timeelapsed,temperature,ages,GPA,weight,stresslevel

• e.g.GPA2.5and2.8meansaboutasmuchasthedifferencebetweenaGPAof3and3.3

Categorical/Nominal

variables

- Variablewithvaluesthatarenamesorcategories(thatis,theyarenamesratherthannumbers)

o Nominalcomesfromtheideathatitsvaluesarenames

o Variableinnameonly.category,numberdon’tnecessarymeananything,justacategory,

e.g.religion,gender(1=male,2=female)

o Doesn’tdenoteanythingabouttherelativemagnitude

Frequencytable - descriptivedata

- showshowfrequentlyeachvalueofavariableoccurs

- usefulforshowingoveralltendencies

- e.g.,stressratingsof30students:8,7,4,10,8,6,8,9,9,7,3,7,6,5,0,9,10,7,7,3,6,7,5,2,1,6,7,10,8,8

Makingafrequency

table

- makealiststartingwiththelowestscoreendingwiththehighest

o includevalueswhichdidn’toccur

- workthroughyourscoresandplaceaticknexttoeachvalueonyourlist

o numberofticks=numberofscores

- makeaneattablewithvaluesdownleftsideandthenumberofticksnexttothem

Groupedfrequency

tables

whentherearemanyvalues

- tablebecomesawkward

- useallvalueswithinaninterval

- useequalintervals

- recordfrequencyofallvaluesineachinterval

Histograms • atypeofbargraph

• awayofgraphingtheinformationinafrequencytable

• theheightofeachbaristhefrequencyofeachintervalinthetable

• canusethedatafromfrequencytableorgroupedfrequencytable

Frequency

polygons

• alinegraphoftheinformationinafrequencytable

• canusethedatafromfrequencytableorgroupedfrequencytable

• theheightofeachpointisthefrequencyofeachvalue(orinterval)

Shapesof

distributions

• frequencytables,histograms,frequencypolygonsdescribethedistribution

o howarescoresdistributedacrossarangeofvalues?

• commonpatternsandfeatures:

o isthereasinglepeak,two,none?

o isitroughlysymmetrical?

o howthickorheavyarethetails?

Numberofpeaks Modality:howmanypeaks?

Isthereasinglepeak,two,none?

• 1peak:unimodal

• 2peaks:bimodal

• >2peaks:multimodal

• withoutanyrealpeaks:rectangular

Strictlyspeaking,adistributionisbimodalormultimodalonlyifthepeaksareexactlyequal;however,

psychologistsusethetermsmoreinformallytodescribethegeneralshape.

Isitroughly

symmetrical?

ifnotsymmetrical,skeweddistribution

- positiveskew:iftailpointstoright

- negativeskew:iftailpointstoleft

Kurtosis

(width)

Howthickorheavyarethetails?

Needtocomparewiththe‘normal’distribution,thisqualityiscalledkurtosis

a) Normal

b) Leptokurtic(Peaked)

o tailsarethickerorheavierthannormalcurve

o moreeasilyrecognisedbytopofcurvebeingmorepeakedthannormalcurve

c) Platykurtic(Playsoundslikeflat)

o tailsarethinnerorlighterthannormalcurve

Discretevariable Variablethathasspecificvaluesandthatcannothavevaluesbetweenthesespecificvalues

Continuous

variable

Variableforwhich,intheory,thereareaninfinitenumberofvaluesbetweenanytwovalues

Flooreffect Situationinwhichmanyscorespilesupatthelowendofadistribution(creatingskewnesstotheright)

becauseitisnotpossibletohavelowerscore

Ceilingeffect Situationinwhichmanyscorespileupatthehighendofadistribution(creatingskewnesstotheleft)

becauseitisnotpossibletohaveahigherscore

Normalcurve Specific,mathematicallydefined,bell-shapedfrequencydistributionthatissymmetricalandunimodal;

distributionsobservedinnatureandinresearchcommonlyapproximateit.

Lecture2–Centraltendencyandvariability

- Measuresofcentraltendency

o mean

o mode

o median

- Measuresofvariability

o range

o variance

o standarddeviation

- Cautionsandadvice

Central

tendency

Mosttypical,commonscore,representativevalueofagroupofscores

Mean • Sensitivetoanyscore

• =theaveragescore.

• =thesumofallthescoresdividedbythenumberofscores.

• =thetypicalorrepresentativescore.

• bestwayofestimatingwhatanindividualunknownscoremightbe.

• influencedbyallscoresinadistribution(sorepresentsallscoresbutcanbeundulyinfluencedby

extremescoresand,thus,canbebiased).

• E.g.Iask10studentshowmuchstudytheyhavedoneinthelastweekandgetthefollowing

results:

10,2,4,3,4,4,6,5,5,7

• thetotalnumberofhoursstudied=50

• thenumberofscores(observations)=10

• themeannumberofhours=50/10=5

Importantconcepts • itislikeabalancingpointinadistribution

• thetotaldistancefromthemeanofallscoreslessthanthemean=thetotaldistancefrom

themeanofallscoresgreaterthanthemean

• belowmeantotal=-8,

abovemeantotal=+8,

sumofdistances=0

• themeancanbeavalueorscorewhichdoesnotexistintheactualsetofscores

Scores Distance

from mean

10 5

7 2

6 1

5 0

5 0

4 -1

4 -1

4 -1

3 -2

2 -3

Meanofthedistributionofthenumberofdreamsduringaweekfor10students.

Calculatingthe

mean

Themathematicalformulaforcalculatingthemean,M(sometimesµor )

å:aGreeklettersigmameans‘thesumof’

X:ascoreinthedistributionofavariableX

N:thenumberofscoresinadistribution

51050

23444556710

==

+++++++++==

M

NNX

M

Mode • =themostcommonscoreinaunimodaldistribution

• =thepeakofahistogramorafrequencypolygon

• inasymmetricalunimodaldistribution(normaldistribution):

• themode=themean

• usefulwhenonlyafewvaluespossibleasmodeonlydescribesonescore

Themodeasthehighpointinadistribution’shistogram,usingtheexampleofthenumberofdreamsduring

aweekfor10students.

Median • =themiddlescorewhenallscoresareranked

• easyifthereareanoddnumberofscores

• ifevennumber,itfallshalfwaybetweenthetwomiddlescores

• sometimesthemedianisabettermeasureofcentraltendencythanthemean

• inskeweddistributions,afewextremescorescanaffectthemean.usewhenthedataisheavily

skewed,e.g.income,houseprices

NXM =

X

• evenversusoddnumberofcases:

• themiddlescorewhenallscoresareranked

• ifthereisanevennumberofscoresthemedianfallshalfwaybetweentotwomiddle

scores

scores 2 3 4 5

median=3.5

• easyifthereareanoddnumberofscores

scores 2 3 4 5 5

median=4

Whichcentral

tendency

measure

• Mode:onlyfewvalues

• Median:skewed

• Mean:normal

inasymmetricalunimodaldistribution,themean=the

mode=themedian

Variability howspreadoutthescoresareinadistribution

Measuresof

variability

• twodistributionsmayhavethesamemeanbutonemayhaveagreaterspread(orvariability)in

values

• indescribingdistributionsnumerically,needtobeabletodiscussthespread(orvariability)of

scores

Range • thesimplestmeasureofspreadistherange

• therangeissimplythedifferencebetweenthehighestandlowestscores

1336778889

• range=highestscore-lowestscore:9-1=8

Interquartilerange

(IQR)

• IQRprovidestheboundariesforthemiddle50%ofscores

• StepstofindIQR

• findmedian

• findmiddlescoreintopandbottomhalves

• 13367X78889

• IQR=8-3=5

• usualtoreportIQRwithmedian

Variance • Rangeonlydescribestwo,possiblyextreme,values

• IQRbetterbutstillnotrepresentativeofallscores

• Preferameasurethatconsidersallvalues-likeourmeanincentraltendency

• Thevariancetellsushowspreadoutasetofscoresisaroundtheirmean

• itistheaverageofeachscore’ssquareddeviationaroundthemean

• Variance:howmuchindividualscoredifferfromthemean,squarethevaluetocanceloutthe

minus

Calculatingthe

variance

• subtractthemeanfromeachscore(onebyone)togetadeviationscore

(X-M)

• square(multiplybyitself)eachofthesedeviationscorestogetasquareddeviationscore

Documents

Lec Notes 110 Stats Sample · Mean of the distribution of differences between means..... 144