# AIOU Solved Assignment 1 & 2 Code 8614 Autumn & Spring 2021

AIOU Solved Assignments 1 & 2 Code 8614 Autumn & Spring 2021. Solved Assignments code 8614 Educational Statistics 2022. Allama iqbal open university old papers.

Contents

Code: 8614 Assignment No.1
Semester: Autumn & Spring 2021
Educational Statistics (8614)

Q.1Whatarelevelsofmeasurement?Explaineachlevelsothatreadercanunderstandthe descriptionoflevelanddifferentiateeachlevelfromotherlevels.Write down10examplesforeachlevelandfurtherexplainoneexamplefrom level.

NominalScaleExamples

Herearesomeoftheexamplesofnominalmeasurementthatwillhelpinunderstand thismeasurementscalebetter.

-Howwouldyoudescribeyourbehavioralpattern?

E-Extroverted I-Introverted.A-Ambivert

-Whatisyourgender?

M-Male F-Female

-Couldyoupleaseselectanoptionfrombelowtodescribeyourhaircolor.

1-Black2-Brown3-Burgundy.4-Auburn.5-Other

-Pleaseselectthedegreeofdiscomfortofthedisease:

1-Mild2-Moderate.3-Severe

Inthisparticularexample,1=Mild,2=Moderate,and3=Severe.Herenumbersaresimply usedastagsandhavenovalue.

OrdinalScaleExamples

“Howsatisfiedareyouwithourproducts?”

1-TotallySatisfied2-Satisfied3-Neutral4-Dissatisfied5-TotallyDissatisfied

“Howhappyareyouwiththecustomerservice?”

1-VeryUnhappy2-Unhappy3-Neutral4-Unhappy5-VeryUnhappy

RatioScaleExamples

Thefollowingarethemostcommonlyusedexamplesforratioscale:

1.Whatisyourheightinfeetandinches?

Lessthan5feet.5feet1inch–5feet5inches5feet6inches-6feetMorethan6feet

2.Whatisyourweightinkgs?

Lessthan50kgs51-70kgs71-90kgs91-110kgsMorethan110kgs

3.Howmuchtimedoyouspenddailywatchingtelevision?

Lessthan2hours3-4hours4-5hours5-6hoursMorethan6hours

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Q.2Whatisimportanceofpresentationofdata?Explaindifferentmethodsof effectivepresentationofdata.Listdifferenttypesofgraphsandwritenoteoneachtype.

Whatisdatapresentationandanalysis?

Datapresentationandanalysisformsanintegralpartofallacademicstudies,commercial,

industrialandmarketingactivitiesaswellasprofessionalpractices.Itisnecessarytomakeuse ofcollecteddatawhichisconsideredtoberawdatawhichmustbeprocessedtoputforany application.Dataanalysishelpsintheinterpretationofdataandtakeadecisionoranswerthe researchquestion.Dataanalysisstartswiththecollectionofdatafollowedbydataprocessing andsortingit.Processeddatahelpsinobtaininginformationfromitastherawdataisnon- comprehensiveinnature.Presentingthedataincludesthepictorialrepresentationofthedata byusinggraphs,charts,mapsandothermethods.Thesemethodshelpinaddingthevisual aspecttodatawhichmakesitmuchmorecomfortableandquickertounderstand

Datapresentationandanalysisplaysanessentialroleineveryfield.Anexcellentpresentation canbeadealmakerordealbreaker.Somepeoplemakeanincrediblyusefulpresentationwith thesamesetoffactsandfigureswhichareavailablewithothers.Attimespeoplewhodidall thehardworkbutfailedtopresentitpresentitproperlyhavelostessentialcontracts,thework whichtheydidisunabletoimpressthedecisionmakers.Sotogetthejobdoneespeciallywhile dealingwithclientsorhigherauthoritiespresentationmatters!Nooneiswillingtospendhours inunderstandingwhatyouhavetoshowandthisispreciselywhypresentationmatters!

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Q.3 a) Explainthemeasuresofcentraltendencyandmeasuresofdispersion.How thesetwoconceptsarerelated?

Measuresofcentraltendency(alsoreferredasmeasuresofcenterofcentrallocation)allow ustosummarizedatawithasinglevalue.Itisatypicalscoreamongagroupofscores(the midpoint).Theygiveusaneasywaytodescribeasetofdatawithasinglenumber.Thissingle numberrepresentsavalueorscorethatgenerallyisinthemiddleofthedataset.

Thegoalofthemeasureofcentraltendencyis:

i)Tocondensedatainasinglevalue.

ii)Tofacilitatecomparisonbetweendata.

Goodmeasureofcentraltendencyshouldbe:

i)Bestrictlydefined.

ii)Besimpletounderstandandeasytocalculate.

iii)Becapableoffurthermathematicaltreatment.

iv)Bebasedonallvaluesofgivendata.

v)Havesamplingstability.

vi)Notbeundulyaffectedbyextremevalues.

Commonlyusedmeasuresofcentraltendencyarethemean,themedianandthemode.

Eachoftheseindicesisusedwithadifferentscaleofmeasurement.

IntroductiontoMeasuresofDispersion

Measuresofcentraltendencyfocusonwhatisanaverageorinthemiddleofthedistributionof scores.Oftentheinformationprovidedbythesemeasuresdoesnotgiveusclearpictureofthe dataandweneedsomethingmore.Itmeansthatknowingthemean,median,andmodeofa distributiondoesallowustodifferentiatebetweentwoormorethantwodistributions;andwe needadditionalinformationaboutthedistribution.Thisadditionalinformationisprovidedbya seriesofmeasureswhicharecommonlyknownasmeasuresofdispersion.Thereisdispersion whenthereisdissimilarityamongthedatavalues.Thegreaterthedissimilarity,thegreaterthe degreeofdispersionwillbe.

Measuresofdispersionareneededforfourbasicpurposes.

i)Todeterminethereliabilityofanaverage.

ii)Toserveasabasisforthecontrolofthevariability.

iii)Tocomparetwoormoreserieswithregardtotheirvariability.

iv)Tofacilitatetheuseifotherstatisticalmeasures.

Measureofdispersionenablesustocomparetwoormoreserieswithregardstotheirvariability. Itisalsolookedasameansofdetermininguniformityorconsistency.Ahighdegreewould meanlittleconsistencyoruniformitywhereaslowdegreeofvariationwouldmeangreater uniformityorconsistencyamongthedataset.Commonlyusedmeasuresofdispersionare range,quartiledeviation,meandeviation,variance,andstandarddeviation.

HowTheseTwoConceptsareRelated?

Explanation:

Measuresofcentraltendencyaremean,modeandmedian.Evenwehavethreetypesofmean, suchasarithmaticmean,geometricmeanandharmonicmean.

Theytellusthecentralvaluearoundwhichthedataisdistributed.Forexampleconsiderthe dataset#6,8,2,4,12,5,8,10,3,4#.Inthissumofnumbersis#62#andastheyaretenin number,meanis#62/10=6.2#

Notethatsmallestnumberis#2#andlargestnumberis#12#.Now,evenifwehadsetof numbersas#5,6,7,5,8#andassumofnumbersis#31#andtheyarefive,meanisstill #31/5=6.2#.But#5,6,7,5,8#arefarmorenarrowlyspreadandhencenatureofdataisnotvery

wellbroughtoutbyjustmean.

Similarly,wecanhavetwodatasetswithsamemedianormode,buttheirspreadmaybe different,asmodeisjustthemorefrequentamongdatapointsandmedianisthevalueof centraldatapoint,whenthesammeisarrangedinincreasingordecreasingorder.

Measuresofdispersiontellusbetteraboutthekindofspread.Inaway,meandeviationor standarddeviationtellusmoreaboutthewaydataisspread.

Forexample,dataset#30,40,50,60,70#anddataset#10,30,50,70,90#havesamemean,mode andmedianbutwhilemeandeviationoffirstdatasetis#12#,thatofseconddatasetis#24#, indicatingthatseconddatasetistoowidespread.

Whatabouttwodatasets#30,40,50,60,70#and#130,140,150,160,170#?Theirmeandeviation issamei.e.#12#,butaretheynotwidelydifferentasmeanoffirstdatasetis#50#,whilethat ofseconddatasetis#150#.

Itisobviousthatmeasuresofcentraltendencyandmeasuresofdispersionarebothimportant andcomplementary.

b) Howthesetwoconceptsarerelated?Suggestonemeasureofdispersionfor eachmeasureofcentraltendencywithlogicalreasons.

HowTheseTwoConceptsareRelated?

Explanation:

Measuresofcentraltendencyaremean,modeandmedian.Evenwehavethreetypesofmean, suchasarithmaticmean,geometricmeanandharmonicmean.

Theytellusthecentralvaluearoundwhichthedataisdistributed.Forexampleconsiderthe dataset#6,8,2,4,12,5,8,10,3,4#.Inthissumofnumbersis#62#andastheyaretenin number,meanis#62/10=6.2#

Notethatsmallestnumberis#2#andlargestnumberis#12#.Now,evenifwehadsetof numbersas#5,6,7,5,8#andassumofnumbersis#31#andtheyarefive,meanisstill #31/5=6.2#.But#5,6,7,5,8#arefarmorenarrowlyspreadandhencenatureofdataisnotvery wellbroughtoutbyjustmean.

Similarly,wecanhavetwodatasetswithsamemedianormode,buttheirspreadmaybe different,asmodeisjustthemorefrequentamongdatapointsandmedianisthevalueof centraldatapoint,whenthesammeisarrangedinincreasingordecreasingorder.

Measuresofdispersiontellusbetteraboutthekindofspread.Inaway,meandeviationor standarddeviationtellusmoreaboutthewaydataisspread.

Forexample,dataset#30,40,50,60,70#anddataset#10,30,50,70,90#havesamemean,mode andmedianbutwhilemeandeviationoffirstdatasetis#12#,thatofseconddatasetis#24#, indicatingthatseconddatasetistoowidespread.

Whatabouttwodatasets#30,40,50,60,70#and#130,140,150,160,170#?Theirmeandeviation issamei.e.#12#,butaretheynotwidelydifferentasmeanoffirstdatasetis#50#,whilethat ofseconddatasetis#150#.

Itisobviousthatmeasuresofcentraltendencyandmeasuresofdispersionarebothimportant andcomplementary..

(2partanswernotavailable)

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Q.4Whatisnormaldistribution?Explaintheroleofnormaldistributionindecisionmakingfor dataanalysis.Writeanoteonskewenessandkurtosisandexplainitscauses.

Anormaldistribution,sometimescalledthebellcurve,isadistributionthatoccursnaturallyin manysituations.Forexample,thebellcurveisseenintestsliketheSATandGRE.Thebulkof studentswillscoretheaverage(C),whilesmallernumbersofstudentswillscoreaBorD.An evensmallerpercentageofstudentsscoreanForanA.Thiscreatesadistributionthat resemblesabell(hencethenickname).Thebellcurveissymmetrical.Halfofthedatawillfallto theleftofthemean;halfwillfalltotheright.

Manygroupsfollowthistypeofpattern.That’swhyit’swidelyusedinbusiness,statisticsandin governmentbodiesliketheFDA:

Heightsofpeople.

Measurementerrors.

Bloodpressure.

Pointsonatest.

IQscores.

Salaries

1) Ithasoneoftheimportantpropertiescalledcentraltheorem.Centraltheoremmeans relationshipbetweenshapeofpopulationdistributionandshapeofsamplingdistributionof mean.Thismeansthatsamplingdistributionofmeanapproachesnormalassamplesize increase.

2) Incasethesamplesizeislargethenormaldistributionservesasgoodapproximation.

3) Duetoitsmathematicalpropertiesitismorepopularandeasytocalculate.

4) Itisusedinstatisticalqualitycontrolinsettingupofcontrollimits.

5) Thewholetheoryofsampletestst,fandchi-squaretestisbasedonthenormal distribution.

Thesearetheimportanceorusesorbenefitsofnormaldistribution.

a)Skewness

Skewnesstellsusabouttheamountanddirectionofthevariationofthedataset.Itisa measureofsymmetry.Adistributionordatasetissymmetricifitlooksthesametotheleftand rightofthecentralpoint.Ifbulkofdataisatthelefti.e.thepeakistowardsleftandtherighttail islonger,wesaythatthedistributionisskewedrightorpositivelyskewed.Ontheotherhandif thebulkofdataistowardsrightor,inotherwords,thepeakistowardsrightandthelefttailis longer,wesaythatthedistributionisskewedleftornegativelyskewed.Iftheskewnessisequal tozero,thedataareperfectlysymmetrical.Butitisquietunlikelyinrealworld.

Herearesomerulesofthumb:

i)Iftheskewnessislessthan–1orgreaterthan+1,thedistributionishighlyskewed.

ii)Iftheskewnessisbetween-1and-orbetween+and+1,thedistributionismoderately skewed.

iii)Iftheskewnessisbetween-and+,thedistributionisapproximatelyskewed.

b)Kurtosis

Kurtosisisaparameterthatdescribestheshapeofvariation.Itisameasurementthattellsus howthegraphofthesetofdataispeakedandhowhighthegraphisaroundthemean.Inother wordswecansaythatkurtosismeasurestheshapeofthedistribution,.i.e.thefatnessofthe tails,itfocusesonhowreturnsarearrangedaroundthemean.Apositivevaluemeansthattoo littledataisinthetailandpositivevaluemeansthattoomuchdataisinthetail.Thisheaviness orthelightnessinthetailmeansthatdatalooksmorepeakedoflesspeaked.Kurtosisis

measuredagainstthestandardnormaldistribution.Astandardnormaldistributionhasa kurtosisof3.Kurtosishasthreetypes,mesokurtic,platykurtic,andleptokurtic.Ifthedistribution haskurtosisofzero,thenthegraphisnearlynormal.Thisnearlynormaldistributioniscalled mesokurtic.Ifthedistributionhasnegativekurtosis,itiscalledplatykurtic.Anexampleof platykurticdistributionisauniformdistribution,whichhasasmuchdataineachtailasitdoes inthepeak.Ifthedistributionhaspositivekurtosis,itiscalledleptokurtic.Suchdistributionhas bulkofdatainthepeak.

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Q.5Explainthefollowingtermswithexamples

a) ContinuousVariable

Avariableisaquantitythathasachangingvalue;thevaluecanvaryfromoneexampletothe next.Acontinuousvariableisavariablethathasaninfinitenumberofpossiblevalues.Inother words,anyvalueispossibleforthevariable.

Acontinuousvariableistheoppositeofadiscretevariable,whichcanonlytakeonacertain numberofvalues.

Acontinuousvariabledoesn’thavetohaveeverypossiblenumber(like-infinityto+infinity),it canalsobecontinuousbetweentwonumbers,like1and2.Forexample,discretevariables couldbe1,2whilethecontinuousvariablescouldbe1,2andeverythinginbetween:1.00,1.01, 1.001,1.0001…

Afewexamplesofcontinuousvariables/data:

Timeittakesacomputertocompleteatask.Youmightthinkyoucancountit,buttimeisoften roundeduptoconvenientintervals,likesecondsormilliseconds.Timeisactuallyacontinuum: itcouldtake1.3secondsoritcouldtake1.333333333333333…seconds.

Aperson’sweight.Someonecouldweigh180pounds,theycouldweigh180.10poundsorthey couldweigh180.1110pounds.Thenumberofpossibilitiesforweightarelimitless.

Income.Youmightthinkthatincomeiscountable(becauseit’sindollars)butwhoistosay someonecan’thaveanincomeofabilliondollarsayear?Twobillion?Fiftyninetrillion?Andso on…Age.So,you’re25years-old.Areyousure?Howabout25years,19daysandamillisecondor two?Liketime,agecantakeonaninfinitenumberofpossibilitiesandsoit’sacontinuous variable.

Thepriceofgas.Sure,itmightbe\$4agallon.Butonetimeinrecenthistoryitwas99cents. Andgiveinflationafewyearsitwillbe\$99.nottomentionthegasstationsalwaysliketouse fractions(i.e.gasisrarely\$4.47agallon,you’llseeinthesmallprintit’sactually\$4.479/10ths

b) CategoricalVariable

Asthenamesuggests,categoricalvariablesarethosevariablesthatfallintoaparticular category.Haircolor,gender,collegemajor,collegeattended,politicalaffiliation,disability,or sexualorientationareallcategoriesthatcouldhavelistsofcategoricalvariables.Usually,the variablestakeononeofanumberoffixedvariablesinaset.

Forexample:

Thecategory“haircolor”couldcontainthecategoricalvariables“black,”“brown,”“blonde,”and “red.”

Thecategory“gender”couldcontainthecategoricalvariables“Male”,“Female”,or“Other.”

Notethat“haircolor”and“gender”arethecategoriesandarenotcategoricalvariables themselves.Acategoricalvariableisavaluethatvariablesinastudytake;thevaluevariesfrom persontoperson.Let’ssayyousurveypeopleandaskthemtotellyoutheirhaircolor.They wouldrespondwithacategoricalvariableofblack,brown,blond,orred.Theywouldn’trespond “haircolor.”

Isthereanordertocategoricalvariables?

Thereisnoordertocategoricalvariables;inotherwords,theyaren’trankedfromhighestto lowestorlowesttohighest.Forexample,thereisnointrinsicordertothecategoriesofmale andfemale.Ifthereissomekindoforder,thenthosevariableswouldbeordinalvariablesand notcategoricalvariables.Forexample,youcouldcategorizehousepricesbycheap,moderate andexpensive.Althoughthesearecategories,thereisaclearorder(withcheaponthebottom andexpensiveontop).

Examplesofcategoricalvariables:

Brandoftoothpaste(Colgate,Aquafresh…)

Collegemajor(English,Math…)

Telephonecompany(BellSouth,AT&T…)

Checkingaccountlocation(Jacksonville,NewYorkCity…)

Schoolattended(LeeHigh,WescottHigh…)

Examplesofquantitativevariables:

Numberoftoothpastetubesusedperyear.

G.P.A.forcollegemajor.

Bytesofdatauploadedonyourphone.

Checkingaccountbalance.

Averagenumberofstudentsinaclass

c) IndependentVariable

IndependentVariableDefinition.

Independentvariablesarevariablesthatstandontheirownandaren’taffectedbyanythingthat you,asaresearcher,do.Youhavecompletecontroloverwhichindependentvariablesyou choose.Duringanexperiment,youusuallychooseindependentvariablesthatyouthinkwill affectdependentvariables.Thosearevariablesthatcanbechangedbyoutsidefactors.Ifa variableisclassifiedasacontrolvariable,itmaybethoughttoaltereithertheindependent variableordependentvariablebutitisn’tthefocusoftheexperiment.

Example:youwanttoknowhowcalorieintakeaffectsweight.Calorieintakeisyour independentvariableandweightisyourdependentvariable.Youcanchoosethecaloriesgiven toparticipants,andyouseehowthatindependentvariableaffectstheweights.Youmaydecide toincludeacontrolvariableofageinyourstudytoseeifitaffectstheoutcome.

Theabovegraphshowstheindependentvariableofmaleorfemaleplottedonthex=axis. “Male”or“Female”isunchangeablebyyou,theresearcher,oranythingyoucanperforminyour experiment.Ontheotherhand,thedependentvariableof“meanvocabularyscores”is potentiallychangedbywhichindependentvariableisassigned.Inotherwords,themean

vocabularyscoresdependontheindependentvariable:whethertheparticipantismaleor female.

Anotherwayoflookingatindependentvariablesisthattheycausesomething(orarethoughtto causesomething).Intheaboveexample,theindependentvariableiscalorieconsumption. That’sthoughttocauseweightgain(orloss).

d) DependentVariable

DependentVariable:

Adependentvariableiswhatyoumeasureintheexperimentandwhatisaffectedduringthe experiment.Thedependentvariablerespondstotheindependentvariable.Itiscalleddependent becauseit”depends”ontheindependentvariable.Inascientificexperiment,youcannothavea dependentvariablewithoutanindependentvariable.

Example:Youareinterestedinhowstressaffectsheartrateinhumans.Yourindependent variablewouldbethestressandthedependentvariablewouldbetheheartrate.Youcan directlymanipulatestresslevelsinyourhumansubjectsandmeasurehowthosestresslevels changeheartrate.

.Covarianceis ameasureofhowmuch tworandom variablesvarytogether. It’ssimilarto variance,butwhere variancetells youhowasinglevariable varies,co variancetellsyouhow twovariables varytogether.

TheCovarianceFormula

Theformulais:

Cov(X,Y)=?E((X-?)E(Y-?))/n-1where:

Xisarandomvariable

E(X)=?istheexpectedvalue(themean)oftherandomvariableXand

E(Y)=?istheexpectedvalue(themean)oftherandomvariableY

n=thenumberofitemsinthedataset

Example

Calculatecovarianceforthefollowingdataset:

x:2.1,2.5,3.6,4.0(mean=3.1)

y:8,10,12,14(mean=11)

Substitutethevaluesintotheformulaandsolve:

Cov(X,Y)=?E((X-?)(Y-?))/n-1

=(2.1-3.1)(8-11)+(2.5-3.1)(10-11)+(3.6-3.1)(12-11)+(4.0-3.1)(14-11)/(4-1)

=(-1)(-3)+(-0.6)(-1)+(.5)(1)+(0.9)(3)/3

=3+0.6+.5+2.7/3

=6.8/3

=2.267

Theresultispositive,meaningthatthevariablesarepositivelyrelated.

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