Download pdf - MODELING DATA—9.2 9.2 Data Distribution Z · 9.2 Data Distribution ... histograms, and box plots since sixth grade. In this course ... SECONDARY MATH I // MODULE 9 MODELING DATA

SECONDARY MATH I // MODULE 3 MODELING DATA—9.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

9.2 Data Distribution A Practice Understanding Task Alotofinformationcanbeobtainedfromlookingatdataplotsandtheirdistributions.Itisimportantwhendescribingdatathatweusecontexttocommunicatetheshape,center,andspread.Shapeandspread:

• Modes:uniform(evenlyspread-noobviousmode),unimodal(onemainpeak),bimodal(twomainpeaks),ormultimodal(multiplelocationswherethedataisrelativelyhigherthanothers).

• Skeweddistribution:whenmostdataistoonesideleavingtheotherwitha‘tail’.Dataisskewedtosideoftail.(iftailisonleftsideofdata,thenitisskewedleft).

• Normaldistributionandstandarddeviation:curveisunimodalandsymmetric.Datathathasanormaldistributioncanalsodescribethedatabyhowfaritisfromthemeanusingstandarddeviation.

• Outliers:valuesthatstandawayfromthebodyofthedistribution.Forabox-and-whiskeroutliersdeterminediftheyaremorethan1.5timestheinterquartilerange(lengthofbox)beyondquartiles1and3.Alsoconsideredanoutlinerifdataismorethantwostandarddeviationsfromthecenterofanormaldistribution.

• Variability:valuesthatareclosetogetherhavelowvariability;valuesthatarespreadaparthavehighvariability.

Center:• Analyzethedataandseeifonevaluecanbeusedtodescribethedataset.Normal

distributionsmakethiseasy.Ifnotanormaldistribution,determineifthereisa‘center’valuethatbestdescribesthedata.Bimodalormultimodaldatamaynothaveacenterthatwouldprovideusefuldata.

Therearerepresentationsoftestscoresfromsixdifferentclassesfoundbelow,foreach:

1. Describethedatadistribution.2. ComparedatadistributionsbetweenAndersonandWilliams.3. ComparedatadistributionsbetweenWilliamsandLemon.4. ComparedatadistributionsbetweenCroftandHurlea.5. ComparedatadistributionsbetweenJones,Spencer,andAnderson.6. ComparedatadistributionsbetweenSpencerandtheotherhistograms.7. Whichdistributionsaremostsimilar?Different?Explainyouranswer.

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DatasetI:Williams’sclass DatasetII:Lemon’sclass

DatasetIII:Croft’sClass DatasetIV:Anderson’sClass

DatasetV:Hurlea’sclass DatasetVI:Jones’class

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DatasetVII:Spencer’sclass

DatasetVIII:OverallAchievementTestScores

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9.2 Data Distribution – Teacher Notes A Practice Understanding Task Purpose:Studentsarealreadyfamiliarwithdotplots,boxplots,andhistograms.Thistaskhasthemdescribedatadistributionsandcompareshape,center,andspreadoftwoormoresetsofdata.

CoreStandardsFocus:

S.ID.2Usestatisticsappropriatetotheshapeofthedatadistributiontocomparecenter(median,mean)andspread(interquartilerange,standarddeviation)oftwoormoredifferentdatasets.S.ID.3Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).

RelatedStandards:S.ID.1

StandardsforMathematicalPractice:

SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP4–Modelwithmathematics

SMP8–Lookforandexpressregularityinrepeatedreasoning

TheTeachingCycle:

Note:Itwouldbegoodtohavethedatayouwanttocompareinaformatthatislargeandvisibleforthewholegroupdiscussion.Forexample,youcouldcopythetwodatasetsyouwishtocompareandplacethemnexttoeachotherinaformatthatcanbeprojectedsothatwhenstudentsaresharingduringwholegroup,thevisualrepresentationisavailableforeveryonetosee.

Note:Studentshavebeenaskedtoidentifyandinterpretunivariatedatausingdotplots,histograms,andboxplotssincesixthgrade.Inthiscourse,studentsareaskedtocomparedatasetsusingtheirknowledgeofshape,center,andspreadandhavebecomemorecomfortablewiththeseattributes.Outliers,skeweddata,andnormaldistributionmaybenewthisyearaswell.Launch(WholeClass):

Havestudentsreadthevocabularytodescribedatadistributionsandaskthemtounderlineinformationthatisnewtothem.Havethemworkindividuallyforawhileonquestion1thathasthemdescribeeachdatasetbeforehavingthemworktogetherwithapartnerorsmallgrouptoanswertheremainingquestions(wheretheycomparedatasets).



Explore(SmallGroup):

Givestudentstimetoanswerthequestionscomparingdatasets.Listenforstudentstousevocabularyindescribingagivendataset,andtocompareshape,center,andspreadoftwoormoredatasets.Listenforstudentstocomparedatasets,notjustlistattributesofeach.Pressstudentstomakecomparisonsshowingtheyunderstandwhentousedatatodescribeandcompareshape,center,andspreadbetweendatasets.Examplesincludenoticingoutliers,variabilityandspreadbetweendata(noticethatHurleaandSpencerhaveascalethatisdifferentthantheothers),andothertrends.Again,makesurestudentsdonotjustlistcharacteristicsofeachdistributionandthinktheyare‘comparing’.Discuss(WholeClass):

Beginthewholegroupdiscussionbyselectingproblemsfromquestionsthatcomparedatasets.Basedonsmallgroupconversations,choosewhichcomparisonstoshareoutinwholegroup.Thefocusofthewholegroupdiscussionistodothefollowing:

o Showstudentunderstandingofusingstatisticsappropriatetotheshapeofthedatadistributiontocomparecenterandspread.

o Showstudentunderstandingofwhatinformationisprovidedwhengivenahistogram,boxplot,dotplot.

AlignedReady,Set,Go:ModelingData9.2

SECONDARY MATH I // MODULE 9

MODELING DATA – 9.2

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

9.2

READY Topic:Drawingconclusionsfromdata.Inproblems1–4youaretoselectthebestanswerbasedonthegivendata.Belowyourchosenanswerisaconfidencescale.Circlethestatementthatbestdescribesyourconfidenceinthecorrectnessoftheansweryouchose.Thegoalistogainawarenessofhowitseemseasiertodrawconclusionsinsomecasesthaninothers. 1.Data:1,2,4,8,16,32, Thenextnumberinthelistwillbe:________

a.largerthan32 b.positive c.exactly64 d.lessthan32

IamcertainIamcorrect. Iamalittleunsure. IhadnoideasoIguessed.

Whataboutthedatamadeyoufeelthewayyoudidabouttheansweryoumarked?

2.Data:47,-13,-8,9,-23,14, Thenextnumberinthelistwillbe:________

a.positive b.negative c.lessthan100 d.lessthan-100



3.Data:-10,¾,38,-10,½,-81,-10,¼,93,-10, Thenextnumberinthelistwillbe:______

a.morethan93 b.negative c.afraction d.awholenumber


4.Data:50,-43,36,-29,22,-15 Thenextnumberinthelistwillbe:______

a.odd b.lessthan9 c.two-digits d.greaterthan-15



READY, SET, GO! Name PeriodDate

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9.2

SET Topic:Creatinghistograms.

Mr.Austingaveaten-pointquiztohis9thgrademathclasses.Atotalof50studentstookthequiz.Mr.Austinscoredthequizzesandlistedthescoresalphabeticallyasfollows.

1stPeriodMath 2ndPeriodMath 3rdPeriodMath

6,4,5,7,5,

9,5,4,6,6,

8,5,7,5,8,

1,8,7,10,9

4,5,8,6,8,

9,5,8,5,1,

5,5,7,5,7

9,8,10,5,9,

7,8,9,8,5,

8,10,8,8,5

5.UseALLofthequizdatatomakeafrequencytablewithintervals.Useanintervalof2.

Score Frequency

0-1

2–3

4–5

6–7

8–9

10-11

6.Useyourfrequencytabletomakeahistogramforthedata

7.Describethedatadistributionofthehistogramyoucreated.Includewordssuchas:mode,skewed,outlier,normal,symmetric,center,andspread,iftheyapply.(Hint:Don’tforgetstandarddeviation.)

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9.2

8.Createagraphofyourchoice(histogram,boxplot,dotplot)for1stand3rdperiod.

9.Whichclassperformedbetter? Justifyyouranswerbycomparingtheshape,center,andspreadofthetwoclasses.(Hint:Don’tforgetstandarddeviation.)

GO

Topic:Figuringpercentages

10.Whatpercentof97is11? 11.Whatpercentof88is132?

12.Whatpercentof84is9? 13.Whatpercentof88.6is70?

14.Whatis270%of60? 15.Whatis84%of25?

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