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Calculating Measures of Dispersion

ReportingCategory StatisticsTopic CalculatingMeanAbsoluteDeviation,Variance,andStandardDeviationPrimarySOL A.9Thestudent,givenasetofdata,willinterpretvariationinrealworld

contextsandcalculateandinterpretmeanabsolutedeviation,standard

deviation,andzscores.

RelatedSOLMaterials

GraphingCalculators Overheadorcomputergraphingcalculatortobeprojectedtotheclass(optional)

Vocabularymean,median,mode,range,measureofcentraltendencydispersion,meanabsolutedeviation,standarddeviation,summationnotation,variance(A.9)

Student/TeacherActionsWhatshouldstudentsbedoing?Whatshouldteachersbedoingtofacilitatelearning?

1. Presentstudentswiththefollowingscenario:StudentsinMrs.SmithsCommunicationClubareinterestedinhowmanyhoursthey

spendwatchingtelevision. Theycollectedthefollowingdatawhichshowsthenumber

ofhours

they

each

watched

television

in

one

week.

3,8.5,9,9,12.5,14,16.5,18,19,20.5

2. Askstudentstothinkaboutwhattheycoulddowiththisdata. Havethemworkinsmallgroupsorwithapartnertodowhattheycanwiththedata. Somewillrepresentit

graphically,findthemean,median,mode,range,etc.

3. Askstudentstosharewhattheychosetodowiththedata. Questionthestudentsaboutwhatthegraphorstatisticsaysabouttheinformationaftereachideaisshared.

Ex. Whatdoesthemeantellusaboutthedata? Whatdoesthishistogramshowus

aboutthedata?

4. Inthediscussionattachthevocabularyofmean,medianandmodebeingmeasuresofcentraltendencyandrangebeingameasureofdispersionorthespreadofthedata.5. Tellstudentsanotherwaytomeasuredispersionistocomparetheelementstothe

mean. Askstudentstohelpyoufindthedeviationofeachelementfromthemean,or

thedistanceeachelementinthedataisfromthemean. Askstudentshowtheywould

findthisdeviation. (Theyshouldsubtractthemeanfromeachvalue.) Createachartfor

allstudentstoseewiththeoriginalelementsononesideandaspaceforthedeviation

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fromthemeanontheother. Recordthemeanbesidethechart,usingandintroducing

thesymbolofforthemean.

6. Havetheclasshelpyoufillinthechart. Askthemtocalculatetheaveragedistancefromthemeanoraveragedeviation. Askfortheirobservations(thedeviationssumwillbe

zero). Askstudentsifthiswillalwaysoccurandwhyorwhynot. Oncestudents

determinethat

this

will

always

occur,

because

of

the

negative

values

ask

students

how

theycouldensurethatanumberisalwayspositive. Continuethediscussionuntilthe

ideaofabsolutevalueemerges.

7. Havestudentsfindthesumofdeviationsusingtheabsolutevalues. Introducestudentstothenameforthisdescriptivestatisticasthemeanabsolutedeviation. Askstudents

tosummarizehowtheyfoundthemeanabsolutedeviation. Presentstudentswiththe

formulabelowandhavethemdiscusshowitmatcheswhattheydidtocalculatethe

meanabsolutedeviationandtheirsummaryoftheprocess. Ifyouhavenotpreviously

discussedsummationnotation,youwillneedtoexplainthisnotation.

x x

3

8.5

9

9

12.5

14

16.5

18

19

20.5

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8. Askstudentswhatthemeanabsolutedeviationcouldtellthemaboutthedata. Whatwouldahighmeanabsolutedeviationindicate? Whataboutalowvalueforthemean

absolutedeviation?

9. Tellstudentsthatthereisanothermethodforfindingthedispersionaboutthemean.Guidestudentstothinkofotherwaystonothavenegativesinasetofnumbers. You

mayaskthemquestionstogetstudentsthinkingabouttheirpastexperienceswith

quadraticsandsquaringnumbers.

10.Havestudentsfindthesquareddeviations(youmaywanttoaddanothercolumntothechartforthis).Thenhavethemfindtheaverageofthesquareddeviations.

11.Introducethisvalueasthevariance,anothermeasureofdispersion. Givethesymbolforthisvalue

2 . Alsoshowtheformulaforfindingthevarianceandhavestudents

explainhowtheformulacorrelatestowhattheydidtocomputethevariance.

12.Askstudentswhattheunitofthevarianceis(hours2). Askhowtheycouldgettheunittobethesameastheunitsinthedata(bytakingthesquareroot). Explainthatthisis

calledthestandarddeviation()andisanothermeasureofdispersion. Provide

studentswiththeformulaandhavethemexplainhowthisformulaaddresses

dispersion.

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13.Askstudentswhatthestandarddeviationcouldtellthemaboutthedata. Whatwouldahighstandarddeviationindicate? Whataboutalowvalueforthestandarddeviation?

14.Usingthedatafromthisproblem,instructstudentsonhowtocalculatethemeanabsolutedeviation,variance,andstandarddeviationintheirgraphingcalculators.

Graphing

calculator

keystrokes

can

be

found

in

the

Technical

Assistance

Document

for

A.9foundontheVDOEWebsiteInstruction>Mathematics>HighSchoolMathematics

InstructionalResources.Theinstructionsincludeanothersetofdatawhichmaybegiven

tostudentsforadditionalpractice. Youmayalsoneedtoaddressthedifferenceinthe

calculatorusingx insteadoftorepresentthearithmeticmean.

Assessment Questions

o Whataresomewaysyoucanmeasurethedispersionofasetofdata?o

Explain

how

to

calculate

each

of

these

measures

without

a

calculator.

Journal/writingpromptso Explainwhythemethodsofmeanabsolutedeviation,variance,andstandard

deviationareusedtomeasuredispersion.

o Compareandcontraststandarddeviationandmeanabsolutedeviation. Other

o Havestudentscollectdatafromtheirclassandcomputestandarddeviationandmeanabsolutedeviation. (Usethegraphingcalculatorifthedatasetislarge. If

calculatingbyhandkeepthenumberofdataelementslessthan10.)

ExtensionsandConnections(forallstudents) Havestudentsremovetheoutlierinthedataandrecomputedthestandarddeviation

andmeanabsolutedeviation. Whichseemstobemoreaffectedbyanoutlier?

StrategiesforDifferentiation Havestudentscreateavisualdiagramtocompareandcontraststandarddeviationand

absolutemeandeviation.

Havestudentscreateagraphicorganizertoarrangethevocabulary,statisticsnotation,and

steps

for

calculating

measures

of

dispersion.

Givestudentswithvisualdiscrimination/processingissuesasheetthatincludestheformulasanddefinitionsinthegrayboxesandexampleproblemsforeachwithdataand

correctnumberspluggedintotheformulas.

Providewarmupproblems(priortothelesson)thatincludesymbolsusedinthevariousstatisticsformulastohavestudentsbecomefamiliarwiththesymboliclanguage. For

example,

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o Evaluatethefollowingif 4,2,6 === zyx . ( )2yx ( )

z

yx2

z

yx

o Evaluatethefollowingif 4,4,9 === x . ( )21 x

1x

1x