Download 01 What is standard deviation teacher notes and answers

What is Standard
Deviation?
Teacher Notes and
Answers
789101112
TI-Nspire
Investigation
Student
30min
Introduction
JamalandRudiplayforalocalcricketclub.Theirfirstfourlotsofscoresfortheseasonare:
Jamal:29,35,29and27
Rudi:7,12,18,and83
Thisexplorationcomparesthescoresforeachcricketerandlooksatthewaystheirscoresaresimilarand
thewaysinwhichtheyaredifferent.Italsoexplorestheusefulnessofanewstatisticalmeasure,the
standarddeviation.
Part 1: Exploring deviation
Question1.
Copyandcompletethefollowingtocalculatethemeanscoreforeachboy.
!29! + !35! + !29! + !27!
a) Jamal’smeanbattingscore=
!
Rudi’smeanbattingscore=
!!!4!
!7!! + !12! + !18! + !83!
!!!4!
=
=
!120
!4!!!!
!120
!!!4!!
= !30! = !30! !
b) Whatdothemeanscorestellusabouttheboys’scores?Whatdothemeanscoresnot
tellus abouttheboys’scores?
Bothhavescoredthesametotalofruns,andthesamenumberofinnings.However,it
doesnottellyouhowconsistentthebatsmenwereduringthefourinnings.
Themeanprovidesameasureofthecentreofastatisticalvariable,butnotinformationabouthow
spreadoutthescoresmaybe.Tomeasurethe‘spread’ofthedata,wewilllookathowmucheach
individualscoredeviates(isdifferent)fromthemeanscore.
©
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
2
WhatisStandardDeviation?–TeacherNotesandAnswers
OntheTI-NspireCAS:
•
•
•
PressHOME-1tocreateaNewDocument,andthenpress
therelevanticontoaddaLists&Spreadsheetpage.
TypescoreasthevariablenameforcolumnA,thenenter
Jamal’sscoresintothiscolumn.
TypedevasthevariablenameforcolumnB,theninthe
cellbelow,typeintheformula=score-mean(score).
Nowuseyourspreadsheettoanswerthefollowingquestions.
Question2.
a) Whichscoredeviatesthemostfromthemeanscore?
35
b) Whichscoreisclosesttothemeanscore?
29
Question3.
a) Apossiblemeasureofthe‘average’deviationisthe‘mean’deviation,whichiscalculated
as
follows(copyandcomplete).
Jamal's!mean!deviation!=
=
Sum!of!deviations
Number!of!scores
!71! + !!5! + !71! + !73!
!!!4!
=
!0!!
!!4!
= !0!
!
b) Isthisausefulmeasureofthe‘average’deviation?Why/whynot?
Thepositiveandnegativedeviationscancelout.Themeanscorewillalwaysbe
thevaluefor whichthiscancellingoccurs.Hencethemeandeviationisnotauseful
measureoftheaverage
amountofdeviation.
Anotherpossiblewayofmeasuringthe‘average’deviationistosquareeachdeviationfirst,findthemean
ofthesquareddeviations,andtofindthesquarerootofthis.
OntheTI-NspireCAS
•
•
TypesqdevasthevariablenameforcolumnC
Inthecellbelow,typeintheformula=dev2.
Notethatallsquareddeviationsarepositive.
Themean‘squared’deviationofJamal’sscorescanbecalculatedinthefollowingmanner.
Question4.
Copyandcompletethefollowing,givingyouranswerinsimplestfractionform.
©
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
3
WhatisStandardDeviation?–TeacherNotesandAnswers
Jamal's!mean!squared!deviation!=
=
Sum!of!squared!deviations
Number!of!scores
!1! + !!25! + !1! + !9!
!
!!!4!
=
!36
!!4!
= !9!
Tocalculatethemean‘squared’deviationofJamal’sscores,
dothefollowing.ontheTI-NspireCAS
•
MovetocellD1,andtypethelabelmeansqdev.
•
MovetocellD2,andtypetheformula
=approx(mean(sqdev)).
Confirmthatitgivesthesameanswerforthemean‘squared’
deviationastheoneyoufoundabove.
Finally,tofindthe‘average’deviationbythismethod,findthesquarerootofthevalueoftheaverage
squareddeviation.
Nowwehaveastandardisedmeasureofthemeandeviation(referredtoasthestandarddeviation),
whichignoreswhethertheindividualdeviationsarepositiveornegative.
TocalculatethestandarddeviationofJamal’sscoresonthe
TI-NspireCAS
•
•
MovetocellD3,andtypethelabelstdev.
MovetocellD4,andtypetheformula=sqrt(D2)
NowletslookatthespreadofRudi’sbattingscores(7,12,18and83runs).
Onthespreadsheet,changethevaluesinColumnAtoRudi’sscores.
Question5.
Forthesenewscores,find(correctto2decimalplaces)
a) themeansquareddeviation
951.5
b) thestandarddeviation
30.8
Question6.
ComparethestandarddeviationofRudi’sscoreswiththestandarddeviationofJamal’sscoresfound
previously.Whatdoyounotice?Whatdoesthissuggest?
ThestandarddeviationofRudi’sscoresismuchgreater,indicatingthathisscoresare,onaverage,
furtherfromthemeanscore.
©
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
4
WhatisStandardDeviation?–TeacherNotesandAnswers
Now,supposeRudiisabouttobatagainforthefifthtime.TrydifferentscoresforRudi’sfifthscore.
Question7.
a) Whatfifthinningsscorewillmakehisstandarddeviationaslowaspossible?
30
b) Whatisthevalueofthisstandarddeviation(correctto2decimalplaces)?
27.59
c) Explainwhythisscorewillachievethesmallestpossiblevalueforthestandarddeviation.
Thenumeratorintheexpressionformeansquareddeviationisunchangedbutthe
denominatordecreasesbyone.Anyotherchangewouldincreasethenumeratoralso,so
thestandarddeviationwouldbelarger.
Part 2: Exploring the standard deviation
Wenowlookatsomeotherdatatoexploretheideaofstandarddeviation,andhowitisaffectedbythe
spreadofthedatavalues.
UsingtheTI-NspireCASspreadsheettemplateyouhavedeveloped,answerthefollowingquestions.
Question8.
a)
b)
c)
d)
e)
f)
g)
©
Find4battingscoresthathaveameanscoreof30andastandarddeviationof0.
30,30,30,30
Find4battingscoresthathaveameanscoreof30andastandarddeviationof10.
20,20,40,40
Find4battingscoresthathaveameanscoreof30andastandarddeviationof20.
10,10,50,50
Explainthemethodyouusedtoarriveatyouranswersabove.
Thesumofthescoresmustbe120(forsamemeanscore).Also,thesizeofeachdeviation
mustbethesame.
Find4battingscores(between0and100inclusive)thathavethelargestpossible
standarddeviation.
0,0,100,100;SD=50
Find4battingscores(between0and100inclusive)thathavethesmallestpossible
standarddeviation.
Any4identicalscores(e.g.23,23,23,23)SD=0
Thinkaboutyouranswerstoquestionseandf.Inyourownwordsexplainhowthe
standarddeviationisrelatedtothespreadofadataset.
Thelargerthestandarddeviation,thegreaterthevariationinthedataset;thesmaller
standarddeviation,thesmallerthevariationinthedataset.
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
5
WhatisStandardDeviation?–TeacherNotesandAnswers
Part 3: Changing the data
Considertheoriginal4scoresforJamal.Ifeachscorehadbeen20runsmore,itisclearthathismean
score(his‘battingaverage’)wouldbehigher,butbyhowmuch?Also,howwouldsuchanincreaseineach
scoreaffectthevalueofthestandarddeviation?
Todothiswiththeaidofthespreadsheettemplate,wewill
addaformulaincellD5thatcalculatesthemeanscore.
•
•
MovetocellD5.
Typetheformula=approx(mean(score).
Nowuseyourspreadsheettemplatetohelpanswerthesequestions.
Question9.
Whateffectdoesadding20runstoeachscorehaveupon
a) themeanscore?
Meanscoreincreasesby20.
b) thestandarddeviationofthescore?
Standarddeviationisunchanged
c) Tryrepeatingthisbyaddinganother20runstoeachscore.
Whatdoyounoticeaboutthevalueinthemeanandthestandarddeviationnow?
Meanscoreincreasesby20againandstandarddeviationisstillunchanged.
d) Whydoyouthinkthishappens?
Allscores(includingmeanscore)areincreasedbyafixedamount,butthedeviationof
eachnewscorefromthenewmeanscoreisunchanged.
Finally,weinvestigatewhathappenstothemeanandstandarddeviationifwemultiplyeachscorebya
givennumber(e.g.×2ordoubling).Whateffectdoesthishaveonthemeanandstandarddeviation?
Useyourspreadsheettemplatetohelpanswerthefollowingquestions.
Question10. WhateffectdoesdoublingeachofJamal’soriginalscoreshaveuponthe
a) themeanscore?
Meanscoreisdoubled.
b) thestandarddeviationofthescore?
Standarddeviationisdoubled
©
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
6
WhatisStandardDeviation?–TeacherNotesandAnswers
Experimentwithsomeothervalues(e.g.triplingeachscore),andthencopyandcompletethefollowing
table.
Originalscores
Originalscoresx2
Originalscoresx3
Originalscoresx0.5
Score1
29
58
87
14.5
Score2
35
70
108
27.5
Score3
29
58
87
14.5
Score4
27
54
81
13.5
Mean
30
60
90
15
Standarddeviation
3
6
9
1.5
Question11. Inyourownwords,summarisewhathappenstothevaluesofthemeanandstandarddeviationwhen
eachscoreismultipliedbyaconstantfactor.Suggestareasonwhythismighthappen.
Boththemeanandthestandarddeviationarealsomultipliedbythatconstantfactor.Thestandard
deviationincreasesbythefactor,sincethedifferenceofeachscorefromthemeanincreasesbythat
factor.
Challenge
Whatwouldhappentothevaluesofthemeanandstandarddeviationifyouweretodoubleeachscore
andthenadd10?TrytodothisbyusingtheresultsfromPart3,andthenchecktoseeifyouarecorrect.
Generaliseyourresult:ifxrepresentsthescores,thenwhatwillhappentothemeanandstandard
deviationifeachscoreismultipliedbyaandthenbisadded(i.e.changextoax+b)?
©
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
7
WhatisStandardDeviation?–TeacherNotesandAnswers
Teacher notes
•
•
•
•
•
•
Inthistask,theconstructionofthestandarddeviationasameasureofspreadisintroduced.The
focusisonitsuseasadescriptivemeasureofspread,andsotheformulausedis
Standarddeviation=
∑(x − x )
n
2
InPart1,theconceptofdeviationfromthemean,andtheproblemofusingsigneddeviationsto
calculateameasureofthespreadareintroduced.Thedevelopmentoftheideaofanunsigned
deviation(bysquaringandthentakingthepositivesquarerootofthedeviations)isintroducedasa
moreusefulmeasureofspread.
Studentsmaynothavebeenexposedtotheformalnotationofastatisticalvariable,andsothiscould
bediscussed:‘Weareinterestedinthecentralvalueofthevariable,andhowwellspreadoutarethe
valuesofthisvariable.’
Notetheuseofthe“approx.”command.ThisisusedtopreventtheTI-NspireCASgivingexact
answersinfractionalformforthemeansquaredvariationandthestandarddeviation,whichmight
obscuretheideaforsomestudents.
Part2exploreshowthefourvaluescouldbechosentoachieveaparticularvalueofastandard
deviation.Thisisfreeexploration—itwouldbeagoodoutcomeifthestudentscanrecognisehowthe
standarddeviationisaffectedbyvaluesthatareverycloseoridentical,aswellasvaluesthatarewell
spreadout.
Part3highlightstheinvarianceofthestandarddeviationwhenaconstantamountisaddedtoa
statisticalvariable,butnotwhenthevariableismultipliedbyaconstantfactor.Thisnotionisoften
takenupmoreformallyinseniormathematicscourses.
–SD(X+b)=SD(X)
–SD(aX)=aSD(X)
–SD(aX+b)=aSD(X)
•
Inseniormathematicscourseswherethestandarddeviationofasampleisusedasanestimateofthe
populationstandarddeviation,itisusualtousen–1ratherthanninthedefinition.
©
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan