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What is Standard
Deviation?
Student Worksheet
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.
!!!!!!! + !!!!!!! + !!!!!!! + !!!!!!!
a) Jamal’smeanbattingscore=
!
Rudi’smeanbattingscore=
!!!!!!!
!!!!!!! + !!!!!!! + !!!!!!! + !!!!!!!
!!!!!!!
=
=
!!!!!!!
= !!!!!!! !!!!!!!
!!!!!!!
!!!!!!!
= !!!!!!! !
b) Whatdothemeanscorestellusabouttheboys’scores?Whatdothemeanscoresnot
tellus abouttheboys’scores?
Bothhav
escoredthesametotalofruns,andthesamenumberofinnings.
Themeanprovidesameasureofthecentreofastatisticalvariable,butnotinformationabouthow
spreadoutthescoresmaybe.Tomeasurethe‘spread’ofthedata,wewilllookathowmucheach
individualscoredeviates(isdifferent)fromthemeanscore.
OntheTI-NspireCAS:
•
•
•
PressHOME-1tocreateaNewDocument,andthenpress
therelevanticontoaddaLists&Spreadsheetpage.
TypescoreasthevariablenameforcolumnA,thenenter
Jamal’sscoresintothiscolumn.
TypedevasthevariablenameforcolumnB,theninthe
cellbelow,typeintheformula=score-mean(score).
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
2
WhatisStandardDeviation?–StudentWorksheet
Nowuseyourspreadsheettoanswerthefollowingquestions.
Question2.
a) Whichscoredeviatesthemostfromthemeanscore?
b) Whichscoreisclosesttothemeanscore?
29
Question3.
a) Apossiblemeasureofthe‘average’deviationisthe‘mean’deviation,whichiscalculated
as
follows(copyandcomplete).
Jamal's!mean!deviation!=
=
Sum!of!deviations
Number!of!scores
!!!! + !!!!! + !!!!! + !!!!!
=
!!!!!
!!!!!
!!!!!
= !!!!!
!
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.
Jamal's!mean!squared!deviation!=
=
!
©
Sum!of!squared!deviations
Number!of!scores
!!!!! + !!!!! + !!!!! + !!!!!
!!!!!
=
!!!!!
!!!!!
= !!!!!
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
3
WhatisStandardDeviation?–StudentWorksheet
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.
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TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
4
WhatisStandardDeviation?–StudentWorksheet
Now,supposeRudiisabouttobatagainforthefifthtime.TrydifferentscoresforRudi’sfifthscore.
Question7.
a) Whatfifthinningsscorewillmakehisstandarddeviationaslowaspossible?
30
b) Whatisthevalueofthisstandarddeviation(correctto2decimalplaces)?
27.59
c) Explainwhythisscorewillachievethesmallestpossiblevalueforthestandarddeviation.
Thenumeratorintheexpressionformeansquareddeviationisunchangedbut
thedenominator
decreasesbyone.Anyotherchangewouldincreasethe
numeratoralso,sothestandard
deviationwouldbelarger.
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,thesizeofeach
deviationmustbethesame.
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;the
smallerstandard
deviation,thesmallerthevariationinthedataset.
TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
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WhatisStandardDeviation?–StudentWorksheet
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,butthe
deviationofeachnew scorefromthenewmeanscoreisunchanged.
Finally,weinvestigatewhathappenstothemeanandstandarddeviationifwemultiplyeachscorebya
givennumber(e.g.×2ordoubling).Whateffectdoesthishaveonthemeanandstandarddeviation?
Useyourspreadsheettemplatetohelpanswerthefollowingquestions.
Question10. WhateffectdoesdoublingeachofJamal’soriginalscoreshaveuponthe
a) themeanscore?
Meanscoreisdoubled.
b) thestandarddeviationofthescore?
Standarddeviationisdoubled
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TexasInstruments2015.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
6
WhatisStandardDeviation?–StudentWorksheet
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