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[ Pobierz całość w formacie PDF ]Numerically
SummarizingData
CHAPTER
Outline
3.1
MeasuresofCentralTendency
3.2
MeasuresofDispersion
3.3
MeasuresofCentralTendencyandDispersion
fromGroupedData
3.4
MeasuresofPosition
3.5
TheFive-NumberSummaryandBoxplots
"
ChapterReview
"
CaseStudy:WhoWas“AMourner”?(OnCD)
DECISIONS
Supposethatyouareinthemarketforausedcar.Tomakean
informeddecisionregardingyourpurchase,youdecidetocollect
asmuchinformationaspossible.Whatinformationisimportantinhelping
youmakethisdecision?SeetheDecisionsprojectonpage164.
PuttingItAllTogether
Whenwelookatadistributionofdata,weshouldconsid-
erthreecharacteristicsofthedistribution:itsshape,its
center,anditsspread.Inthelastchapter,wediscussed
methodsfororganizingrawdataintotablesandgraphs.
Thesegraphs(suchasthehistogram)allowustoidentify
theshapeofthedistribution.Recallthatwedescribethe
shapeofadistributionassymmetric(inparticular,bell
shapedoruniform),skewedright,orskewedleft.
Thecenterandspreadarenumericalsummariesof
thedata.Thecenterofadatasetiscommonlycalledthe
average.Therearemanywaystodescribetheaverage
valueofadistribution.Inaddition,therearemanyways
tomeasurethespreadofadistribution.Themostappro-
priatemeasureofcenterandspreaddependsonthe
shapeofthedistribution.
Oncethesethreecharacteristicsofthedistribution
areknown,wecananalyzethedataforinterestingfea-
tures,includingunusualdatavalues,called
outliers
.
106
Section3.1Measuresof:CentralTendency
107
3.1
MeasuresofCentralTendency
PreparingforThisSection
Beforegettingstarted,reviewthefollowing:
•
Quantitativedata(Section1.1,p.8)
•
Qualitativedata(Section1.1,p.8)
•
Populationversussample(Section1.1,p.4)
•
Simplerandomsampling(Section1.2,pp.16–19)
Objectives
Determinethearithmeticmeanofavariablefromraw
data
Determinethemedianofavariablefromrawdata
Determinethemodeofavariablefromrawdata
Usethemeanandthemediantohelpidentifytheshape
ofadistribution
Ameasureofcentraltendencynumericallydescribestheaverageortypical
datavalueofavariable.Weheartheword
average
inthenewsallthetime:
•Theaveragemilespergallonofgasolineofthe2006ChevroletCamaroin
citydrivingis19miles.
•AccordingtotheU.S.CensusBureau,thenationalaveragecommutetime
toworkin2005was24.3minutes.
•AccordingtotheU.S.CensusBureau,theaveragehouseholdincomein
2003was$43,527.
•TheaverageAmericanwomanis tallandweighs142pounds.
CAUTION
Wheneveryouheartheword
average
,beawarethatthewordmay
notalwaysbereferringtothemean.
Oneaveragecouldbeusedtosupport
oneposition,whileanotheraverage
couldbeusedtosupportadifferent
position.
Inthischapter,wediscussthreemeasuresofcentraltendency:the
mean
,
the
median
,andthe
mode
.Whileothermeasuresofcentraltendencyexist,
thesethreearethemostwidelyused.Whentheword
average
isusedinthe
media(newspapers,reporters,andsoon)itusuallyreferstothemean.Butbe-
ware!Somereportersusetheterm
average
torefertothemedianormode.As
weshallsee,thesethreemeasuresofcentraltendencycangiveverydifferent
results!
Beforewediscussmeasuresofcentraltendency,wemustconsiderwhether
ornotwearecomputingameasureofcentraltendencythatdescribesapopula-
tionoronethatdescribesasample.
Definitions
A
parameter
isadescriptivemeasureofapopulation.
A
statistic
isadescriptivemeasureofasample.
InOtherWords
Tohelpyourememberthedifference
betweenaparameterandastatistic,
thinkofthefollowing:
Forexample,ifwedeterminetheaveragetestscorefor
all
thestudentsina
statisticsclass,ourpopulation,theaverageisaparameter.Ifwecomputetheaver-
agebasedonasimplerandomsampleoffivestudents,theaverageisastatistic.
parameter =
population
DeterminetheArithmeticMeanofaVariable
fromRawData
Whenusedineverydaylanguage,theword
average
oftenstandsforthearith-
meticmean.Tocomputethearithmeticmeanofasetofdata,thedatamustbe
quantitative.
statistic=
sample
Definitions
The
arithmeticmean
ofavariableiscomputedbydeterminingthesum
ofallthevaluesofthevariableinthedataset,dividedbythenumberofob-
servations.The
populationarithmeticmean
,(pronounced“mew”),
iscomputedusingalltheindividualsinapopulation.Thepopulationmeanis
aparameter.The
samplearithmeticmean
,(pronounced“x-bar”),is
computedusingsampledata.Thesamplemeanisastatistic.
108
Chapter3NumericallySummarizingData
Whileothertypesofmeansexist(seeProblems51and52),thearithmetic
meanisgenerallyreferredtoasthe
mean
.Wewillfollowthispracticeforthere-
mainderofthetext.
Instatistics,Greeklettersareusedtorepresentparameters,andRomanlet-
tersareusedtorepresentstatistics.Statisticiansusemathematicalexpressions
todescribethemethodforcomputingmeans.
Definitions
If arethe
N
observationsofavariablefromapopulation,
thenthepopulationmean, is
1
,
x
Á
,
x
(1)
InOtherWords
Tofindthemeanofasetofdata,add
upalltheobservationsanddividebythe
numberofobservations.
If are
n
observationsofavariablefromasample,thenthe
samplemean, is
1
,
x
Á
,
x
(2)
Notethat
N
representsthesizeofthepopulation,while
n
representsthe
sizeofthesample.Thesymbol (theGreeklettercapitalsigma)tellsusthe
termsaretobeadded.Thesubscript
i
isusedtomakethevariousvaluesdistinct
anddoesnotserveasamathematicaloperation.Forexample,
isthefirstdata
value, isthesecond,andsoon.
Let’slookatanexampletohelpdistinguishthepopulationmeanandsam-
plemean.
EXAMPLE1
ComputingaPopulationMeanandaSampleMean
Problem
:
ThedatainTable1representthefirstexamscoreof10students
enrolledinasectionofIntroductoryStatistics.
Table1
Student Score
1.Michelle 82
2.Ryanne 77
3.Bilal 90
4.Pam 71
5.Jennifer 62
6.Dave 68
7.Joel 74
8.Sam 84
9.Justine
(a)
Computethepopulationmean.
(b)
Findasimplerandomsampleofsize students.
(c)
Computethesamplemeanofthesampleobtainedinpart(b).
Approach
(a)
Tocomputethepopulationmean,weaddupallthedatavalues(testscores)
andthendividebythenumberofindividualsinthepopulation.
(b)
RecallfromSection1.2thatwecanuseeitherTableIinAppendixA,acal-
culatorwitharandom-numbergenerator,orcomputersoftwaretoobtain
simplerandomsamples.WewilluseaTI-84Plusgraphingcalculator.
(c)
Thesamplemeanisfoundbyaddingthedatavaluesthatcorrespondtothe
individualsselectedinthesampleandthendividingby thesamplesize.
94
=
4,
10.Juan
88
Solution
(a)
Wecomputethepopulationmeanbyaddingthescoresofall10students:
3
+
Á
+ x
10
a
x
i
= x
1
+ x
2
+ x
=
82
+
77
+
90
+
71
+
62
+
68
+
74
+
84
+
94
+
88
=
790
Dividethisresultby10,thenumberofstudentsintheclass.
=
790
10
=
79
Section3.1MeasuresofCentralTendency
109
Althoughitwasnotnecessaryinthisproblem,wewillagreetoroundthe
meantoonemoredecimalplacethanthatintherawdata.
(b)
Tofindasimplerandomsampleofsize fromapopulationwhosesize
is wewillusetheTI-84Plusrandom-numbergeneratorwithaseed
of54.(Recallthatthisgivesthestartingpointthatthecalculatorusesto
generatethelistofrandomnumbers.)Figure1showsthestudentsinthe
sample.Bilal(90),Ryanne(77),Pam(71),andMichelle(82)areinthesample.
(c)
Wecomputethesamplemeanbyfirstaddingthescoresoftheindividuals
inthesample.
=
10,
Figure1
=
90
+
77
+
71
+
82
=
320
Dividethisresultby4,thenumberofindividualsinthesample.
=
320
4
=
80
NowWorkProblem25.
In-ClassActivity:PopulationMeanversusSampleMean
Treatthestudentsintheclassasapopulation.Allthestudentsintheclassshould
determinetheirpulserates.
(a)Computethepopulationmeanpulserate.
(b)Obtainasimplerandomsampleof studentsandcomputethesample
mean.Doesthesamplemeanequalthepopulationmean?
(c)Obtainasecondsimplerandomsampleof studentsandcomputethe
samplemean.Doesthesamplemeanequalthepopulationmean?
(d)Arethesamplemeansthesame?Why?
Itishelpfultothinkofthemeanofadatasetasthecenterofgravity.In
otherwords,themeanisthevaluesuchthatahistogramofthedataisperfect-
lybalanced,withequalweightoneachsideofthemean.Figure2showsa
histogramofthedatainTable1withthemeanlabeled.Thehistogrambal-
ancesat
m
=
79.
Figure2
ScoresonFirstExam
3
2
60 65 70 75 80 85 90 95100
m
$
79
Score
110
Chapter3NumericallySummarizingData
In-ClassActivity:TheMeanastheCenterofGravity
Findayardstick,afulcrum,andthreeobjectsofequalweight(maybe1-kilogram
weightsfromthephysicsdepartment).Placethefulcrumat18inchessothatthe
yardstickbalanceslikeateeter-totter.Nowplaceoneweightontheyardstickat12
inches,anotherat15inches,andthethirdat27inches.SeeFigure3.
Figure3
12
15
18
27
Doestheyardstickbalance?Nowcomputethemeanofthelocationofthethree
weights.Comparethisresultwiththelocationofthefulcrum.Concludethatthe
meanisthecenterofgravityofthedataset.
DeterminetheMedianofaVariablefromRaw
Data
Asecondmeasureofcentraltendencyisthemedian.Tocomputethemedianof
asetofdata,thedatamustbequantitative.
Definition
The
median
ofavariableisthevaluethatliesinthemiddleofthedata
whenarrangedinascendingorder.Thatis,halfthedataarebelowtheme-
dianandhalfthedataareabovethemedian.Weuse
M
torepresentthe
median.
InOtherWords
Tohelpremembertheideabehindthe
median,thinkofthemedianofahighway;
itdividesthehighwayinhalf.
Tocomputethemedianofasetofdata,weusethefollowingsteps:
StepsinComputingtheMedianofaDataSet
Step1
:
Arrangethedatainascendingorder.
Step2
:
Determinethenumberofobservations,
n
.
Step3
:
Determinetheobservationinthemiddleofthedataset.
•Ifthenumberofobservationsisodd,thenthemedianisthedatavalue
thatisexactlyinthemiddleofthedataset.Thatis,themedianisthe
observationthatliesinthe position.
•Ifthenumberofobservationsiseven,thenthemedianisthemeanof
thetwomiddleobservationsinthedataset.Thatis,themedianisthe
meanofthedatavaluesoneithersideoftheobservationthatliesinthe
position.
EXAMPLE2
ComputingtheMedianofaDataSetwithanOdd
NumberofObservations
Problem
:
ThedatainTable2representthelength(inseconds)ofarandom
sampleofsongsreleasedinthe1970s.Findthemedianlengthofthesongs.
Approach
:
Wewillfollowthestepslistedabove.
Solution
Step1
:
Arrangethedatainascendingorder:
179,201,206,208,217,222,240,257,284
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