The basic practice of statistics

152 Pages·2003·1.68 MB·english

by  David S. Moore

W. H. Freeman Publishers - The Basic Practi... http://www.whfreeman.com/highschool/book.as...
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The Basic Practice of Statistics
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David S. Moore (Purdue U.) Go To Companion Site

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Exploring Data: Variables and Distributions
Companion Site
Chapter 1 - Picturing Distributions with Graphs (CH 01.pdf; 300KB)
Chapter 2 - Describing Distributions with Numbers (CH 02.pdf; 212KB)
Summary
Features Chapter 3 - Normal Distributions (CH 03.pdf; 328KB)
New to This
Edition Exploring Data: Relationships
Media Chapter 4 - Scatterplots and Correlation (CH 04.pdf; 300KB)
Supplements Chapter 5 - Regression (CH 05.pdf; 212KB)
Table of
Chapter 6 - Two-Way Tables (CH 06.pdf; 328KB)
Contents
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Exploring Data
T
hefirststepinunderstandingdataistohearwhatthedatasay,to“let
the statistics speak for themselves.” But numbers speak clearly only
whenwehelpthemspeakbyorganizing,displaying,summarizing,and
asking questions. That’s data analysis. The six chapters in Part I present the
ideas and tools of statistical data analysis. They equip you with skills that are
immediatelyusefulwheneveryoudealwithnumbers.
Thesechaptersreflectthestrongemphasisonexploringdatathatcharacter-
izesmodernstatistics.Althoughcarefulexplorationofdataisessentialifweare
totrusttheresultsofinference,dataanalysisisn’tjustpreparationforinference.
Tothinkaboutinference,wecarefullydistinguishbetweenthedataweactually
haveandthelargeruniversewewantconclusionsabout.TheBureauofLabor
Statistics, for example, has data about employment in the 55,000 households
contactedbyitsCurrentPopulationSurvey.Thebureauwantstodrawconclu-
sionsaboutemploymentinall110millionU.S.households.That’sacomplex
problem. From the viewpoint of data analysis, things are simpler. We want to
exploreandunderstandonlythedatainhand.Thedistinctionsthatinference
requires don’t concern us in Chapters 1 to 6. What does concern us is a sys-
tematicstrategyforexaminingdataandthetoolsthatweusetocarryoutthat
strategy.
Partofthatstrategyistofirstlookatonethingatatimeandthenatrelation-
ships.InChapters1,2,and3youwillstudyvariablesand theirdistributions.
Chapters4,5,and6concernrelationships among variables.
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PART
I
EXPLORING DATA: VARIABLES AND DISTRIBUTIONS
Chapter 1 Picturing Distributions with Graphs
Chapter 2 Describing Distributions with Numbers
Chapter 3 The Normal Distributions
EXPLORING DATA: RELATIONSHIPS
Chapter 4 Scatterplots and Correlation
Chapter 5 Regression
Chapter 6 Two-Way Tables
EXPLORING DATA REVIEW
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CHAPTER
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Picturing Distributions Inthischapterwecover...
Individualsandvariables
with Graphs Categoricalvariables:
piechartsandbargraphs
Quantitativevariables:
histograms
Interpretinghistograms
Quantitativevariables:
Statistics is the science of data. The volume of data available to us is over-
stemplots
whelming.EachMarch,forexample,theCensusBureaucollectseconomicand
Timeplots
employmentdatafrommorethan200,000people.Fromthebureau’sWebsite
youcanchoosetoexaminemorethan300itemsofdataforeachperson(and
more for households): child care assistance, child care support, hours worked,
usual weekly earnings, and much more. The first step in dealing with such a
floodofdataistoorganizeourthinkingaboutdata.
Individualsandvariables
Anysetofdatacontainsinformationaboutsomegroupofindividuals.Thein-
formationisorganizedinvariables.
INDIVIDUALSANDVARIABLES
Individualsaretheobjectsdescribedbyasetofdata.Individualsmaybe
people,buttheymayalsobeanimalsorthings.
Avariableisanycharacteristicofanindividual.Avariablecantake
differentvaluesfordifferentindividuals.
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4 CHAPTER1 PicturingDistributionswithGraphs
A college’s student data base, for example, includes data about every cur-
rentlyenrolledstudent.Thestudentsaretheindividualsdescribedbythedata
set. For each individual, the data contain the values of variables such as date
ofbirth,gender(femaleormale),choiceofmajor,andgradepointaverage.In
practice,anysetofdataisaccompaniedbybackgroundinformationthathelps
usunderstandthedata.Whenyouplanastatisticalstudyorexploredatafrom
someoneelse’swork,askyourselfthefollowingquestions:
1. Who? What individuals do the data describe? How many individuals
appearinthedata?
2. What? How many variables do the data contain? What are the exact
definitions of these variables? In what units of measurement is each
variable recorded? Weights, for example, might be recorded in pounds,
Aredataartistic? inthousandsofpounds,orinkilograms.
DavidGalenson,aneconomist 3. Why? What purpose do the data have? Do we hope to answer some
attheUniversityofChicago, specific questions? Do we want to draw conclusions about individuals
usesdataandstatistical otherthantheonesweactuallyhavedatafor?Arethevariablessuitable
analysistostudyinnovation
fortheintendedpurpose?
amongpaintersfromthe
nineteenthcenturytothe Somevariables,likegenderandcollegemajor,simplyplaceindividualsinto
present.Economicsjournals categories.Others,likeheightandgradepointaverage,takenumericalvalues
publishhiswork.Arthistory forwhichwecandoarithmetic.Itmakessensetogiveanaverageincomefora
journalssenditback
company’semployees,butitdoesnotmakesensetogivean“average”gender.
unread.“Fundamentally
We can, however, count the numbers of female and male employees and do
antagonistictotheway
humanistsdotheirwork,”said arithmeticwiththesecounts.
thechairofarthistoryat
Chicago.Ifyouareastudentof
thehumanities,readingthis CATEGORICALANDQUANTITATIVEVARIABLES
statisticstextmayhelpyou
startanewwaveinyourfield. Acategorical variableplacesanindividualintooneofseveralgroupsor
categories.
Aquantitative variabletakesnumericalvaluesforwhicharithmetic
operationssuchasaddingandaveragingmakesense.
Thedistributionofavariabletellsuswhatvaluesittakesandhowoften
ittakesthesevalues.
EXAMPLE1.1 Aprofessor’sdataset
Hereispartofthedatasetinwhichaprofessorrecordsinformationaboutstudent
performanceinacourse:
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Individualsandvariables 5
Theindividualsdescribedarethestudents.Eachrowrecordsdataononeindividual.
Eachcolumncontainsthevaluesofonevariableforalltheindividuals.Inaddition
tothestudent’sname,thereare7variables.Schoolandmajorarecategoricalvari-
ables. Scores on homework, the midterm, and the final exam and the total score
arequantitative.Gradeisrecordedasacategory(A,B,andsoon),buteachgrade
alsocorrespondstoaquantitativescore(A=4,B=3,andsoon)thatisusedto
calculatestudentgradepointaverages.
Most data tables follow this format—each row is an individual, and each col-
umnisavariable.Thisdatasetappearsinaspreadsheetprogramthathasrowsand spreadsheet
columnsreadyforyouruse.Spreadsheetsarecommonlyusedtoenterandtransmit
data and to do simple calculations such as adding homework, midterm, and final
scorestogettotalpoints.
APPLYYOURKNOWLEDGE
1.1 Fuel economy. Hereisasmallpartofadatasetthatdescribesthefuel
economy(inmilespergallon)of2002modelmotorvehicles:
Makeand Vehicle Transmission Numberof City Highway
model type type cylinders MPG MPG
·
·
·
AcuraNSX Two-seater Automatic 6 17 24
AudiA4 Compact Manual 4 22 31
BuickCentury Midsize Automatic 6 20 29
DodgeRam1500 Standardpickuptruck Automatic 8 15 20
·
·
·
(a) Whataretheindividualsinthisdataset?
(b) Foreachindividual,whatvariablesaregiven?Whichofthese
variablesarecategoricalandwhicharequantitative?
1.2 A medical study. Datafromamedicalstudycontainvaluesofmany
variablesforeachofthepeoplewhowerethesubjectsofthestudy.
Whichofthefollowingvariablesarecategoricalandwhichare
quantitative?
(a) Gender(femaleormale)
(b) Age(years)
(c) Race(Asian,black,white,orother)
(d) Smoker(yesorno)
(e) Systolicbloodpressure(millimetersofmercury)
(f) Levelofcalciumintheblood(microgramspermilliliter)
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6 CHAPTER1 PicturingDistributionswithGraphs
Categoricalvariables:piechartsandbargraphs
Statisticaltoolsandideashelpusexaminedatainordertodescribetheirmain
exploratory data analysis features.Thisexaminationiscalledexploratorydataanalysis.Likeanexplorer
crossingunknownlands,wewantfirsttosimplydescribewhatwesee.Hereare
twobasicstrategiesthathelpusorganizeourexplorationofasetofdata:
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Beginbyexaminingeachvariablebyitself.Thenmoveontostudythe
relationshipsamongthevariables.
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Beginwithagraphorgraphs.Thenaddnumericalsummariesofspecific
aspectsofthedata.
We will follow these principles in organizing our learning. Chapters 1 to 3
presentmethodsfordescribingasinglevariable.Westudyrelationshipsamong
severalvariablesinChapters4to6.Ineachcase,webeginwithgraphicaldis-
plays,thenaddnumericalsummariesformorecompletedescription.
Theproperchoiceofgraphdependsonthenatureofthevariable.Theval-
ues of a categorical variable are labels for the categories, such as “male”and
“female.” The distribution of a categorical variable lists the categories and
giveseitherthecountorthepercentofindividualswhofallineachcategory.
EXAMPLE1.2 Garbage
Theformalnameforgarbageis“municipalsolidwaste.”Hereisabreakdownofthe
materialsthatmadeupAmericanmunicipalsolidwastein2000.1
Weight
Material (milliontons) Percentoftotal
Foodscraps 25.9 11.2%
Glass 12.8 5.5%
Metals 18.0 7.8%
Paper,paperboard 86.7 37.4%
Plastics 24.7 10.7%
Rubber,leather,textiles 15.8 6.8%
Wood 12.7 5.5%
Yardtrimmings 27.7 11.9%
Other 7.5 3.2%
Total 231.9 100.0
It’s a good idea to check data for consistency. The weights of the nine materials
addto231.8milliontons,notexactlyequaltothetotalof231.9milliontonsgiven
roundoff error inthetable.Whathappened?Roundofferror:Eachentryisroundedtothenearest
tenth,andthetotalisroundedseparately.Theexactvalueswouldaddexactly,but
theroundedvaluesdon’tquite.
pie chart The pie chart in Figure 1.1 shows us each material as a part of the whole.
For example, the “plastics”slice makes up 10.7% of the pie because 10.7% of
municipal solid waste consists of plastics. The graph shows more clearly than
the numbers the predominance of paper and the importance of food scraps,
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Categoricalvariables:piechartsandbargraphs 7
Figure1.1 Piechartof Metals
materialsinmunicipalsolid
Glass
waste,byweight.
Paper
Food scraps
Other
Yard trimmings
Plastics
Wood
Rubber, leather, textiles
plastics,andyardtrimmingsinourgarbage.Piechartsareawkwardtomakeby
hand,butsoftwarewilldothejobforyou.
We could also make a bar graph that represents each material’s weight by bar graph
the height of a bar. To make a pie chart, you must include all the categories
thatmakeupawhole.Bargraphsaremoreflexible.Figure1.2(a)isabargraph
ofthepercentofeachmaterialthatwasrecycledorcompostedin2000.These
percentsarenotpartofawholebecauseeachreferstoadifferentmaterial.We
couldreplacethepiechartinFigure1.1byabargraph,butwecan’tmakeapie
charttoreplaceFigure1.2(a).Wecanoftenimproveabargraphbychanging
theorderofthegroupswearecomparing.Figure1.2(b)displaystherecycling
datawiththematerialsinorderofpercentrecycledorcomposted.Figures1.1
and1.2togethersuggestthatwemightpaymoreattentiontorecyclingplastics.
Bar graphs and pie charts help an audience grasp the distribution quickly.
Theyare,however,oflimitedusefordataanalysisbecauseitiseasytounder-
stand data on a single categorical variable without a graph. We will move on
toquantitativevariables,wheregraphsareessentialtools.
APPLYYOURKNOWLEDGE
1.3 The color of your car. Hereisabreakdownofthemostpopularcolors
forvehiclesmadeinNorthAmericaduringthe2001modelyear:2
Color Percent Color Percent
Silver 21.0% Mediumred 6.9%
White 15.6% Brown 5.6%
Black 11.2% Gold 4.5%
Blue 9.9% Brightred 4.3%
Green 7.6% Grey 2.0%
(a) Whatpercentofvehiclesaresomeothercolor?
(b) Makeabargraphofthecolordata.Woulditbecorrecttomakea
piechartifyouaddedan“Other”category?
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8 CHAPTER1 PicturingDistributionswithGraphs
The height of this bar is 45.4
0
because 45.4% of paper 6
municipal waste was recycled.
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Food Glass Metals Paper Plastics Textiles Wood Yard Other
Material
(a)
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Yard Paper Metals Glass Textiles Other Plastics Wood Food
Material
(b)
Figure1.2 Bargraphscomparingthepercentsofeachmaterialinmunicipalsolid
wastethatwererecycledorcomposted.