Chapter 3 Frequency Distributions - Kendall Hunt Publishing · 34 Chapter 3 Frequency Distributions Learning ... 5 How are these terms used to describe the shape of distributions:

Frequency Distributions

3.1 Distributions as Tables

3.2 Distributions as GraphsHistogram • Frequency Polygon

3.3 Eyeball-estimation

3.4 The Shape of DistributionsDescribing Distributions • The Normal Distribution

3.5 Eyeball-calibration

3.6 Bar Graphs of Nominal and Ordinal Variables

3.7 ConnectionsCumulative Review • Computers • Homework Tips

Exercises for Chapter 3

Personal Trainer LECTLET 3A: Frequency Distributions as Tables LECTLET 3B: Frequency Distributions as Graphs LABS: Lab for Chapter 3 REViEwMASTER 3A: RESOuRCE 3A: Stem and Leaf Displays RESOuRCE 3X: Additional Exercises

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Chapter 3: Frequency Distributions from Comprehending Behavioral Statistics by Russell Hurlburt 978-1-4652-0178-2 | 5th Edition | 2012 Copyright

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34 Chapter 3 Frequency Distributions

Learning ObjectivesQ Howisatabularfrequencydistributionconstructed?2 Whatareclassintervalsandwhatroledotheyplayinthedevelopmentofgrouped

frequencydistributions?3 Whataretwographicalmethodsofrepresentinginterval/ratiodatafromgrouped

frequencydistributions?4 Whatstepsareinvolvedineyeball-estimatingfrequencydistributions?5 Howarethesetermsusedtodescribetheshapeofdistributions:unimodal,bimodal,

symmetric,positivelyskewed,negativelyskewed,asymptotic,normal?6 Howisabargraphsimilartoanddifferentfromahistogram?Forwhatkindof

dataiseachused?

There are three main concepts in statistics: Frequency distributions, thesamplingdistributionofthemean,andtheteststatistic.Wefocusinthischapteron frequencydistributionsandhow theyaredisplayed: in tables

(asungroupedandgroupedfrequencydistributions)andingraphs(ashistogramsandfrequencypolygons).Youwill learn toeyeball-estimate frequencydistributionsandterminologyfordescribingtheshapesofdistributions:unimodal,bimodal,symmetric,positivelyandnegativelyskewed,asymptotic,andnormal.

Table3.1showsthegroupedfrequencydistributionsandFigure3.1showsthehis-togramsfortheintellectualgrowthscoresdescribedinChapter1.[YoumayrecallthatRosenthalandJacobson(1968)ledteacherstobelievethatsecond-gradersidentifiedas“bloomers”wouldshowIQspurts,butthe“other”childrenwerenotexpectedtospurt.Actually,thebloomerswerearandomsampleofsecond-graders—nodifferent,onaver-age,fromtheotherchildren.]Thegroupedfrequencydistributions(Table3.1)andhisto-grams(Figure3.1)showthatthemostfrequentlyoccurringIQgainamongthebloomerswasbetween10and20IQpoints,whereasthemostfrequentlyoccurringIQgainamongtheotherchildrenwasbetween0and10IQpoints.Itisalsoeasytoseethatthereiscon-siderableoverlapbetweenthebloomerandotherchildren’sintellectualgrowth.

FrequencyIQ Gain Bloomers Others

61– 70 1 0 51– 60 0 0 41– 50 0 0 31– 40 0 1 21– 30 2 2 11– 20 6 14 1– 10 1 18 –9– 0 2 9–19––10 0 3

12 47

*BasedonastudybyRosenthal&Jacobson(1968)

TaBle 3.1 Grouped frequency distributions of intellectual growth (IQ gain) for Oak School bloomers and other children*



Chapter 3 Frequency Distributions 35

20 –

15 –

10 –

5 –

0 – –20 –10 0 10 20 30 40 50 60 70

– – – – – – – – – –

f

IQ Gain of Bloomers

20 –

15 –

10 –

5 –

0 – –20 –10 0 10 20 30 40 50 60 70

– – – – – – – – – –

f

IQ Gain of Others

FIGURe 3.1 Histograms of intellectual growth (IQ gain) for Oak School bloomers and other children (Based on Rosenthal & Jacobson, 1968)

The break in the X-axis shows that the Y-axis does not intersect at X = 0 as is customary.

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WecouldhavemadethesameobservationsfrominspectingthedatainChapter1,butthefrequencydistributionsthatyouwilllearnaboutinthischaptermakeiteasiertoseethecharacteristicsofadataset.

RecallthatinChapter1wepointedoutthethreemajorconceptsinstatistics:(1)whata“distributionofavariable”isandhowtodescribeit,(2)whata“samplingdistributionofmeans”isandhowitisrelatedtothedistribution

ofavariable,and(3)whata“teststatistic”isandhowitisrelatedtothesamplingdistributionofmeans.Thischapter(andalsoChapters4and5)focusesonthefirstconcept,describingthedistributionsofvariables.Ourfirsttaskistoconvinceour-selvesthatunderstanding frequency distributions will make it simpler for us to think about and communicate about data.

Suppose,forexample,thatweareinterestedintheweightsofmalestudentsinourstatisticsclass.Thereare25menintheclass,andwemeasureeachman’sweight,withtheresultsshowninFigure3.2,whereeachmanisrepresentedbyasquare.




135 180 190 137 154

149 164 185 173 163

162 157 161 173 180

179 197 159 182 164

164 144 152 150 163

FIGURe 3.2 Weights (lb) of male students as they sit in the classroom

TaBle 3.2 Enumeration of male students’ weights (lb)

135 180 190 137 154

149 164 185 173 163

162 157 161 173 180

179 197 159 182 164

164 144 152 150 163

In an enumerated list, you have to hunt for the largest or smallest values.

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NowsupposeourfriendJackaskshowheavythemalestudentsinourclassare.Wecouldsimplylistthedata:“Thefirststudentweighs135pounds,thenextstudentweighs180pounds,thethirdstudentweighs. . . ,andthelaststudentweighs163pounds.”Thiskindoflistingiscalledanenumeration.

Enumerationscanbeprinted,asshowninTable3.2.AnenumerationisaperfectlyaccurateanswertoJack’squestionaboutstudentweights,butitisprobablyanundesir-ableanswerbothbecauseitistoolong(wegiveJackmoreinformationthanhewants)andbecauseitdoesnothighlighttheimportantcharacteristicsofthedistribution(itdoesnot,forexample,makeiteasytodiscoverthelargestorthesmallestweight,ortoseewhichweightsoccurmostfrequently).

Statisticiansmaketheanswers toquestionssuchasJack’smore informativebyusingfrequencydistributionsintheformoftablesorgraphs.

ClickLectletsandthen3AinthePersonal TrainerforanaudiovisualdiscussionofSections3.1and3.2.

3.1 Distributions as TablesA tabular frequency distribution isa table that lists thenumericalvaluesofavari-ableinalogicalorderalongwithfrequencyofeachvalue.Avariable,aswesawinChapter 2,isthatcharacteristicofinterestthatcantakeondifferentvaluesforeach

enumeration Listingallpointsinadataset

tabular frequency distribution Anorderedlistingofallvaluesofavariableandtheirfrequencies

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3.1 Distributions as Tables 37

subjectunderconsideration,andthe frequency(usuallyabbreviated f ) is thenum-berof timesaparticularvalueof thevariableoccurs.Table3.3shows the tabularfrequencydistributionofmalestudents’weights.Thevalue163occurstwice(“hasfrequency2”),asdo173and180,andthevalue164isoccursthreetimes(“hasfre-quency3”).Allotherfrequenciesfortheweightslistedinthetableare1.IfaweightdoesnotoccurinthedataofTable 3.2(forexample,142or181),thenitsfrequencyis0,andweomititentirelyintheTable3.3frequencydistribution.

Notethatthesumofthefrequencies(25inourexample)isshownatthebottomofthefrequencycolumnandisalwaysequaltothenumberofentriesintheoriginaldataset.

Notealsothatthefrequencydistributionshowsthevaluesofthevariableunderconsideration(weight)in order; theconventionistoputthelargestvaluefirst.Thispresentationsimplifiesourcommunicationaboutthedata.NowifJackwantstoknowhowheavy themen inour class are,we can immediately say, “The lightest is 135pounds,thesecondlightestis137pounds,thethirdlightestis144pounds,...,andtheheaviestis197pounds.”Thisisstillalongandcumbersomeanswer,buttheorderingmakesiteasierforJacktogainsomeappreciationofhowtheweightsstandinrelationtoeachother.However,althoughthisfrequencydistributionismoreinformativethanasimpleenumerationoftheindividualweights,ithasdisadvantages:Itstillprovidesa long list of weights and their frequencies, and it is relatively difficult to identifyweightswith0frequencies.

Agrouped frequency distribution(alsocalleda“frequencydistributionusingclassintervals”)ismorecompactandefficient.Successiveweightsaregroupedtogetherintoclassintervalsandthefrequenciesarereportedfortheintervals,notfortheindividualweights,asshowninTable3.4.Notethat thesumofthefrequenciesisstill25, thenumberoforiginalobservations.

Theuseof class intervals (130–139pounds, etc.) generallymakes a frequencydistributioneasiertounderstand.Forexample,itisnowclearthatthemostfrequentweightisinthe160s.

Classintervalsmustfollowtheserules:Eachintervalmustbethesamewidth(inTable3.4,10pounds);theclassintervalsmustbenonoverlappingandconsecutive(forexample,iftheupperlimitofoneintervalis159pounds,thenthelowerlimitofthenextmustbe160pounds);anditisgenerallydesirabletohavebetween6and20class

TaBle 3.3 Frequency distribution of male students’ weights

Weight (Ib) f Weight (lb) f

197 1 161 1190 1 159 1185 1 157 1182 1 154 1180 2 152 1179 1 150 1173 2 149 1164 3 144 1163 2 137 1162 1 135 1

25

In a frequency distribution, largest and smallest values are easy to spot.

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grouped frequency distribution Afrequencydistributionwithadjacentvaluesofthevariablegroupedtogether

frequency Thenumberoftimestheparticularvalueofavariableoccurs

(continues)




intervals(exactlyhowmanyintervalsisamatterofjudgment,decidingwhichnumberofclassintervalspresentsthedatainthemostinformativemanner).

Sixstepsareinvolvedincreatingagroupedfrequencydistribution:

STep 1 Range. Findtherangeofthescores(highestscoreminuslowestscore).

STep 2 Number. Makeapreliminarychoiceofthedesirednumberofclassintervals.

STep 3 Width. Determinetheintervalwidthbydividingtherangebythenumberofclassintervals.Roundtheintervalwidthineitherdirectiontoaconve-nientnumber,evenif thatmeansadjusting thenumberofclass intervalsyouselectedinStep2.

STep 4 Lowest. Determine the lower limit of the lowest interval.This must bechosensothatthelowestdatapointfallssomewhereinthefirstinterval;theintervalsshouldhaveconvenientlimits.

STep 5 Limits. Preparealistofthelimitsofeachclassinterval,beginningatthebottomofthetablewiththelowestscoreandproceedingupward.Besurethattheintervalsareallthesamewidth,areconsecutive,andarenonover-lapping.Besurethatthehighestintervalcontainsthehighestscore.

STep 6 Frequencies. Countthenumberofobservationsthatoccurineachintervalandenterthatcountasthefrequencyoftheinterval.

NotethatthewordconvenientordesiredappearsinSteps2,3,and4.Thisindi-cates that there is no simple rule governing the choices to be made; instead, somejudgmentmustbemadeabouttheintendedcommunicationandtheselectionschosentomaketheclearestcommunication.

Weillustratethisprocedurebypreparinganothergroupedfrequencydistributionforourweightdata.SupposeweinspectTable3.4andwonderwhetherwewouldhaveprovidedmoreinformationifwehadusedmoreclassintervalsthanthesevenshownthere.Wedecidetoprepareagroupedfrequencydistributionusingtwiceasmanyclassintervals,sowewilluse14.Weusethesamesteps.

InStep1,wenotethatthehighestweightis197poundsandthelowestis135pounds,sotherangeis197–135=62.

TaBle 3.4 Grouped frequency distribution of male students’ weights using 10-pound class intervals

Class Interval f

190–199 2180–189 4170–179 3160–169 7150–159 5140–149 2130–139 2

25

Steps for creating a grouped frequency distribution

The largest values of X are at the top of all frequency distributions.

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3.1 Distributions as Tables 39

InStep2,ourpreliminarychoiceofthenumberofclassintervalsthistimeis14.

InStep3,theintervalwidthwillbeapproximatelytherangedividedbythenumberofintervals,or62/14=4.43pounds.Wecouldroundthatto4orto5pounds,butwedecidethat5ismoreconvenientbecausethelowerlimitofeverysecondintervalthenendsin0.Thatmeanswewillhave13insteadof14intervals,butourstatisticaljudgmentisthatusingroundnumbersasintervallimitsmorethanoffsetsthatdisadvantage.

InStep4,thelowestintervalmustcontainthelowestpoint(135),sowecouldchooseanyofthefollowingvaluesasthelowerlimitofthatinterval:131,132,133,134,or135(oranyintermediatedecimalvalueifwedesired,but thatmightimplythatwemeasuredweightstoagreaterprecisionthanwholepounds).Weselect135poundsbecauseitisaroundnumber.Thelowestintervalisthere-fore135–139pounds.Notethatthatintervalis5poundswide(not4asitmightappear)becauseitcontainsthefivevalues135,136,137,138,and139.

InStep5,thefirstintervalis135–139.Thesecondintervalmusthavethesamewidthandbeconsecutive,soitis140–144.WecontinuecreatingintervalsasshowninTable3.5,endingwiththehighestinterval(195–199),whichincludesthehighestpoint(197pounds).

InStep6,werefertoouroriginalfrequencydistribution(Table3.3)andcountthenumberofindividualswhoseweightsfallintothelistedclasses.Twoindividu-als(135and137)haveweightsinthefirstclass(135–139),forexample,sothefrequencyofthatclassis2.Weenterthosefrequenciesintheright-handcolumnofTable3.5,andthegroupedfrequencydistributionusing5-poundintervalsiscompleted.Doublechecking,weadddownthelistoffrequencies;thesummuststillbe25,thetotalnumberofdatapoints.

Note thatagrouped frequencydistribution includesclass intervalseven if theyhave0frequency(forexample, theinterval165–169is includedinTable3.5).Thisisincontrasttoanungroupedfrequencydistribution(forexample,Table3.3),wherevalueswith0frequencyareomitted.

TaBle 3.5 Grouped frequency distribution of male students’ weights using 5-pound class intervals

Class Interval f

195–199 1190–194 1185–189 1180–184 3175–179 1170–174 2165–169 0160–164 7155–159 2150–154 3145–149 1140–144 1135–139 2

25




Choosing between these two grouped frequency distributions, the one with10-poundintervalsandtheonewith5-poundintervals,isamatterofjudgment.Ifwewishtoconveythatthefrequencyofweightsrisestoamaximuminthe160-poundrange and then decreases again, we would probably prefer the 10-pound intervals(Table3.4).Ontheotherhand,ifwewishtodemonstratethatnoindividualsweighbetween165and170pounds,wewouldpreferthe5-poundintervals(Table3.5).Thechoiceisamatterofjudgmentandcommunication.

Note that some information is lost in agrouped frequencydistribution suchasTable3.4orTable3.5.AlthoughTable3.4tellsusthatfiveweightsarebetween150and159,wehavenowayofknowingwhetherthosefiveindividualsallweigh150orallweigh159orarespreadoverthatrange.Tables3.4and3.5haveimprovedtheclar-ityandeaseofcomprehendingthedataincomparisonwithTable3.3,buttheyhavesacrificedsomedetailinsodoing.

ClickLectletsandthen3BinthePersonal TrainerforanaudiovisualdiscussionofSections3.2through3.6.

3.2 Distributions as GraphsHistogramThe grouped frequency distributions we have shown as tables can also be repre-sentedgraphicallybyhistogramsorfrequencypolygons.Ahistogramisaplotoftheclassintervalsofthevariable(weight,inourexample)onthehorizontalaxis(some-timescalledtheX-axisorabscissa)andthefrequencyofeachintervalontheverticalaxis(Y-axisorordinate)ofagraph.Figure3.3showsthehistogramobtainedfromTable 3.4,whichused10-poundclassintervals.Notethateachinterval’sfrequencyisdisplayedasarectangleor“bar,”andthereisnospace(onlyasingleverticalline)separatingthebars.

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histogram Agraphicalpresentationofagroupedfrequencydistributionwithfrequenciesrepresentedasverticalbars;itisappropriateforinterval/ratiodata

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 130–139 140–149 150–159 160–169 170–179 180–189 190–199

– – – – – – – –

f

Weight (lb)

FIGURe 3.3 Histogram of male students’ weights

Check: Axes labeled (including units)Class intervals under barsBars share common borders

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3.2 Distributions as Graphs 41

Herearethestepsusedincreatingahistogram:

STep 1 Begin. Begin with a grouped frequency distribution as described in thepreceding section.For ourweight example, that grouped frequencydis-tribution,using10-poundintervals(wecouldhaveused5-poundintervalsjustaswell),isshowninTable3.4.

STep 2 Axes. Draw and label the axes of the histogram. It is conventional tomaketheverticalaxisabouttwo-thirdstothree-quartersthelengthofthehorizontalaxis.Theverticalaxisshouldbelabeled“ f.”Thehorizontalaxis should be labeled with the name of the variable, “Weight” in ourexample,withtheunitofmeasurementinparentheses,“(lb).”Entertheclassintervalrangesonthehorizontalaxis.Theyshouldbeequallyspacedbecause the intervals in the grouped frequency distribution should havebeenequalsized.

STep 3 Break? Itisconventionaltohavetheverticalaxisintersectthehorizontalaxisatthepointwherethevalueofthevariableis0.Ifthisisnotthecaseinyourhistogram,thenindicatethatwithabreakinthehorizontalaxis.TheverticalaxisinFigure3.3seemstointersectthehorizontalaxisatabout120pounds,not0,soabreakisindicatedintheaxis.

STep 4 Edges. Drawverticallinesattheedgesoftheclassintervalstoformtheedgesofthehistogrambars.Notethattheedgesofeachbararesinglelines,indicatingthatthereisnospacebetweenthevaluesthatthebarsrepresent.

STep 5 Heights. Theheightofeachverticalbarshouldequalthefrequencyofthevaluesinthatinterval.

NowwhenJackasksusabouttheweightsofthemalestudentsinourclass,wecananswerbyshowinghimthishistogram.ThismaybeamoreconcisesummaryforJackbecausehecanseeataglancethatthelowestweightisabout130pounds,thehighestweightisabout200pounds,andmostoftheweightsclusteraround160or170pounds.

The histogram based on class intervals presents a relatively complete, easilycomprehensiblepictureofourdata,andinfacthistogramsareamongthemostwidelyusedwaysofdescribingsetsofdata.

Frequency PolygonAnother,entirelyequivalent,graphicalpresentationofdataisthefrequency polygon.Recallthatahistogramisaplotofaseriesofrectangularbars,whereeachbar’swidthisthewidthoftheclassintervalandthebar’sheightisthefrequencyofthatclassinter-val(seeFigure3.3).Afrequencypolygonissimplyatransformationofthehistogramobtainedbysubstitutingasinglepointforeachbarandthenconnectingthepointswithstraightlinesegments.Figure3.4isthehistogramofFigure3.3withtheequivalentfrequencypolygonsuperimposed.Figure3.5thenshowsthehistogramremoved,leav-ingthefrequencypolygonbyitself.Itiscustomarytoconnectthefirstandlastdotstothehorizontalaxiswithdiagonaldashedlines.Thefrequencypolygonisthusasecond

Steps for creating a histogram

frequency polygon Agraphicalpresentationofagroupedfrequencydistributionwithfrequenciesrepresentedaspoints;itisappropriateforinterval/ratiodata




8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 130–139 140–149 150–159 160–169 170–179 180–189 190–199

– – – – – – – –

f

Weight (lb)

FIGURe 3.4 Transforming a histogram into a frequency polygon (The completed frequency polygon is shown in Figure 3.5.)

8 –

7 –

6 –

5 –

4 –

3 –

2 –

1 –

0 – 134.5 144.5 154.5 164.5 174.5 184.5 194.5

– – – – – – –

f

Weight (lb)

FIGURe 3.5 Frequency polygon of male students’ weights

formofgraphicalfrequencydistribution;itcontainsexactlythesameinformationasdoesthehistogram.

Note that the weight values on the X-axis of a frequency polygon are singlepointsplottedatthemidpointsoftheclassintervalsratherthandisplayingtheclassintervalsbeneaththebarsaswasthecaseinthehistogram.Forexample,theinterval160–169poundshasthemidpoint(160+169)/2=164.5.Thedotofthefrequencypolygonisplotteddirectlyabove164.5.

Wemakethreeimportantobservationsabouthistogramsandfrequencypolygons.First, theybothcontain the identical informationfoundinagroupedfrequencydis-tributiontable.Therefore,thechoiceofusingahistogram,afrequencypolygon,oragroupedfrequencydistributiontableisbasedentirelyonclarityofpresentation.Usewhicheverformpresentsyourdatamosteffectively.

Check:Axes labeled (including units) Midpoints at tic marksDots above midpointsDots connected with straight

lines Dashed lines diagonally to 0

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The histogram and frequency polygon are both shown so that you can see the relationships. In actual practice, choose one or the other, not both.

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3.3 Eyeball-estimation 43

Second,wehaveplottedthefrequenciesontheverticalaxisofthehistogramsandfrequencypolygons.Insomecases,particularlywhenthenumberofobservationsislarge,itisdesirabletoplotameasureofrelative frequency,suchasaproportionorpercentage,onthataxis.Forexample,becausethereare25meninourweightdata,eachmanrepresents1/25,or4%,ofthetotalsample.Thus, ifwewished,thelabelontheverticalaxisinFigures3.3,3.4,or3.5couldbealteredbyreplacing“ f ”with“Percentage”andreplacingthevalues“l,2,3,...”with“4%,8%,16%,...”Thenewgraphwouldconveythesameinformationastheoriginal.

Third,assamplesizesbecomelarger,thewidthofeachbarinahistogram(orthedistancebetweeneachpointinafrequencypolygon)canbemadenarrowerandnarrower.Thentheshapeofthefigurebecomes,ingeneral,smootherandsmoother.Figure 3.6showssuchahistogram.Ifweimagineaninfinitelylargesamplewithinfinitelynarrowclassintervals,thenthehistogramorfrequencypolygonbecomesasmoothcurve.

ClickResourcesandthen3AinthePersonal Trainerforonemoreimportantbutoptionalwayofpresentingfrequencydistributionsofinterval/ratiodata.Thestemandleafdisplaycombinessomeofthebestpropertiesoftablesandgraphs,butisnotasyetwidelyused.

ClickLectletsandthen3BinthePersonal TrainerforanaudiovisualdiscussionofSections3.3through3.6.

3.3 Eyeball-estimationWehavefollowedonesetofdata,theweightsofagroupofstudents,fromtherawdatastage (Table3.2) to thehistogram(Figure3.3)and frequencypolygon (Figure 3.5)graphical presentations of those data.We have seen that histograms and frequencypolygonsaresimpler,moreinformativewaysofcommunicatingaboutadistributionofdatathanistheenumerationoftherawdatathemselves.Nowitisourtasktobecomeskilledinvisualizingdistributionsevenwhenwedonothavetherawdata tobegin

relative frequency Frequencydividedbythesizeofthegroup,expressedasaproportionorpercentage

The concept of relative frequency will be important in Chapter 6.

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110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260

– – – – – – – – – – – – – – – –

f

Weight (lb)

FIGURe 3.6 Histogram of male students’ weights with a very large sample size and very narrow class intervals

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Generally, large samples and narrow intervals give smoother histograms.

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with.Wewanttobeabletosketchareasonablyaccuratefrequencydistribution“byeyeball,”basedonwhatweknowabouttheworldingeneral.(Inthistextbook,wewillhighlight descriptions of the process of “eyeball-estimation” by using the eyeglasslogoshownatthebeginningofthissection.)

Nowwearestartingoverwiththeweightsofmalestudents.ForgetforthemomentthatweeversawthedatainFigure3.2.Ournewtaskistoattempttosketchthedistri-butionofweightsofstudentsinastatisticsclassbased on what we know about students in general,notbasedonanydataaboutanyparticularclassthatwemighthaveseen.Thestepsinthissketching-by-eyeballprocedurearelistednext.

STep 1 Axes. Drawand label theaxes.Afrequencydistributionalwayshas thevariable thatweare interested inplottedon theX-axis; inour case, thevariableis“Weight(lb),”whichshouldbethelabeloftheX-axis.Donotforgettheunitofmeasurement(“lb”)inthevariablelabelbecausewewillseeinlaterchaptersthatkeepingtheunitsclearlyinmindisessentialtounderstanding statistics. The vertical axis of a frequency distribution isalwaysthefrequencyandiscustomarilylabeled“ f.”Theverticalaxisiscustomarilydrawn tobeabout two-thirds to three-quarters the lengthofthehorizontalaxis.AttheendofStep1,ourdistributionshouldappearasshowninFigure3.7.Notethatoureyeballeddistributionisanabstractioninthesensethatitdoesnotrefertoanyparticularsampleofvaluesortoanyparticularsamplesize,sothef-axisdoesnothavenumericalvalues.

STep 2 Min and Max. Plotyourestimatesoftheminimumandmaximumvaluesofthevariable.Whatdoyouthinkisthesmallestman’sweightinacol-lege class? My estimate is about 120 pounds; your estimate might be abithigherorlowerdependingonyourexperience.Ourtaskhereistobeapproximate,notexact.Labelthelowestweight(120pounds)neartheleftendoftheX-axis.Ifwethinkthatthelowestweightis120pounds,thenthefrequencyofanyvalue(weight)below120mustbe0,andtheplotoffrequenciesshouldbegintoriseataboutthevalue120pounds.Repeattheprocessforthelargestweightlikelytobefoundinacollegeclass,puttingthat value near the right end of the X-axis. My estimate is 220 pounds.Figure3.8showstheplotwithminimumandmaximumvalues.

STep 3 Intermediate. Addintermediatevalues to theX-axis,usingroundnum-bers.Keeptheintervalsbetweenthevaluesequal,andmakethedistancesbetweenthosevaluesontheX-axisequal.Inourexample,weneedroundnumbersbetween120and220withequal intervalsbetween them; thus,140,160,180,and200areappropriateandareequallyspacedontheX-axisasshowninFigure3.9.

STep 4 Mode. Plotestimate(s)ofthemode(s)ofthefrequencydistribution.Themodeisthevalueofthevariable(weightinourexample)thatoccursmostfrequentlyinadistribution;therecansometimesbemorethanonemode.What weight or small range of weights do you think is likely to occurmostfrequently?Itseemstomethatthemostfrequentlyoccurringweightamongcollegemenisabout170pounds(yourestimate—andthetruth!—mightbesomewhathigherorlower).Withthisassumption,thefrequency

Steps for eyeball-estimating a frequency distribution



3.3 Eyeball-estimation 45

f

Weight (lb)

f

Weight (lb) 220

– –

FIGURe 3.7 Eyeball-estimating the frequency distribution: Step 1, drawing and labeling the axes

FIGURe 3.8 Eyeball-estimating the frequency distribution: Step 2, plotting minimum and maximum values

distributionmustreachitsmaximumat170anddecreaseinbothdirectionsfrom there. Therefore, we locate a point above 170 pounds on our fre-quencydistributionandsketchalittlepeakatthatpointtoindicatethatthefrequencywilldecreaseasweighteitherincreasesordecreasesfrom170pounds(seeFigure3.10).Theabsoluteheightofthispeakisnotimportantinthissketch.Iftherearetwoormoremodes,thentherelativeheightisimportant:Themorefrequentlyoccurringmodeshouldbehigher.

STep 5 Complete. Connectthepartsofthesketchthatyouhavejustcreated,keep-ingthedistributionsmoothandcontinuous(unlessthereisgoodreasontohaveadiscontinuityinthedistribution).Natureasaruledoesnotchangeabruptly,anddistributionsofvariables thatoccur innatureare thereforeusuallysmooth.Figure3.11showstheresultofoursketch.

f

Weight (lb) 120 140 160 180 200 220

– – – – – – – – – – –

f

Weight (lb) 120 140 160 180 200 220

– – – – – – – – – – –

FIGURe 3.9 Eyeball-estimating the frequency distribution: Step 3, labeling intermediate values on the X-axis

FIGURe 3.10 Eyeball-estimating the frequency distribution: Step 4, sketching the mode




Oursketchofthedistributionofmen’sweightsincollegeclassesisnowcomplete.Howdoesoursketch(whichwecreatedwithoutconsideringanyparticulardata)com-parewiththehistogramandfrequencypolygonbuiltfromactualdataandshowninFigures3.3and3.5?First,oursketchedcurveissmootherthaneitherthehistogramorthefrequencypolygon.TheroughedgesorsuddenchangesindirectionintheplotsofFigures3.3and3.5areartifactsofthesmallsamplesizeandwideclassintervals.AswesawinFigure3.6,thelargerthesamplesizeandthenarrowertheclassintervals,thesmootherthehistogramandfrequencypolygon.Oureyeball-estimatedfrequencydistributionissmooth,asifwehadextremelynarrowclassintervals.

Second,theminimumandmaximumvaluesarenotexactlythesame(butclose)in our estimate as those in the data.The difference could be because the data ofTable 3.2representedjustoneclass,whereasoursketchrepresentedclassesingen-eral.IfTable 3.2hadbeenofadifferentsingleclass,thentheminimumandmaxi-mumvaluesofthedatawouldlikelyhavebeendifferent.Alternatively,itispossiblethat our eyeball is somewhatmistaken in estimating theminimumandmaximumvaluesofthisvariable.

Third,boththesketchandthedatahavethemodeatabout170pounds.Ouresti-mateofthepeakformen’sweightswascorroboratedbythedata.Ingeneral,oursketchhasroughlythesamecharacteristicsasthedata.

3.4 The Shape of Distributions Describing DistributionsAswesketchfrequencydistributions, itwillbeconvenient tohavesometermsthatdescribetheshapesofthedistributions.

Thefirsttwotermsdescribethenumberofmodesthedistributionhas.Recallthatthemodeisthemostfrequentlyoccurringvalueinadistribution;itliesdirectlybelow

f

Weight (lb) 120 140 160 180 200 220

– – – – – – – – – – –

FIGURe 3.11 Eyeball-estimating the frequency distribution: Step 5, completing the sketch with a smooth curve

The eyeball philosophy: Quick, easy approximations exercise your comprehension. You don’t have to be exact (that’s why we do computations).

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3.4 The Shape of Distributions 47

the highest point on a graphical frequency distribution.A distribution that has onemodeiscalledunimodal.Oursketchofthedistributionofmen’sweightsinFigure 3.11isunimodal.

Abimodal distributionhas twomodes.For example,wemight expect thedis-tributionofweightsofcollegestudentsingeneraltohavetwomodes,onemostfre-quentlyoccurringweightforthemenandonemostfrequentlyoccurringweightforthewomen,asshowninFigure3.12.Mostdistributionsinnatureareunimodal;bimodaldistributionsoccuronlywhenthereisasplittingofapopulationintotworelativelydistinctparts.Beingbimodaldoesnotimplynooverlapbetweenthedistributions(forexample,inourweightexamplesomemenarelighterthanthefemalemodeandsomewomenareheavierthanthemalemode).

We call a distribution symmetric if the right side is a mirror image of the leftside.The unimodal distribution of men’s weights in Figure 3.11 is symmetric; thebimodaldistributionofmen’sandwomen’sweights inFigure3.12 isalsoapproxi-matelysymmetric.

f

Weight (lb)

– – – – – – – – – – – – – –

100 120 140 160 180 200 220

FIGURe 3.12 A bimodal distribution: weights of college students (both men and women)

unimodal distribution Adistributionthathasonemostfrequentlyoccurringvalue

bimodal distribution Adistributionthathastwomostfrequentlyoccurringvalues

symmetric distribution Adistributionwhoseleftsideisamirrorimageofitsrightside

f

X

f

X

FIGURe 3.13 A positively skewed distribution FIGURe 3.14 A negatively skewed distribution




Wecall adistributionskewed ifone“tail” (the sideof adistribution) is longerthan theother: thus, skeweddistributionsmustbeasymmetric.Distributionswhoseright-handtailislongerthantheleftarepositively skewed(seeFigure3.13),andthosewhoseleft-handtailislongerarenegatively skewed(seeFigure3.14).

WecallthetailofadistributionasymptoticifitgraduallyapproachestheX-axisbutneveractuallytouchesit.(Tobetrulyasymptoticisthereforeamathematicalabstrac-tion;allrealdistributionshavealowestvaluebelowwhichthefrequencyis0andthuswhere thedistributionactuallymeets theX-axis.)ThedistributionofFigure 3.14 isasymptoticinthelefttailbutnotasymptoticintherighttail.

The Normal DistributionForreasonsthatwillbecomeclearerinChapter7,thedistributionsofmanyvariablesthat occur in nature have a characteristic form called the normal distribution. Theheightsofpeople(ortreesordandelions),IQsofpeople,maze-runningspeedsofrats,andgainsorlossesonthestockexchangeareallapproximatelynormallydistributedvariables.Thissectionwillacquaintyouwith thegeneralshapeofnormaldistribu-tions,andlaterchapterswillfillinthecomputationaldetails.

Normaldistributionsareunimodal, symmetric,andasymptotic,whichmeansthattheyhavethegeneralbellshapeshowninFigures3.15,3.16,and3.17.Normaldistribu-tionscanbetallorshort,narroworwide,buttheymusthavethesamegeneralbellshape.

YoumaywonderwhythenormaldistributioniscalledasymptoticwhenthecurvesdrawninFigures3.15,3.16,and3.17seemtotouchtheX-axis.Beingasymptoticisamathematicalideal,impossibletorepresentaccuratelyinthoseportionsofthecurvewherethecorrectheightislessthanthewidthoftheinklines.Figures3.15,3.16,and3.17areinfactpreciselydrawnnormaldistributions.(Thereisawidespreadmiscon-ception,however,thatnormaldistributionsshouldseemto“float”abovetheX-axis;amisconceptioncreatedbytextbooksthatincorrectlyillustratenormaldistributions.)

Forcontrast,Figures3.18through3.20presentnonnormaldistributions.Astheircaptionsshow,thesecurveseachhaveonlytwoofthethreecharacteristicsrequiredfornormaldistributions.Figure3.21showsacurvewithallthreecharacteristics,andyetitisstillnotactuallynormal.

f

X

f

X

FIGURe 3.15 A normal distribution FIGURe 3.16 Another (wider) normal distribution

positively skewed distribution Adistributionwhoserighttailislongerthanitslefttail

negatively skewed distribution Adistributionwhoselefttailislongerthanitsrighttail

asymptotic GraduallyapproachingtheX-axis

normal distribution Aclassofdistributionsthatareunimodal,symmetric,andasymptotic



3.4 The Shape of Distributions 49

f

X

f

X

FIGURe 3.17 A third (narrower) normal distribution FIGURe 3.18 Nonnormal distribution (unimodal and symmetric but nonasymptotic)

f

X

f

X

FIGURe 3.19 Nonnormal distribution (unimodal and asymptotic but not symmetric)

FIGURe 3.20 Nonnormal distribution (symmetric and symptotic but bimodal)

f

X

FIGURe 3.21 Nonnormal distribution (unimodal, symmetric, and asymptotic but too flat on top)




3.5 Eyeball-calibrationLetushavesomepracticeineyeball-estimatingfrequencydistributions.(Inthistext,wecallsuchpractice“eyeball-calibration”andindicateitwiththeglasses-and-rulerlogo shown at left.) Suppose we are interested in the distribution of the speeds ofautomobilesonasectionofaninterstatehighwaywherethespeedlimitis65milesperhour(mph).InStep1ofourprocedure,wedrawandlabeltheaxes.Thehorizontalaxisislabeled“Speed(mph)”becausethatisthevariableofinterest,andtheverticalaxis is the“frequency,”whichwecustomarilyabbreviateas“ f.” InStep2,weaskourselveswhatistheslowestlikelyspeedonsuchaninterstate,andwemightguessabout50mph.(Notethatwearemakingsomeimplicitassumptionshere,suchastheabsenceofhillsorsnowyroadconditions.Oneofthetasksofscienceistomakesuchassumptions explicit, but that is a topic of research design, not of statistics itself.)The maximum speed is perhaps about 80 mph.We mark the minimum and maxi-mumspeedsontheX-axis.InStep3,weaddtheintermediatevaluestotheX-axis:60and70mph.InStep4,weenterourestimateofthemode,whichwemightguesstobeabout65mph.TheninStep5,weconnectthepointstoformthecompletesketchshowninFigure3.22.Oursketcheddistributionismoreorlessnormallyshaped.

Asanotherexample,letuseyeball-estimatethedistributionofthelengthsofbladesofgrassinalawnthathasjustbeenmowedtoaheightof3inches.InStep1,wedrawandlabeltheaxes:TheX-axisis“Lengthofblade(in.)”andtheY-axisis“frequency”or“ f.”InStep2,weplottheminimumlength,whichweguesstobeabout2inches,andthemaximumlength,whichwetaketobejustslightlygreaterthan3inches.Thelawnhasjustbeencutandanylongerbladeswillhavebeensnippedoff,butweallowforsomeslightlylongerbladesthathappenedtobegrowingatanangle.InStep3,weentertheintermediatevalues:2.25,2.50,and2.75inches.InStep4,weeyeball-estimate the mode. The most frequently occurring length in this case seems to be3inchesbecauseanylongerbladeswouldhavebeencutto3inches.Weenterthemodealmostdirectlyabovethemaximumvalue.InStep5,weconnectthepoints,andhere

The general steps for this procedure are found on pages 38–40.

c

f

Speed (mph) 50 60 70 80

– – – –

FIGURe 3.22 Eyeball-estimate of the frequency distribution of speeds on an interstate highway



3.5 Eyeball-calibration 51

wehavereasontoacceptarathersharpdiscontinuityintheshapeofthedistribution.Therewillbeanabruptdecreaseinthefrequencyofthelengthsofbladesat3inchesbecausethemowercutsanybladesthatarelongerthanthat.TheresultisshowninFigure3.23.Thisdistributionisextremelynegativelyskewed.

Theexercisesforthischapterwillgiveyouthechancetoconvinceyourselfthatyoucan sketchbyeyeball relativelyaccurate frequencydistributionsof all sortsofvariables,rangingfromtheheightsofbasketballplayerstothelengthsofMcDonald’sfrenchfries.Beingabletosketchanapproximatelyaccuratefrequencydistributionisevidencethatyouunderstandwhatafrequencydistributionisandthatyouhavesomeknowledgeofthewayvariablesaredistributedintherealworld.

Statistics,tobesure,arefrequentlyusedtodescribevariableswhosedistributionswedo notknow;that’sonewayscientistslearnabouttherealworld.However,atthisstageinlearningtounderstandstatistics,wearefocusingonvariablesthatarefamiliar.Your task is to recognize that therearevariableswithdistributionseverywhereyoulook,andthatifyouarefamiliarwiththevariable,youcandescribeitsdistributionwithatleastsomeaccuracy.Iwantyou,asyou’rewalkingacrosscampus,toidentifyvariablessuchastheheightofpinetrees,orthelengthofwomen’shair,orthenumberofbooksstudentscarry,ortheamountoftimestudentstaketowalkfromthelibrarytothestudentunion,orthenumberofmilesontheodometersofcarsintheparkinglot,orthenumberoftimesthewordtheappearsperpageinyourtextbooks,or....Iwantyoutovisualizethefrequencydistributionsofthosevariables.Iwantyoutosaytoyourselfasyouwalk,“Theshortestpinetreesonthiscampusseemtobeabout4feetandthetallestabout40feet,andthedistributionseemstobesymmetricwiththemodeatabout20feet.”“Thewomen’sshortesthairlengthisabout3inches,thelongestisabout30inches,andthemodeisabout8inches,andIcanseeinmymind’seyethefrequencydistribution:Itbeginstoriseat3inches,reachesapeakat8inches,andthentapersoff,reaching0frequencyat30inches.”Thatis,Iwantyou,forthemoment,topracticeseeingdistributionsofheightsoftrees,ratherthanthetreesthemselves,anddistributionsofthelengthsofhair,ratherthanthehairitself.

f

Length of Blade (in.) 2.00 2.25 2.50 2.75 3.00

– – – – –

FIGURe 3.23 Eyeball-estimate of the frequency distribution of lengths of blades of grass in a recently mowed lawn

Review: Three main concepts:

1. Distribution of a variable2. Sampling distribution of

means3. Test statistic

We are focusing here on distributions of variables.

c




Weshall see throughout the remainderof this textbook that statistics is almostentirely the study of distributions and their characteristics. If we are to understandstatistics,wemustbecomeconfidentlyfamiliarwithdistributions,beingableto“see”themaccuratelywhereverwelook.

3.6 Bar Graphs of Nominal and Ordinal VariablesOur discussions so far have focused on the graphical display of interval/ratio vari-ables—thatis,onhistogramsandfrequencypolygons.Ifavariableisnominalorordi-nal,however,theuseofahistogramorfrequencypolygoncanbequitemisleading.

Suppose the campus newspaper polls students about their political preference,askingthemwhethertheyareRepublican,Democrat,Independent,ordeclinetostate.This isclearlyanominalvariable,and the frequenciesareshowninTable3.6. Ifwe(mistakenly)weretocreateafrequencypolygonforthesedata,wewouldplotapointforRepublicans,apointforDemocrats,andsoon,andthenconnectthepointswithcontinu-ouslines.ThelinebetweentheRepublicanandtheDemocratpointswouldgivetheerro-neousimpressionthatmeasurementsweremadeatintermediatepointsbetweenthosetwoparties.Toavoidsuchmisleadingdisplays,wedonotusehistogramsorfrequencypolygonsfornominalorordinaldata,preferringinsteadthebargraph.

Abar graphisagraphicaldisplayofafrequencydistributionwherefrequenciesarerepresentedasseparatedverticalbars,asshowninFigure3.24.Notehowthesepa-rationofthebarsmakesitclearthatnointermediatevaluesweremeasured.Notealsothattheorderofthebarscouldberearranged,withDemocratsplacedattheleft,forexample.Thisisbecausethepoliticalpreferencevariableismeasuredatthenominallevel,whereorderisnotimportant.

The frequency distributions of ordinal data are displayed as bar graphs, histo-grams,orfrequencypolygons.Usingthebargraphforordinaldatahastheadvantageofmaking itclear that the intervalsbetweenpointsarenotnecessarilyequal sized,

TaBle 3.6 Frequency distribution of students’ political preferences

preference f

Republican 375Democrat 510Independent 141Declinetostate 84

bar graph Agraphicalpresentationofafrequencydistributionofnominalorordinaldata

600 –

500 –

400 –

300 –

200 –

100 –

0 – Republican Democrat Independent Declined

f

Political Party Preference

FIGURe 3.24 Bar graph of political party preference

Spaces between bars in a bar graph indicate that the bars could be reordered.

c



3.7 Connections 53

butithasthedisadvantageofimplyingthattherearenointermediatevaluesbetweenthebars.Thechoicebetweenthemethodsofdisplayforordinaldata isa judgmentcallmadefortheindividualfrequencydistribution,consideringwhichkindofdisplaygivestheclearestcommunicationaboutthedata.

Toreview:Whereashistogramsandfrequencypolygonsaresuitableforrepresent-ingdatameasuredonlyat the interval/ratio level, bargraphsare suitable for eithernominalorordinaldata.Andwhereasthebarsinhistogramsaredirectlyadjacent,thebarsinbargraphsareseparatedsomedistancefromoneanother.

Connections3.7

Cumulative ReviewChapters 1 and 2 distinguished between populations (the entire group of interest)andsamples(somesubsetofthatgroup)andbetweenparameters(characteristicsofpopulations)andstatistics(characteristicsofsamples).Variablesweredefinedaschar-acteristicsthatcouldtakeonanyofseveralvalues,andwedistinguishedamongnomi-nal(categorizationonly),ordinal(orderedcategorization),andinterval/ratio(orderedcategorization where the categories all have equal sizes) levels of measurement.Summationnotationandprobabilitywereintroducedastwooftheprimarymathemati-calskillsunderlyingstatistics.

ComputersUseDataGentocreateafrequencydistributionofthedatainTable3.2:

1. InthePersonal Trainer,clickDataGen (Win)ifyouareaWindowsuserorDataGen (Mac) ifyouareaMacintoshuser.DataGen(Win) requires thatyou have Microsoft Office Excel 2007 or later installed on yourWindowscomputer.DataGen(Mac)requiresthatyouhaveMicrosoftOfficeExcel2011orlaterinstalledonyourMacintoshcomputer.If you have a choice, use the Windows version of DataGen.

2. EnterthedataofTable3.2.Notethatlogicallythisisonecolumnofdata,soenterall25valuesdownDataGen’sVariable 1column.

3. NotethatDataGendoesnotcreatefrequencydistributionsorhistograms,sothiscannotbedonedirectly.YoucanaskDataGentosortthedata,sothatcre-atingahistogramiseasier.Highlightthe25valuesinVariable 1.ClickEdit variableandthenclickSort (this variable only).

4. Savethesedatabecausewewillusethemlater:ClickSave As.ChooseafolderintheSave Aswindow.IntheFile name:celltypetable 3-2.Thefilenameatthetopofthedatagenspreadsheetwindowshouldnowreadtable 3-2.xlsx.

5. Openthefileyoujustsavedtomakesureitwassavedcorrectly:ClickOpen.Makesuretable 3-2.xlsxappearsintheFile name:cell,andthenclickOpen.

Personal Trainer

DataGen




UseDataGentoadd100toallthevaluesofavariable:

1. Enter thefivevalues2,4,6,8, and10 into theVariable 1 columnof thespreadsheet.

2. Highlightthosefivecells,clickEdit variable,enter100intothetextentrycellnexttotheAddbuttonandthis valueandclickAdd.

SPSS

S

%a

UseSPSStocreateafrequencydistributionofthedatainTable3.2:

1. Enter thedata fromTable3.2by typing thevaluesdowna singlecolumn,usingEnterbetweenvalues.(Do notputthedatainfivedifferentcolumns.)

2. Savethesedatasothatyoucanusethemlater:ClickFile,thenclickSave As.... In theFile name:cell, typetable 3-2,whichwillmake thenewfilenametable 3-2.sav.ThenclickSave.

3. Recallthisfiletomakesureitissavedcorrectly:ClickFile,thenOpen,andthenData....Clicktable 3-2.sav.ClickOpen.

4. ClickAnalyze,thenDescriptive Statistics,thenFrequencies....

5. Click5andthenclickOK.

6. Note that the frequency distribution appears in the Output1-IBM SPSS Statistics Viewerwindow.Check tomake sure that the frequencycolumnsums to25and that the tablematchesTable3.3 in the textbook.Note thatSPSSchoosestolistthesmallestvaluesfirst,unliketheconventiondescribedinthetext.

UseSPSStocreateahistogramofthesedata:

7. ClickGraphsandthenChart Builder….

8. Ifawindowthatreads“Beforeyouusethisdialog...”appears,clickOK.

9. IntheChoose from:dialog,selectHistogram.

10. DragtheiconforthehistogramintothewindowthatsaysDrag a Gallery chart here to use it as your starting point.

11. Drag VAR00001 from the Variables: dialog to the X-Axis? label of thehistogram.

12. ClickOK.

13. NotethatthehistogramappearsintheOutput1-IBM SPSS Statistics Viewerwindow.

UseSPSStoadd100toallthevaluesofavariable:

1. ClearthedatabyclickingFile>New Data.

2. Enter2,4,6,8,and10intothefirstcolumnofthespreadsheet.

3. ClickTransformandthenclickCompute variable....

4. EnterVAR00001intheTarget Variable:textentrycell.

5. Click5,click+,click1,click0,click0,andthenclickOK.

6. ClickOKtoChange existing variable?

7. NotetheresultintheUntitled-IBM SPSS Statistics Data Editorwindow.



3.6 Bar Graphs of Nominal and Ordinal Variables 55

Excel

H0: m1 = m2

X

UseExcel tocreateagroupedfrequencydistribution(likeTable3.4)of thedata inTable3.2(Excelwillnoteasilycreateanungroupedfrequencydistribution,butseetheDataGenhintabove):

1. EnterthedatafromTable3.2intothefirst5cellsofspreadsheetcolumnsAthroughE.

2. Savethesedatasothatyoucanusethemlater:ClickFile > Save As.IntheFile name:celltypetable 3.2,whichwillmakethenewfilename“table3.2.xlsx.”ThenclickSave.

3. EnterthetopvaluesoftheclassintervalsfromTable3.4(199,189,179,etc.)intothefirst7cellsofcolumnF.

4. Highlightthefirst7cellsofcolumnG(that’swherethefrequencieswillgo).

5. IntheFormulastabclickInsert Function.

6. IntheOr select a category:cellselectStatistical.

7. ScrolldownandselectFREQUENCYandclickOK.

8. ClicktheRefEditcontrol attheendoftheData_arraycell.Highlightthe25valuesincolumnsAthroughE.ClicktheRefEditcontrolagain.

9. ClicktheRefEditcontrolattheendoftheBins_arraycell.Highlightthe7valuesincolumnF.ClicktheRefEditcontrolagain.

10. OnthekeyboardpresssimultaneouslyCtrl-Shift-Enter.ThiswillentertheFREQUENCYfunctionintoall7cellsincolumnG.

11. ComparethecontentsofcolumnGtothefrequenciescolumninTable3.4.

UseExceltocreateahistogramofthesedata:

12. ClicktheDatatabandthenclickData Analysis.IfthereisnoData Analysis entryintheribbon,dostep12A.Otherwisegotostep13.

12A. IfthereisnoData Analysisentry,youwillhavetoinstalltheExcelAnalysisToolpak,whichcomeswithExcelbutisnotautomaticallyinstalled.ClickFile > Options > Add-Ins.HighlightAnalysis ToolPak.ClickGo.Thatproduces the Add-Ins window. Check Analysis ToolPak and click OK.Then,intheDatatabclickData Analysis.

13. ScrolldowntoHistogram,highlightit,andclickOK.

14. ClicktheRefEditcontrolattheendoftheInput range:cell.Highlightthe25valuesincolumnsAthroughE.ClicktheRefEditcontrolagain.

15. ClicktheRefEditcontrolattheendoftheBin Range:cell.Highlightthe7valuesincolumnF.ClicktheRefEditcontrolagain.

16. ChecktheChart OutputboxandclickOK.

17. ComparethehistogramthatExcelcreatestoFigure3.3.

18. Note thatExcelplaces thehistogram inanewworksheet.To return to theoriginaldata,clickSheet1atthebottomofthewindow.




UseExceltoadd100toallthevaluesofavariable:

19. ClearthedatabyclickingFile > New > Blankworkbook.

20. Enterthefivevalues2,4,6,8,10intothefirstcellsofcolumnA:.

21. ClickoncellB1, type=A1+100.Don’tforgettotypetheequalsignat thebeginningofthisentry.ThenpressEnter.Notethatthevalue102(=2+100)appearsincellB1.

22. Click on cell B1. Grab the fill handle ( , the little square box thatappearsatthelowerrightcorneroftheB1cell)anddragitdownuntilthefirst5cellsincolumnBarehighlighted.Thenreleasethehandle.

Homework Tips1. Checkthelistoflearningobjectivesatthebeginningofthischapter.Doyouunder-

standeachone?

2. Whenyouarepreparingagroupedfrequencydistribution,makesurethatalltheclassintervalshavethesamewidth.Themostfrequentmistakeistomakethefirstorthelastintervalwiderornarrowerthantheremainingintervals.Iftheintervalsareequal,thenthelastdigitintheintervallimits(includingthefirstandlastinter-val)willformsomeuniformsequence,suchas0,5,0,5,...or0,4,8,2,6,0,4,8,2,6,...

3. Remembertoplotthepointsinafrequencypolygonatthemidpointsoftheclassintervalsandtoconnectthefirstandlastdotstotheaxiswithdiagonaldashedlines.

ClickReviewMasterandthenChapter 3inthePersonal TrainerforanelectronicinteractivereviewoftheconceptsinChapter3.

ClickLabsandthenChapter 3inthePersonal TrainerforinteractivepracticeoftheskillsinChapter3andaquiztotestyourunderstanding.

Personal Trainer

ReviewMaster

Personal Trainer

Labs



Chapter 3 Exercises 57

CHAPTER3 E xErc i s EsSection A: Basic Exercises (Answers in Appendix D, page 527)

[HiNT:Saveyouranswerstotheseexercises.YouwillusethemagainfortheexercisesinChapters4and5.Whensketchingdistributions,followthestepsofFigures3.7–3.11. Do not forget to label your axes (including theunitofmeasurement).]

1. Tomeasurethefitnessofstudents,EdithDrabushenselects20studentsatrandomandasksthemtorideastationarybicycleasfastastheycanfor30min-utes.Thebicyclehasanodometer,andsherecordsthenumberofmileseachstudent rides.Thedata(inmiles)are8,15,22,17,9,16,15,15,21,12,14,16,22,17,16,11,13,15,20,15.HelpEdithpreparethesedistributions:

(a) Frequencydistribution (b) Groupedfrequencydistributionwithabouteight

groups (c) Histogram (d) Frequencypolygon

ExErcisE i workEd out(a) Frequencydistribution ✓ Labels“Distanceridden”and“f ” ✓ Units“(mi)”shown ✓ Largestvaluesfirst

Distance Ridden (mi) f

22 221 120 117 216 315 514 113 112 111 1 9 1 8 1

20

✓ Nozerosinfcolumn ✓ fcolumnsumshown ✓ fcolumnsumequalsn

(b) Groupedfrequencydistribution ✓ Labels“Distanceridden(mi)”and“f ” ✓ Topintervalincludeslargestpoint ✓ Bottomintervalincludessmallestpoint ✓ Allgroupssamewidth(especiallyfirstandlast!) ✓ Nogapsbetweengroups ✓ fcolumnsumstillequalsn

Distance Ridden (mi) f

22–23 220–21 218–19 016–17 514–15 612–13 210–11 18–9 2

20

(c) Histogram ✓ Labels“Distanceridden”and“f ” ✓ Units“(mi)”shown ✓ Nospacebetweenbars ✓ Breakinhorizontalaxisbecauseverticalaxisis

notat0 ✓ IntervalsshownonXaxis

7 –6 –5 –4 –3 –2 –1 –0 –

8–9 10–11 12–13 14–15 16–17 18–19 20–21 22–23

f

Distance Ridden (mi.)

– – – – – – – – –




(d) Frequencypolygon ✓ Labels“Distanceridden”and“f ” ✓ Units“(mi)”shown ✓ Horizontalaxisvaluesatmidpointsofclassintervals ✓ Dotsatmidpointsofintervals ✓ Dashedlinesfromlastdotstoaxis

thedistributionofvibratofrequencies.Whattermscanbeusedtodescribethisdistribution?

6. Coca-ColabottlingcompaniesmanufacturebottlesofCokethatsay“16ounces”onthelabel.Bottlersknow that consumer groups will complain if toomanybottlescontainlessthan16ounces,buttheyalso don’t want to waste money by putting toomuch Coke in the bottles. They know that theirbottlingmachinesarenotperfectlyaccurateindis-pensing exact amounts of liquid. They thereforeset their machines to dispense Coke so that themodalamountofCokeis16.5ounces.SketchbyeyeballthedistributionofamountsofCokein“16ounce”Coca-Colabottles.Whattermscanbeusedtodescribethisdistribution?

7. Sketch by eyeball the distribution of lengths ofMcDonald’sfrenchfries.Whattermscanbeusedtodescribethisdistribution?

8. RoyalPerfectoisacompetitorofMcDonald’s,andit,likeMcDonald’s,growsitsownpotatoesfromwhichfrenchfriesaremade.RoyalPerfectohasdiscoveredawayofgrowingpotatoesthatareperfectlycubical,5inchesoneachside.Furthermore,RoyalPerfectopotatoes,whencutintostripsanddeepfried,retaintheirexact5-inchlengths.Sketchthedistributionoflengths of Royal Perfecto french fries.What termscanbeusedtodescribethisdistribution?

9. Big Brothers is an organization that pairs youngboyswithadultmen.TheLosAngelesBigBrothersClubhasaluncheonwhereboysaged7cancomewiththeiradultBigBrothers.Sketchthedistribu-tion of heights of all people (boys and their BigBrothers) who attend this luncheon. What termscanbeusedtodescribethisdistribution?

10. Sketchthedistributionoflengthsofbladesofgrassonthelawnnearthebuildingwhereyourstatisticsclass meets.What terms can be used to describethisdistribution?

11. Supposethecollegegroundcrewwentonstrikefortwoweeksandthelawndoesnotgetmoweddur-ingthattime.Sketchbyeyeballthedistributionoflengthsofbladesofgrassattheendofthestrike.How is this distribution similar to and differentfromthedistributionyousketchedinExercise10?

7 –6 –5 –4 –3 –2 –1 –0 –

8.5 10.5 12.5 14.5 16.5 18.5 20.5 22.5

f

Distance Ridden (mi.)

– – – – – – – –

2. (a) ForthedataofExercise1,prepareahistogramusingaboutfourgroups.

(b) Whichhistogramismoreinformative,theonewitheightgroupsortheonewithfourgroups?

3. Thesedataarethescoresofallthestudentsonthefirstexaminastatisticsclass:81,86,91,75,96,82,88,88,71,68,72,61,84,86,95,91,84,83,91,83,89,90.Preparethesedistributions:

(a) Frequencydistribution (b) Grouped frequency distribution with about

eight groups. The lower limit of the lowestgroup could be 60, 61, 62, or 63. Which ofthesechoicesmakesthemostsense?Why?

(c) Histogram (d) Frequencypolygon

4. Most professional basketball players are about 6feet8inches(80inches)tall.Afewareastallas7feet2inches,andafewareasshortas5feet8inches.Sketchbyeyeballthedistributionofheightsofprofessionalbasketballplayers.Whattermscanbeusedtodescribethisdistribution?

5. Vibratoistheslightwaveringinpitchthatmusiciansadd to musical notes to give them warmth. Mostmusicians’ vibratohas a frequencyof about 7Hz(hertz,or cyclesper second).A fewareas fast as9Hz,andafewareasslowas5Hz.Sketchbyeyeball



Chapter 3 Exercises 59

12. Suppose that the registrar at Hower Universityreports these enrollments in each of the colleges:Liberal Arts 3024, Science and Mathematics1127,PerformingArts752,HealthSciences1452,BusinessandEconomics4320,andEducation2431.

(a) Whatkindofgraphicalpresentationisappropri-ateforthefrequencydistributionatHowerU?

(b) Createthatgraphicaldistribution.

Section B: Supplementary Exercises13. Fora studyonhuman learning,a researchercre-

atedalistof25pairsofnonsensesyllables(suchasdak, fom).Thefirstsyllableofeachpairispre-sentedtothesubjectoneatatime,andthesubjectisasked tosay itspairedassociate. If thesubjectisincorrect,thecorrectsyllableisthenpresented.Oneofthevariablesofinterestinthisstudyisthenumber of times such a list has to be presenteduntil the subject cancorrectlygive all 25 associ-ates.Theresearcherrandomlyselects50studentsfromanintroductorypsychologycourseandtrainseachstudentonthelistuntilthe100%criterionisreached.Thenumbersofpresentationsof the listforthe50studentsare8,9,7,8,16,7,10,11,16,14,12,13,12,13,12,14,8,9,17,12,5,18,14,14,12,8,11,11,9,9,18,15,11,7,9,5,6,8,10,11,11,10,14,16,6,11,15,9,19,12.Preparethesedistributions:

(a) Frequencydistribution (b) Groupedfrequencydistributionwithsixtoeight

groups (c) Histogram (d) Frequencypolygon

14. Acollegeisconsideringsupplyingpersonalcom-putersforitsstudentsandwishestoknowhowfastthestudentscantype.Ittakesarandomsampleof55studentsandadministersatypingtest.Thesearetheresults(inwordsperminute):8,24,20,20,17,18,16,19,17,29,25,27,14,5,21,36,11,16,29,20,11,17,26,10,11,5,5,19,28,7,15,8,14,32,32,12,7,12,13,30,19,16,42,26,16,30,21,8,4,23,5,15,19,9,30.Preparethesedistributions:

(a) Frequencydistribution

(b) Groupedfrequencydistributionwithsixtoeightgroups


15. AsociologistisinterestedintheeconomicimpactofschooleventsatWaytooHighSchool.Sheinter-viewscoupleswhoattendtheseniorpromandaskshow much money each couple spent on this oneevening,includingtickets,clothing,andentertain-ment.Thedata(indollars)are245,190,330,225,140,120,410,395,218,264,256,302,330,310,275,272,188,380,95,160,260,265,387,342,340.Preparethesedistributions:

(a) FrequencydistributionofmoneyspentonthepromatWaytooHigh

(b) Grouped frequency distribution with abouteightgroups


16. The president of a sorority reports these grade-pointaveragesofsororitymembers:3.2,3.5,3.2,4.0,2.7,3.1,3.1,2.9,3.7,2.8,3.6,3.4,3.8,3.8,2.8,3.6,3.9,3.6,3.3,3.4,3.3,3.5,3.4,3.3,3.7,3.5,3.9,3.8,3.1,3.2.Preparethesedistributions:

(a) Frequencydistribution (b) Grouped frequency distribution with about

eightgroups (c) Histogram (d) Frequencypolygon

17.Sketch by eyeball the distribution of lengths ofautomobiles parked in your college parking lot.Describe(inwords)thisdistribution.

18. Sketch by eyeball the distribution of lengths ofsongs played on a rock and roll radio station.Describe(inwords)thisdistribution.

19. Sketchbyeyeballthedistributionofthepricesofmen’sshirtsinalocaldepartmentstore.Describe(inwords)thisdistribution.

20. Sketchbyeyeballthedistributionofthepricesofmen’sshirtsinalocaldiscountstore.Describe(inwords)thisdistribution.




Section C: Cumulative Review (Answers in Appendix D, page 529)

21. IntheAustraliangamblinggameofTwo-Up,twocoinsare“spun”(thrown)simultaneouslyintotheair.Thepersonrunningthegamecallsouttheout-come of the two coins using one of three terms:“heads”(bothareheads),“tails”(botharetails),or“odds”(oneheadandonetail).

(a) Whatistheprobabilityof“heads”? (b) Whatistheprobabilityof“tails”? (c) Whatistheprobabilityof“odds”?

22. ConsiderthedataXi:14,11,12,17,13,15.Find: (a) X4

(b) ∑(4X2i+2Xi–3)

23. Sketchbyeyeballthedistributionoflengthsoflastnames (number of letters in last name, includinghyphenatedsurnames)forstudentsinyourcollege.

24. Identifythelevelofmeasurement(nominal,ordi-nal,orinterval/ratio)foreachofthesevariables:

(a) Numberofpagesinatextbook (b) House address number as a measure of dis-

tancefromthecenteroftown (c) Religiousaffiliation(1=Protestant,2=Catholic,

3=Jewish,etc.)

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Chapter 3 Frequency Distributions - Kendall Hunt Publishing · 34 Chapter 3 Frequency Distributions Learning ... 5 How are these terms used to describe the shape of distributions: