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STAT206:Chapter2(OrganizingandVisualizingVariables)

• MethodstoOrganizeandVisualizeVariables• ForCategoricalVariables:

• SummaryTable;contingencytable(2.1)• Barchart,piechart,Paretochart,side-by-sidebarchart(2.2)

• ForNumericalVariables• (Array),OrderedArray,frequencydistribution,relativefrequencydistribution,percentage

distribution,cumulativepercentagedistribution(2.3)• Stem-and-Leafdisplay,histogram,polygon,cumulativepercentagepolygon(2.4)• Othermethodslater…

• 2.1OrganizingCategoricalVariables

• MustidentifyvariabletypetodeterminetheappropriateorganizationandvisualizationtoolsèRecallVariableTypes• Categorical(Category)

• Nominal–NameofaCategory• Ordinal–Hasanaturalordering

• Numerical/Quantitative(Quantity)• Discrete–distinctcutoffsbetweenvalues• Continuous–onacontinuum

• Definitions:• SummaryTable

• ContingencyTable

• Eachresponsecounted/talliedintooneandonlyonecategory/cell• Example(Problem2.2,p.40):Thefollowingdatarepresenttheresponsestotwoquestionsasked

inasurveyof40collegestudentsmajoringinbusiness:• Whatisyourgender?(M=male;F=female)• Whatisyourmajor?(A=Accounting;C=ComputerInformation;M=Marketing)

Gender: M M M F M F F M F M

Major: A C C M A C A A C CGender: F M M M M F F M F F

Major: A A A M C M A A A CGender: M M M M F M F F M M

Major: C C A A M M C A A A

Gender: F M M M M F M F M M

Major: C C A A A A C C A C

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• Nowtocombinethetwovariables(GenderandMajor):

MAJORCATEGORIES

GENDER A(Accounting) C(Computer) M(Marketing) TOTALSMale(M) 14 9 2 25Female(F) 6 6 3 15TOTALS 20 15 5 40• TablebasedonTotalpercentages:

MAJORCATEGORIES

GENDER A(Accounting) C(Computer) M(Marketing) TOTALSMale(M) Female(F) TOTALS • TablebasedonRowpercentages:

MAJORCATEGORIES

GENDER A(Accounting) C(Computer) M(Marketing) TOTALSMale(M) Female(F) TOTALS • TablebasedonColumnpercentages:

MAJORCATEGORIES

GENDER A(Accounting) C(Computer) M(Marketing) TOTALSMale(M) Female(F) TOTALS

• Questions:

• HowmanyofthesurveyedstudentswerefemalesmajoringinMarketing?

• WhatpercentageofthesurveyedstudentswerefemalesmajoringinMarketing?

• WhatpercentageofthemalestudentssurveyedweremajoringinComputer?

• OfthestudentsmajoringinAccounting,whatpercentagewasmale?

value frequencyrelative.

frequency percentageA"(Accounting) 20 0.500 50.0C"(Computer) 15 0.375 37.5M"(Marketing) 5 0.125 12.5

TOTALS 40 1.000 100.0

Summary.Table.(Major):

value frequencyrelative.

frequency percentageMale%(M) 25 0.625 62.5Female%(F) 15 0.375 37.5

TOTALS 40 1.000 100.0

Summary.Table.(Gender):

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• 2.3VisualizingCategoricalVariables• Piechart–usessectionsofacircletorepresentthetallies/frequencies/percentagesforeach

category• Barchart–aseriesofbars,witheachbarsrepresentingthetallies/frequencies/percentagesfora

singlecategory• Considerourpreviousexample

forMajorCategory:

• PieChart:

• BarChart:

• Question:Whichmajorhasthelowestconcentrationofstudents?

• Discussion:Preferencefortypeofchart?

PercentagebyMajorCategory

A(Accounting)

C(Computer)

M(Marketing)

0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0%

A(Accounting)

C(Computer) M(Marketing)

PercentagebyMajorCategory

SummaryTable(Major):

value frequency relativefrequency percentageA(Accounting) 20 0.500 50.0C(Computer) 15 0.375 37.5M(Marketing) 5 0.125 12.5

TOTALS 40 1.000 100.0

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• Paretochart–aseriesofverticalbarsshowingtallies/frequencies/percentagesindescendingorder

• Example:

• Discussion:HoworwhydoyouthinkthataParetochartwouldbeusefulinthebusinessworld?

• Side-by-SideBarcharts–Usessetsofbarstoshowthejointresponsefromtwocategoricalvariables

• Example:

• Discussion:Whatcanyoudetermineaboutproductutilizationforthisside-by-sidebarchartthatyoumightnotbeabletotellotherwise?

050100150200250300350400

ParetoChart

SummaryTableofCausesofIncompleteATMTransactions

Cause Frequency Percentage

ATMmalfunctions 32 4.42%

ATMoutofcash 28 3.87%

Invalidamountrequested 23 3.18%

Lackoffundsinaccount 19 2.62%

Cardunreadable 234 32.32%

Warpedcardjammed 365 50.41%

Wrongkeystroke 23 3.18%

TOTAL 724 100.00%

A-Q295

A-Q395

A-Q495

A-Q196

A-Q296

B-Q295

B-Q395

B-Q495

B-Q196

B-Q296

C-Q295

C-Q395

C-Q495

C-Q196

C-Q296

D-Q295

D-Q395

D-Q495

D-Q196

D-Q296

0

50,000

100,000

150,000

200,000

250,000

300,000

A-Q295

A-Q395

A-Q495

A-Q196

A-Q296

B-Q295

B-Q395

B-Q495

B-Q196

B-Q296

C-Q295

C-Q395

C-Q495

C-Q196

C-Q296

D-Q295

D-Q395

D-Q495

D-Q196

D-Q296

A4256 - Calibration Solution / Chips

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• 2.2OrganizingNumericalVariables• Orderedarrayarrangesthevaluesofanumericalvariableinrankorder(smallestvaluetolargest

value)ArrayèOrderedArray• Example(Table2.8A&B,p.42):

• FrequencyDistributiontalliesthevaluesofanumericalvariableintoasetofnumericallyorderedclasses,calledaclassinterval• Howmanyclasses?

• Determinetheintervalwidthbythefollowing:

• UsingourMealCostdata,weestimatethatwewant__________classessotheintervalwidthis:

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• RelativeFrequencyDistributionpresentsrelativefrequency,orproportionofthetotalforeachgroup

• Proportionorrelativefrequency,ineachgroupisequaltothenumberofvaluesineachclassdividedbythetotalnumberofvalues• Example:

CITY SUBURBAN

MealCost($)

FrequencyRelativeFrequency Percentage Frequency

RelativeFrequency Percentage

20,but<30 4 4 30,but<40 10 17 40,but<50 12 13 50,but<60 11 10 60,but<70 7 4 70,but<80 5 2 80,but<90 1 0

TOTALS 50 1 100.00% 50 1 100.00%

• TOTALoftherelativefrequencycolumnMUSTBE1.00• TOTALofthepercentagecolumnMUSTBE100.00

• CumulativePercentageDistributionprovidesawayofpresentinginformationaboutthe

percentageofvaluesthatlessthanaspecificamount

CITYandSUBURBAN

MealCost($)Frequency

RelativeFrequency Percentage

<lowerboundary CumulativePercentage<lowerboundary

20,but<30 8 0.08 8.0% <20 0(nomealscostlessthan$20)

30,but<40 27 0.27 27.0% <30 8%=0+8%

40,but<50 25 0.25 25.0% <40 35%=0+8%+27%50,but<60 21 0.21 21.0% <50 60%=0+8%+27%+25%

60,but<70 11 0.11 11.0% <60 81%=0+8%+27%+25%+21%

70,but<80 7 0.07 7.0% <70 92%=0+8%+27%+25%+21%+11%80,but<90 1 0.01 1.0% <80 99%=0+8%+27%+25%+21%+11%+7%

TOTALS 100 1.00 100.0% <90 100%=0+8%+27%+25%+21%+11%+7%+1%

• Question:Whatpercentageofmealcostswaslessthan$50?

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• 2.4VisualizingNumericalVariables• Stem-and-LeafDisplay–Howtocreate:

1. Separateeachobservationinto• Stem(allbutfinaldigit(s))and• Leaf(finaldigit(s)).

2. Writestemsinverticalcolumn–smallestontop3. Writeeachleaf,inincreasingnumericalorder,

inrownexttoappropriatestem• Example:Foreachstate,percentage(withonedecimal

place)ofresidents65andolder• Noticestemof“7”doesnothavealeafèweconcludeno

valueof7.xthereshouldbethesamenumberofleavesasobservations!IncludeALLstemsevenifnovalues/leaves• Leaveaspaceholderifnoleafforastem• Nopunctuation(i.e.,nodecimalpoints,nocommas)• Leavesshouldbelinedontopofoneanotherto

determineSHAPE• Simplewaytodeliveralotofdetailedinformation

FORTHISEXAMPLEreaddatavaluesas: 6.8 - 8.8 9.8,9.9 10.0,10.8 11.1,11.5,11.5,11.6,11.6 12.0,12.1,12.2,12.2,12.2,12.4,12.4,12.4,12.4,12,4,12.5,12.7,12.8,12.8,12.9,12.9,12.913.0,13.1,12.3,13.3,13.3,13.3,13.4,13.4,13.4,13.4,13.8,13.9,13.9 14.0,14.2,14.6,14.6.14.6 15.2,15.3 16.8

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• Histogram:Displaysaquantitativevariableacrossdifferentgroupingsofvalues• Carefulwhenchoosinghowtogrouptogethervalues!

• Groupingsmustcoverthesamerangesohaveofequalwidth• Heightusedtocomparethefrequencyofeachrangeofvalues• Stepstocreateafrequencyhistogram:

• Createequalwidthclasses(groupings)

• Countnumberofvaluesineachclass

• Drawhistogramwithabarforeachclass

• Heightofabarrepresentsthecountforthatbar’sclass

• BarstouchsincethereareNOGAPSbetweenclasses

• Becareful:• Numberofcategoriescan’tbe

toolargeortoosmall• Don’tskipanycategories• Beclearaboutcontentsof

eachcategory• HistogramExample:usingAgeatTimeofFirstOscarAward:

• Groupingschosenhereare:

[20,25)[25,30)[30,35)[35,40)[45,50),…• Where“[“meansthenumberisINCLUDEDintheinterval,

but“)”meansthenumberisNOTincludedintheinterval• Question:IfJackNicholsonwonBestActoratage70,whichcategoryfrequencywould

increase?A. [60,65)B. [65,70)C. [70,75)D. [75,80)

HISTOGRAMofMealCostLocation=CITY

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• Percentagepolygon–usedforvisualizationwhendividingthedataofanumericalvariableintotwoormoregroups

• Usesmidpointsofeachclasstorepresentthedataintheclass• Combinesdatafromtwogroupstoalloweasiercomparison

Let’s&examine&what&it&means&to&turn&a&frequency&into&a&rela6ve&frequency&by&looking&at&the&age&at&Oscar&data&

•  Rela6ve&frequency&histogram&depicts&the&rela6ve&frequency&rather&than&the&raw&frequency&(count)&of&categories&

•  Do&shapes&of&the&frequency&and&rela6ve&frequency&histograms&differ?&

TOTAL &&&76&

Conclusions?

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• CumulativePercentagePolygon(Ogive)usesthecumulativepercentagedistribution(discussedpreviously)toplotthecumulativepercentagesalongtheYaxis

• LOWERBOUNDSoftheclassintervalsareplottedontheXaxis

Conclusions?