Representing*Categorical* Data*
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1 Statistics RepresentingCategorical Data Thisstudyguidecovers: FrequencyTables Joint,Marginal,andConditionalFrequencies PieCharts BarCharts ComparingDistributions
2 FrequencyTables Whatisthedifferencebetweencategoricalandquantitativedata? Whyaretheredifferentwaystosummarizeanddisplaybothtypesofdata? Oncecategoricalorquantitativedatahasbeencollected,itmustbesummarized.The ofadatasetshowsthevaluesofthevariableandhowoftentheyoccur. Afrequencytableisconstructedbyarrangingasetofcollecteddatavalueswiththeir correspondingfrequencies. 1. Suppose50localhighschoolstudentswererandomlyselectedandeachstudentwasasked toratehisorherstudyhabits.shownbelowarethefrequencytableandrelativefrequency tablesummarizingtheresults.interpretthetables. FrequencyTable Rating Frequency Excellent 7 VeryGood 14 Good 16 Poor 9 VeryPoor 4 Total 50 RelativeFrequencyTable Rating RelativeFrequency Excellent 7/50=0.14 VeryGood 14/50=0.28 Good 16/50=0.32 Poor 9/50=0.18 VeryPoor 4/50=0.08 Total 50/50=1 1 StudyEdge studyedge.com
3 2. Suppose15randomadultsinalocalcommunitywererandomlyselected,andeachperson wasaskedtoratehowconcernedheorshewasaboutanewrestaurantthat sbeingbuilt. Theresponsesaredisplayedbelow.Completethefrequencytableandrelativefrequency tablesummarizingtheresults. 1 NotatAllConcerned Concerned Extremely Concerned Somewhat Concerned Somewhat Concerned Moderately Concerned 7 ExtremelyConcerned 12 8 ExtremelyConcerned 13 9 Moderately Concerned 5 SlightlyConcerned 10 ExtremelyConcerned 15 FrequencyTable Extremely Moderately Somewhat Slightly NotatAll Total Frequency Concerned 11 ExtremelyConcerned Moderately Concerned Somewhat Concerned 14 NotatAllConcerned Moderately Concerned RelativeFrequencyTable Extremely Moderately Somewhat Slightly NotatAll Total RelativeFrequency 2 StudyEdge studyedge.com
4 3. Completethefollowingfrequencytableandrelativefrequencytablesummarizingthe resultsfromahypotheticalstudythatevaluatedtheprioritylocalvotersplacedon educationincomparisontootherpoliciesintheupcominglocalelections. Priority FrequencyTable Essential 314 HighPriority MediumPriority LowPriority 86 NotaPriority Frequency Total 1000 Priority RelativeFrequencyTable Essential HighPriority 0.25 MediumPriority 0.28 LowPriority NotaPriority 0.07 Total RelativeFrequency 3 StudyEdge studyedge.com
5 Joint,Marginal,andConditionalFrequencies SupposeJoannasurveys20ofherfriendstodeterminethemostpopulargeometryteacherin theschool.sheasksherfriends,12boysandeightgirls,whichgeometryteachertheywould prefer:mr.lewinsky,ms.ramirez,ms.gundy,ormr.garnet.joannausesatwo_waytableto analyzeherdata. Atwo8waytableisusedtoexaminerelationshipsamongcategoricaldata.Theentriesinthe cellsofatwo_waytablecanbe or.this typeoftableisalsoknownasa table. Joanna stwo_waytableisshownbelow: Mr. Ms. Ms. Mr. Lewinsky Ramirez Gundy Garnet Total Boys Girls Total Entriesinthe"Total"columnand"Total"rowarecalled frequencies. Entriesinthebodyofthetablearecalled frequencies. Marginal arefoundbydividingtherowand/orcolumntotalsby thetotalsamplesize. 1. Completethefollowingtables: Gender MarginalDistribution Probability Boys 12/20= Girls 8/20= Total 20/20=1 Teacher MarginalDistribution Probability Mr.Lewinsky 5/20= Ms.Ramirez 3/20= Ms.Gundy 5/20= Mr.Garnet 7/20= Total 20/20=1 4 StudyEdge studyedge.com
6 Also, frequenciesarefoundbydividingthecellcountsbytheirrowand/or columntotals. 2. Completethefollowingtables: DistributionforBoys Teacher Probability Mr.Lewinsky 4/12= Ms.Ramirez 2/12= Ms.Gundy 3/12= Mr.Garnet 3/12= Total 12/12=1 DistributionforGirls Teacher Probability Mr.Lewinsky 1/8= Ms.Ramirez 1/8= Ms.Gundy 2/8= Mr.Garnet 4/8= Total 8/8=1 5 StudyEdge studyedge.com
7 3. ShownbelowisacontingencytableclassifyingresponsesforthevariablesSocialLifeRating andgender. Excellent Very Very Good Poor Good Poor Total Male Female Total i. DeterminethemarginalrelativeprobabilitiesforGenderandSocialLifeRatingby completingthetablesbelow. Gender MarginalRelativeProbabilities Male Female Probability Total 50/50=1 SocialLife MarginalRelativeProbabilities Excellent VeryGood Good Poor VeryPoor Probability Total 50/50=1 ii. Whatisthemarginalrelativefrequencyofpeoplewitha Good socialliferating? iii. Whatisthejointfrequencyofmaleswitha VeryGood socialliferating? iv. Whatistheconditionalrelativefrequencyoffemaleswithan Excellent sociallife rating? 6 StudyEdge studyedge.com
8 PieCharts Apiechartdisplaysdataina graph. Thepiecesofthegraphare tothefractionofthewholeineach category. Theareaofeachsliceisdeterminedbyeachcategory scountorpercentage. 1. Considerthefollowingpiechart,whichrepresentstheeatinghabitsof50employeesata localcompany. PieChartofEmployeeEatingHabits Poor 18% VeryPoor 8% Excellent 14% Good 32% VeryGood 28% Howcanweinterpretthispiechart? 7 StudyEdge studyedge.com
9 2. Assumethatthefollowingdatarepresentstheeducationalattainmentofpersonsina localcounty. i. Completetherelativefrequencytable. FrequencyTable Educational Frequency Attainment NoHighSchool 15,000 HighSchool 25,000 SomeCollege 15,000 Associate s 25,000 Bachelor s 100,000 Advanced 20,000 Total 200,000 ii. Constructapiechartofthedata. RelativeFrequencyTable Educational Relative Attainment Frequency NoHighSchool HighSchool SomeCollege Associate s Bachelor s Advanced Total 8 StudyEdge studyedge.com
10 3. Considerthepiechartbelow: AverysInvestmentPortfolio Preferred Stocks,25% Common Stocks,12% MutualFunds, 30% Bonds,33% SupposethispiechartdisplaysthedetailsofAverysinvestmentportfolio.Theportfoliosvalue is$300,000.therearethreevarietiesofmutualfunds:equityfunds(25%),fixed_incomefunds (35%),andmoneymarketfunds(40%). i. Whatpercentageoftheportfolioconsistsofequityfunds? ii. HowmuchmoneydoesAveryhaveinvestedinbonds? iii. HowmuchmoneydoesAveryhaveinvestedinmoneymarketfunds? 9 StudyEdge studyedge.com
11 BarCharts Abarchartdisplaysthefrequencyorrelativefrequencyforeachcategoryofacategorical variableusingbarsonagraph. Thebarsarepositionedwithalabelthatrepresentsa. Thebarscanbedrawneitherverticallyor. Theheightorlengthofthebarindicatesthe ofthegroup,definedbythe columnorrowlabel. Formostcategoricalvariables,thebarorderis. 1. Assume500teenswereinterviewedandtheresultsareshowninthebarchartbelow: 40% HowFamiliarAreMillenialswith90sPopMusic? 30% 20% 10% 0% Extremely familiar Veryfamiliar Howmanyteensinthesamplewereveryfamiliarwith90spopularmusic? Somewhat familiar Slightly familiar Notatall familiar 10 StudyEdge studyedge.com
12 2. Suppose50APStatisticsstudentswererandomlyselected,andeachstudentwasaskedto ratehisorherteacher.shownbelowisthefrequencytablesummarizingtheresults. Constructabarchartofthedata. Rating FrequencyTable Frequency Excellent 15 VeryGood 14 Good 10 Poor 6 VeryPoor 5 Total APStatisticsTeacherRatings Excellent VeryGood Good Poor VeryPoor 11 StudyEdge studyedge.com
13 ComparingDistributions 1. Suppose100peopleinthe26 35agegroupwererandomlyselected,andeachpersonwas askedtoratehisorhersociallife.ifsocialliferatingisindependentofgender,aside_by_ sidebarchartoftheconditionaldistributionswouldlookliketheonebelow. SocialLifeRatingbyGender 35% 30% 25% 20% 15% 10% 5% 0% Excellent VeryGood Good Poor VeryPoor SocialLifeRating Male Female Whatconclusioncanbedrawnaboutthedataabove? 12 StudyEdge studyedge.com
14 2. Now,let sconsiderstudents attitudestowardaswitchtoblockscheduling.suppose500 studentswererandomlyselected:50%malesand50%females.thebargraphdisplaying theresultsofthesurveyisshownbelow. StudentsAttitudesTowardtheSwitchtoBlockScheduling 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0% Strongly Approve Approve Neutral Disapprove Strongly Disapprove Male Female Whatconclusionscanbedrawn? 13 StudyEdge studyedge.com
15 3. Suppose100peoplewereinterviewedandclassifiedaccordingtotheirattitudestoward publictransportationintheircity,aswellastheirpersonalitytypes.theresultsareshownin thebarchartbelow AttitudeTowardPublicTransportationbyPersonality Type Positive Neutral Negative TypeA Whichofthefollowingstatementsiscorrect? TypeB A. Ofthepeopleinterviewed,35%haveapositiveattitudetowardpublictransportation. B. Therewere50peoplewithaneutralattitudetowardpublictransportation. C. FortypeoplewithTypeBpersonalitywereinterviewed. D. PeoplewithTypeApersonalitytendtohaveanegativeattitudetowardpublic transportation. E. Ofthepeopleinterviewed,50%haveTypeApersonality. 14 StudyEdge studyedge.com
16 4. Supposearandomsampleof100voterswastaken.Eachvoterwasaskedhisorherpolitical affiliationandwhetherheorshefavoredacertainpoliticalissue.theirresponsesarecross_ classifiedinthecontingencytableandpiechartsbelow. FavorIssue PartyAffiliation Yes No Total Democrat Republican Independent Total VoterswhoResponded"Yes" VoterswhoResponded"No" Democrat Republican Independent Democrat Republican Independent Whatconclusionscanbedrawn? 15 StudyEdge studyedge.com
17 Statistics RepresentingQuantitative Data Thisstudyguidecovers: FrequencyTables CumulativeFrequencyTables DotPlotsandComparingDistributions StemplotsandComparingDistributions HistogramsandComparingDistributions BoxPlotsandComparingDistributions
18 FrequencyTables Quantitativedata Datadescribedusingnumbersthatserveasameasurement Oncedataiscollected,itneedstobesummarized.The ofa datasetshowsthevaluesofthevariableandhowoftentheyoccur. Quantitativedatacanbesummarized (usingtablesorpictures)or (usingnumbers). Graphicallysummarizingquantitativedataallowsustoseeshape,center,spread,and possibleoutliers. andrelative tablesdisplaythefrequencyor relativefrequencyofeachinterval. Considerthefollowingfrequencyandrelativefrequencytableandmakeobservationsaboutit. Interval Frequency RelativeFrequency [$40,$60) /1000=48% [$60,$80) /1000=39% [$80,$120) /1000=13% Total /1000=100% Toconstructafrequencytable,wedividetheobservationsinto. Weuseintervalswith endpointsorchoose intervals. 1 StudyEdge studyedge.com
19 1. Suppose20nursesatMedicalCityDallasHospitalwererandomlysampledandaskedhow muchtheyspentonfoodlastweekwhileatwork.theamountsspentareshownbelow, sortedinascendingorder. $5 $6 $6 $6 $9 $9 $10 $11 $11 $16 $16 $18 $19 $19 $19 $21 $22 $22 $24 $25 i. Completethefrequencyandrelativefrequencytablesbelow. Interval Frequency RelativeFrequency Total ii. Howmanynursesspentlessthan$15? 2 StudyEdge studyedge.com
20 CumulativeFrequencyTables Acumulativefrequencydistributionisusedtodeterminethenumberofobservationsbelowor aboveacertainvalue. Completethecumulativeandrelativecumulativefrequencytablesbelow. Interval Frequency Relative Frequency Interval Cumulative Frequency RelativeCumulative Frequency Total Thecumulativefrequencydistributioncanbedisplayedusingacumulativerelativefrequency plot,asintheexamplebelow Cumulative Relative Frequency Plot 3 StudyEdge studyedge.com
21 1. Suppose20peopleatthelocalKrogersupermarketwererandomlyselectedonhowmuch theyspent(indollars).theresultsareshownbelow: Completethetablebelow.Thencreateacumulativerelativefrequencyplotoftheabove data. Interval Frequency Relative Frequency Interval Cumulative Frequency RelativeCumulative Frequency Total StudyEdge studyedge.com
22 DotPlotsandComparingDistributions Adotplotdisplayseachindividualobservationusingadot. Whyaredotplotsnotusefulfordisplayingreallylargedatasets? Imaginethat50localhighschoolstudentswererandomlysampledandaskedhowmanytexts theysentyesterday.thedataisshownbelow Adotplotofthedataisshownbelow. Number of Text Messages Sent Number of Text Messages Thedistributioncanbedescribedwithrespecttoshape,center,spread,andpotentialoutliers. Shape: Center: Spread: Outliers: 5 StudyEdge studyedge.com
23 Thefollowingarevariousshapesofdistributions.Describeeachdotplotinthespaceprovided. Symmetric (bell-shaped) Symmetric (U-shaped) Right-Skewed Left-Skewed Bimodal Uniform (or rectangular) StudyEdge studyedge.com
24 1. Supposetwolocalhighschools,SchoolAandSchoolB,competeatMathLeagueeveryyear. SchoolAhaswoneveryyear.SchoolBbelievestheycontinuetolosebecausetheyhavea teamwithverydiverseages,insteadofhavingagroupwithanarroweragecohort. Thedatabelowgivetheagesofthe20studentsenrolledinMathLeagueateachschool. Agesofthe20StudentsinMathLeagueatSchoolA Agesofthe20StudentsinMathLeagueatSchoolB i. Createadotplotforeachgrouptocomparethedistributionoftheagesofthe20 studentsateachschool.commentonthesimilaritiesanddifferencesbetweenthetwo distributions. SchoolA SchoolB ii. Usetheinformationinthedotplotsthatyoucreatedtodiscussthevalidityoftheclaims madebyschoolb. 7 StudyEdge studyedge.com
25 2. Supposemaleandfemaleprofessorsfromthesamecollegewererandomlysampledand askedhowmanycupsofcoffeetheydrinkperweek.comparethedistributions Males Females Number of Cups of Coffee 8 StudyEdge studyedge.com
26 StemplotsandComparingDistributions Astemplotdisplaysquantitativedata,usuallyfromsmalldatasets,usingstemsandleaves. Stemplotsalwaysincludeakeytodecodetheunitsofthestemandleaf,toprevent confusion. Example:4 1couldrepresent41,or4 1couldrepresent4.1. Whyarestemplotsnotusefulfordisplayingreallylargedatasets? Considerthefollowingstemplotsandcomparetheirdistributions. StemplotA StemplotB StemplotC Key:2 6= Suppose20peopleatthelocalKrogersupermarketwererandomlyselectedonhowmuch theyspent(indollars).theresultsareshownbelow: Constructandproperlylabelastemplotofthedataset. 9 StudyEdge studyedge.com
27 2. Suppose50collegestudentswererandomlysampledandaskedhowmanytextstheysent thepreviousday.theresultsareshownbelow Shownbelowisastemplotandsplitstemplotofthedata. StemplotofTextMessages SplitStemplotofTextMessages (4 1represents41texts) (4 1represents41texts) Whatisthedifferencebetweenastemplotandasplitstemplot? 10 StudyEdge studyedge.com
28 3. ShownbelowisabackctocbacksplitstemplotdisplayingthenumberofhoursofTVwatched perweekforahypotheticalsampleofmalesandfemales. Males Females (4 0represents40) i. Whatdothestemsandleavesrepresentinthestemplot? ii. Comparetheshape,center,andspreadofthedistributions.Arethereanyoutliers? 11 StudyEdge studyedge.com
29 HistogramsandComparingDistributions Histogramsdisplaythefrequencyorrelativefrequencyofdataoverintervals. Arehistogramsusefulfordisplayingsmalldatasets?Explain. Suppose20cityresidentswererandomlysampledandaskedhowmuchtheyspentoncity parkinglastmonth.theamountsspentareshownbelowsortedfromlowesttohighest. $15 $17 $18 $18 $19 $22 $25 $25 $26 $27 $28 $28 $29 $31 $33 $34 $43 $56 $68 $72 Shownbelowisa and oftheamountsspent. Interval Frequency RelativeFrequency [$10,$20) 5 5/20=25% [$20,$30) 8 8/20=40% [$30,$40) 3 3/20=15% [$40,$50) 1 1/20=5% [$50,$60) 1 1/20=5% [$60,$70) 1 1/20=5% [$70,$80) 1 1/20=5% Total 20 20/20=100% 12 StudyEdge studyedge.com
30 1. Constructafrequencyhistogramandrelativefrequencyhistogramfromtheparking expensedata. Frequency Histogram of Amounts Spent on Parking Relative Frequency Histogram of Amounts Spent on Parking 13 StudyEdge studyedge.com
31 1. Thehistogramsbelowsummarizethesameparkingexpensedatafromtheprevious question.explainthedifferencesbetweenhistogramaandhistogramb. HistogramA Percent Amount0Spent HistogramB Percent Amount/spent 14 StudyEdge studyedge.com
32 2. TherelativefrequencyhistogramsbelowdisplaythedelaytimesofAmericanAirlinesflights departingfromhouston,texas,ontwodifferentnewyear sdays:25flightsonnewyear s Day2016and18flightsonNewYear sday2017(bureauoftransportationstatistics,n.d.). NewYear sday2016 NewYear sday2017 Percent Percent Departuredelay(minutes) Departuredelay(minutes) i. Comparethedistributions. ii. WhatisyouradvicetoagroupoffriendswhoplantoflyfromHoustononNewYear s Day,basedonthisdata? 15 StudyEdge studyedge.com
33 BoxPlotsandComparingDistributions Aboxplotdisplaysthedistributionofadatasetbasedonthefivecnumbersummary:minimum, firstquartile,median,thirdquartile,andmaximum. Whatisthefivecnumbersummary? :thesmallestobservationinthedataset :the25thpercentile(bottomhalfmedian) :the50thpercentile(middleorderedobservation) :the75thpercentile(tophalfmedian) :thelargestobservationinthedataset extendtotheminimumandmaximumvaluesthatarenotoutliers. areobservationsthatfallbelow"1 1.5 )"orabove" )", whereiqristheinterquartilerange,definedas)" = "3 "1. 1. Labelthecomponentsoftheboxplotbelow StudyEdge studyedge.com
34 2. Considerthefollowingboxplotsandtheirdata srespectivehistograms. i. Writethreeobservationstocompareandcontrastthefollowingdistributions. ii. Provideexamplesofrealclifescenariosthatcouldberepresentedbyeachdistribution. 3. UsetheTIc84calculatoroutputontherighttoconstructaboxplotforthesorteddataset ontheleft StudyEdge studyedge.com
35 StudyEdge studyedge.com Matcheachhistogram(A,B,C,D,andE)toitscorrespondingboxplot(I,II,III,IV,andV) A Frequency B C D E V IV III II I Frequency
36 References( BureauofTransportationStatistics.(n.d.).AirlineOn)TimeStatistics:DetailedStatistics Departures.Retrievedfromhttps:// 19 StudyEdge studyedge.com
37 Statistics MeasuringCenterandSpread Thisstudyguidecovers: MeasuringCenterofaDistribution MeasuringSpreadofaDistribution Outliers VisualizingandTransformingData: ImplicationsforMeanandStandard Deviation IntroductiontotheEmpiricalRule TheNormalDistribution
38 MeasuringCenterofaDistribution Threecommonmeasurementsofcenter,orcentraltendency,aremean,median,andmode. Definition Mean Median Mode 1. Calculatethemean,median,andmodeforthedatainthefollowingdotplot. Dot Plot StudyEdge studyedge.com
39 2. Calculatethemeanandmedianforeachdataset.Whatdoyounoticeaboutthe relationshipbetweenthemeanandmedianineachcase? i ii Basedontheshapeofthedistribution,themeanandmedianarerelatedasfollows: RightCskewed LeftCskewed BellCshaped Uniform MeanMedian MeanMedian MeanMedian MeanMedian 2 StudyEdge studyedge.com
40 MeasuringSpreadofaDistribution Fourcommonmeasuresofspreadarerange,variance,standarddeviation,andinterquartile range(iqr).completethefollowingtableforthemeasuresofspread. Definition Formula Range Thedifferencebetweenthe maximumandminimumvalues (Note:Rangeisthemeasureof spreadmostaffectedby outliers.) Variance " # = 1 & 1 ( ) ( # Standarddeviation The typical deviationfrom themean Interquartilerange (IQR) +, = / 3 StudyEdge studyedge.com
41 1. Thestartingsalaries(inthousandsofdollars)of9recentcollegegraduatesareshown below,sortedinascendingorder. i. Calculatethemean. ii. Findthemedian. iii. Findthemode. iv. Calculatetherange. v. Calculatetheinterquartilerange. vi. Calculatethestandarddeviation StudyEdge studyedge.com
42 2. Belowisastemplotofthetimeittookaclassroomof25studentstocompletetheirfirst statisticsexamtothenearestminute. i. Calculatethemean. ii. Findthemedian. iii. Findthemode. iv. Calculatetherange. Time(tothenearestminute)tocompleteExam1 0 v. Calculatetheinterquartilerange. vi. Calculatethevariance. vii.calculatethestandarddeviation (1 8represents18) 5 StudyEdge studyedge.com
43 Outliers Anoutlierisanextremeobservationwithavaluefaraboveorbelowtherestofthedata. 1. Calculatethemeanandmedianforeachdataset: DataSet1: DataSet2: , Indataset2,thevalues71and1,000areoutliers.Howdotheyaffectthemeanand median? 3. Whichmeasureismoresensitivetooutliers:themeanormedian?Explainyourreasoning. 6 StudyEdge studyedge.com
44 HowtoCheckforOutliers Using+ / and+ - fromthefive\numbersummary,observationsthatfallbelow+ / 1.5 +, orabove ,areconsideredoutliers. 4. ThedelayindepartureforeachofJetBlue sfirst15flightsintheyear2017fromthe OrlandoInternationalAirportareshownbelow,whereanegativevaluemeansaflight departedaheadofscheduleandapositivevaluerepresentsadelayindeparture(bureauof TransportationStatistics,n.d.) Usetheoutlierchecktoidentifyanyoutliersinthedataset. 7 StudyEdge studyedge.com
45 VisualizingandTransformingData: ImplicationsforMeanandStandardDeviation Considerthenumbers5,5,5,and5.Whatarethemeanandstandarddeviation? Considerthenumbers3.5,4.5,5.5,and6.5.Whatarethemeanandstandarddeviation? Considerthenumbers0,0,10,and10.Whatarethemeanandstandarddeviation? Themeanmeasureswherethecenterofthedatavaluesis.Thestandarddeviationmeasures howspreadoutthevaluesarearoundthemean. 1. Drawconclusionsfromthedatasetsabove. 8 StudyEdge studyedge.com
46 2. Considerthenumbers2,5,and8.Determinethemeanandstandarddeviationofthe originaldatasetandthetransformationsofthedataset(asshowninthetablebelow). Originaldata Add4 Multiplyby2 2,5,and8 Mean: StandardDeviation: 6,9,and12 Mean: StandardDeviation: 4,10,and16 Mean: StandardDeviation: 3. Howdoestransformingadatasetwithadditionandsubtractionaffectthemeanand standarddeviation? 4. Howdoestransformingadatasetwithmultiplicationanddivisionaffectthemeanand standarddeviation? 9 StudyEdge studyedge.com
47 5. Supposethestartingsalaries(inthousandsofdollars)of 12recentcollegegraduatesarerecorded,andthe summarystatisticsareshowntotheright.whatwould thenewmeanandstandarddeviationbeifeveryone weretoreceivea$2,000raise? 6. Considerallintegersfrom25to75. i. Whichfourvalueshavethesmalleststandarddeviation?Explain. ii. Whichfourvalueshavethelargeststandarddeviation?Explain. 10 StudyEdge studyedge.com
48 IntroductiontotheEmpiricalRule Howcanwecomparetheheightsofamanandawomantoseewhoistaller,relativetotheir gender? Weoftenneedtostandardizethevaluesofapopulationdistributiontocomparevalueswithin adistributionorbetweendistributions. Anormaldistributionisacontinuousdistributionthatisunimodalandsymmetric,or approximatelysymmetric,withabellshape.withthisdistribution,thestandarddeviationcan beusedforamoreprecisemeasurementofthepercentageofdatainthedistribution. Anormaldistributionisnotatedas:(<, >),wherethepopulationmeanis<andthepopulation standarddeviationis>. Theempiricalrule,alsoknownasthe rule,appliestothenormaldistribution.It statesthat 68%ofobservationsfallwithinonestandarddeviationofthemean, 95%ofobservationsfallwithintwostandarddeviationsofthemean,and 99.7%ofobservationsfallwithinthreestandarddeviationsofthemean. VisualizingtheEmpiricalRule < ± 1> 68% < ± 2> 95% < ± 3> 99.7% 1. Supposetheamountofsodaina12\ozcanofsodahasanormaldistributionwithameanof 12.25ozandastandarddeviationof0.15oz. 11 StudyEdge studyedge.com
49 i. Themiddle95%ofweightsfallbetweenwhattwovalues? ii. Whatpercentageofcanscontainbetween11.95and12.40oz? iii. Whatpercentageofcanscontainmorethan12.40oz? iv. Whatpercentageofcanscontainexactly12.10oz? 2. Supposethelifetimeofacertainbrandoftireisnormallydistributedwithameanof60,000 milesandastandarddeviationof5,000miles. 12 StudyEdge studyedge.com
50 i. Thecentral99.7%oflifetimesarebetweenwhattwomileages? ii. Approximatelywhatpercentageoftireslastbeyond55,000miles? iii. Themanufacturerdoesnotwanttoprovideawarrantyfortiresoncetheyreachthetop 2.5%oftheirlifetime.Whatisthemileagelimitofthewarranty? 13 StudyEdge studyedge.com
51 TheNormalDistribution Whatpercentageofvaluesarewithin1.5standarddeviationsfromthemean?Whenwecannot applythe rule,weuseaz=tableofvalues. Az=tabledisplayscumulativeprobabilitiesforthestandardnormaldistribution,F~:(0,1). Whenweusenormaldistributionswithothermeansandstandarddeviations,az=scorecan determinethenumberofstandarddeviationsanobservationisfromitsmean.wecanthen findthecorrespondingcumulativeprobabilityforthatobservationusingthez\table. F = ( < > 1. Usethez\tabletofindthefollowingvalues. i. Theareabelowaz\scoreof1.5 ii. Theprobabilityofobtainingaz\scoregreaterthan StudyEdge studyedge.com
52 iii. Theprobabilityofobtainingaz\scorebetween 1.11and0.57 iv. Thez\scorethatcutsoffthetop30% v. Thez\scoresthatcapturethemiddle60% 15 StudyEdge studyedge.com
53 2. Supposetheaverageheightofamalehighschoolseniorfollowsanormaldistributionwith ameanof67inchesandastandarddeviationof3inches. i. Findtheprobabilitythatarandomlyselectedmalestudentislessthan65inchestall. ii. Findtheprobabilitythatarandomlyselectedmalestudentisatleast68inchestall. iii. Findtheprobabilitythatarandomlyselectedmalestudentisexactly67inchestall. iv. Whatheightmarksthestartofthetallest10%ofmales? 16 StudyEdge studyedge.com
54 3. Basedonthedatainthepreviousquestion,ifamalehighschoolseniorhasaz\scoreof 1.55,whatconclusioncanbedrawn? A. Heisabovetheaverageheight. B. Heisbelowtheaverageheight. C. Heistheaverageheight. D. Cannotbedetermined. 4. SupposetheheightofmeninNorthAmericafollowsanormaldistributionwithameanof 68inchesandstandarddeviationof4inches,whiletheheightofwomeninNorthAmerica followsanormaldistributionwithameanof63inchesandstandarddeviationof3inches. Whoistallerforhisorhersex:aman73inchestallorawoman68inchestall? 17 StudyEdge studyedge.com
55 Ztable StudyEdge studyedge.com
56 StudyEdge studyedge.com
57 References BureauofTransportationStatistics.(n.d.).AirlineOn2TimeStatistics:DetailedStatistics Departures.Retrievedfromhttps:// 20 StudyEdge studyedge.com
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