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Section4:RepresentingQuantitaiveDataThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.47(c).4.01FrequencyTables
• Statistics(1)(E)• Statistics(4)(B)
4.02CumulativeFrequencyTables
• Statistics(4)(B)4.03DotPlotsandComparingDistributions
• Statistics(1)(D)• Statistics(1)(E)• Statistics(4)(B)
4.04StemplotsandComparingDistributions
• Statistics(1)(D)• Statistics(1)(E)• Statistics(4)(B)
4.05HistogramsandComparingDistributions
• Statistics(1)(D)• Statistics(1)(E)• Statistics(4)(B)• Statistics(4)(D)
4.06BoxPlotsandComparingDistributions
• Statistics(1)(D)• Statistics(1)(E)• Statistics(4)(B)• Statistics(4)(D)
Note:Unlessstatedotherwise,anysampledataisfictitiousandusedsolelyforthepurposeofinstruction.
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4.01FrequencyTables
Quantitativedata–Datadescribedusingnumbersthatserveasameasurement
• Oncedataiscollected,itneedstobesummarized.The______________________ofadatasetshowsthevaluesofthevariableandhowoftentheyoccur.
• Quantitativedatacanbesummarized__________________(usingtablesorpictures)or___________________(usingnumbers).
• Graphicallysummarizingquantitativedataallowsustoseeshape,center,spread,andpossibleoutliers.
• _______________andrelative_______________tablesdisplaythefrequencyorrelativefrequencyofeachinterval.
Considerthefollowingfrequencyandrelativefrequencytableandmakeobservationsaboutit.
Interval Frequency RelativeFrequency
[$40,$60) 480 480/1000=48%
[$60,$80) 390 390/1000=39%
[$80,$120) 130 130/1000=13%
Total 1000 1000/1000=100%
Toconstructafrequencytable,wedividetheobservationsinto_______________.
Weuseintervalswith_____________endpointsorchoose______________intervals.
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1. Suppose20nursesatMedicalCityDallasHospitalwererandomlysampledandaskedhowmuchtheyspentonfoodlastweekwhileatwork.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?
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4.02CumulativeFrequencyTables
Acumulativefrequencydistributionisusedtodeterminethenumberofobservationsbeloworaboveacertainvalue.
Completethecumulativeandrelativecumulativefrequencytablesbelow.
Interval Frequency RelativeFrequency Interval Cumulative
FrequencyRelativeCumulative
Frequency10–14 7 0.175 ≤14
15–19 3 0.075 ≤19
20–24 14 0.350 ≤24
25–29 11 0.275 ≤29
30–34 5 0.125 ≤34
Total 40 1.000
Thecumulativefrequencydistributioncanbedisplayedusingacumulativerelativefrequencyplot,asintheexamplebelow.
00.10.20.30.40.50.60.70.80.9
1
14 19 24 29 34
Cumulative Relative Frequency Plot
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1. Suppose20peopleatthelocalKrogersupermarketwererandomlyselectedonhowmuchtheyspent(indollars).Theresultsareshownbelow:
79 53 95 48 78 87 102 101 53 58
82 91 40 41 51 61 36 41 85 71
Completethetablebelow.Thencreateacumulativerelativefrequencyplotoftheabovedata.
Interval Frequency RelativeFrequency Interval Cumulative
FrequencyRelativeCumulative
Frequency
Total
0
0.2
0.4
0.6
0.8
1
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4.03DotPlotsandComparingDistributions
Adotplotdisplayseachindividualobservationusingadot.
Whyaredotplotsnotusefulfordisplayingreallylargedatasets?
Imaginethat50localhighschoolstudentswererandomlysampledandaskedhowmanytextstheysentyesterday.Thedataisshownbelow.
4 37 9 34 33 11 21 12 10 14
7 38 0 13 27 15 8 13 10 41
14 32 32 25 31 35 31 23 17 18
16 30 10 39 9 11 18 19 12 30
38 16 29 33 13 17 38 26 37 7
Adotplotofthedataisshownbelow.
Thedistributioncanbedescribedwithrespecttoshape,center,spread,andpotentialoutliers.
• Shape:• Center:• Spread:• Outliers:
0 10 20 30 40 50Number of Text Messages
Number of Text Messages Sent
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Thefollowingarevariousshapesofdistributions.Describeeachdotplotinthespaceprovided.
10 20 30 40 50 60 70
Symmetric (bell-shaped)
10 20 30 40 50 60 70
Symmetric (U-shaped)
10 20 30 40 50 60 70
Right-Skewed
10 20 30 40 50 60 70
Left-Skewed
10 20 30 40 50 60 70
Bimodal
10 20 30 40 50 60 70
Uniform (or rectangular)
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1. Supposetwolocalhighschools,SchoolAandSchoolB,competeatMathLeagueeveryyear.SchoolAhaswoneveryyear.SchoolBbelievestheycontinuetolosebecausetheyhaveateamwithverydiverseages,insteadofhavingagroupwithanarroweragecohort.
Thedatabelowgivetheagesofthe20studentsenrolledinMathLeagueateachschool.
Agesofthe20StudentsinMathLeagueatSchoolA
19 18 12 17 17 18 14 16 18 17
13 14 18 17 15 16 15 13 19 16
Agesofthe20StudentsinMathLeagueatSchoolB
12 18 13 14 17 12 16 12 13 18
14 13 12 12 15 16 18 15 13 12
i. Createadotplotforeachgrouptocomparethedistributionoftheagesofthe20studentsateachschool.Commentonthesimilaritiesanddifferencesbetweenthetwodistributions.
SchoolA
SchoolB
ii. UsetheinformationinthedotplotsthatyoucreatedtodiscussthevalidityoftheclaimsmadebySchoolB.
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2. Supposemaleandfemaleprofessorsfromthesamecollegewererandomlysampledandaskedhowmanycupsofcoffeetheydrinkperweek.Comparethedistributions.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Fem
ale
s
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ma
les
Number of Cups of Coffee
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4.04StemplotsandComparingDistributions
Astemplotdisplaysquantitativedata,usuallyfromsmalldatasets,usingstemsandleaves.
• Stemplotsalwaysincludeakeytodecodetheunitsofthestemandleaf,topreventconfusion.
• Example:4|1couldrepresent41,or4|1couldrepresent4.1.
Whyarestemplotsnotusefulfordisplayingreallylargedatasets?
Considerthefollowingstemplotsandcomparetheirdistributions.
StemplotA
2 6789
3 02367
4 25
5
6 9
StemplotB
1 4
2 589
3 12334
4 67
5 1
StemplotC
1 2
2 7
3 3
4 56799
5 0234
Key:2|6=26
1.Suppose20peopleatthelocalKrogersupermarketwererandomlyselectedonhowmuchtheyspent(indollars).Theresultsareshownbelow:
79 53 95 48 78
82 91 40 41 51
87 102 101 53 58
61 36 41 85 71
Constructandproperlylabelastemplotofthedataset.
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2. Suppose50collegestudentswererandomlysampledandaskedhowmanytextstheysentthepreviousday.Theresultsareshownbelow.
4 37 9 34 33 11 21 12 10 14
7 38 0 13 27 15 8 13 10 41
14 32 32 25 31 35 31 23 17 18
16 30 10 39 9 11 18 19 12 30
38 16 29 33 13 17 38 26 37 7
Shownbelowisastemplotandsplitstemplotofthedata.
StemplotofTextMessages
0 0477899
1 0001122333344566777889
2 135679
3 0011223345778889
4 1
(4|1represents41texts)
SplitStemplotofTextMessages
0 04
0 77899
1 000112233344
1 56677889
2 13
2 5679
3 001122334
3 5778889
4 1
(4|1represents41texts)
Whatisthedifferencebetweenastemplotandasplitstemplot?
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3. Shownbelowisaback-to-backsplitstemplotdisplayingthenumberofhoursofTVwatchedperweekforahypotheticalsampleofmalesandfemales.
Males Females
4443321100 0 3
9777655 0
422100 1 44
65 1 567778
11 2 11112223
2 5566777889
0 3 00000111124
3 001
4 0
(4|0represents40)
i. Whatdothestemsandleavesrepresentinthestemplot?
ii. Comparetheshape,center,andspreadofthedistributions.Arethereanyoutliers?
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4.05HistogramsandComparingDistributions
Histogramsdisplaythefrequencyorrelativefrequencyofdataoverintervals.Arehistogramsusefulfordisplayingsmalldatasets?Explain.
Suppose20cityresidentswererandomlysampledandaskedhowmuchtheyspentoncityparkinglastmonth.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%
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1. Constructafrequencyhistogramandrelativefrequencyhistogramfromtheparkingexpensedata.
Frequency Histogram of Amounts Spent on Parking
Relative Frequency Histogram of Amounts Spent on Parking
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1. Thehistogramsbelowsummarizethesameparkingexpensedatafromthepreviousquestion.ExplainthedifferencesbetweenHistogramAandHistogramB.
HistogramA
HistogramB
8070605040302010
40
30
20
10
0
Amount Spent
Perc
ent
Histogram of Amounts Spent
706050403020
50
40
30
20
10
0
Amount spent
Perc
ent
Histogram of Amounts Spent
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2. TherelativefrequencyhistogramsbelowdisplaythedelaytimesofAmericanAirlinesflightsdepartingfromHouston,Texas,ontwodifferentNewYear’sDays:25flightsonNewYear’sDay2016and18flightsonNewYear’sDay2017(BureauofTransportationStatistics,n.d.).
NewYear’sDay2016 NewYear’sDay2017
i. Comparethedistributions.
ii. WhatisyouradvicetoagroupoffriendswhoplantoflyfromHoustononNewYear’sDay,basedonthisdata?
0
10
20
30
40
50
60
70
Perc
ent
Departure delay (minutes)
05
1015202530354045
Perc
ent
Departure delay (minutes)
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4.06BoxPlotsandComparingDistributions
Aboxplotdisplaysthedistributionofadatasetbasedonthefive-numbersummary:minimum,firstquartile,median,thirdquartile,andmaximum.
Whatisthefive-numbersummary?
• ______________:thesmallestobservationinthedataset
• ______________:the25thpercentile(bottomhalfmedian)
• ______________:the50thpercentile(middleorderedobservation)
• ______________:the75thpercentile(tophalfmedian)
• ______________:thelargestobservationinthedataset
______________extendtotheminimumandmaximumvaluesthatarenotoutliers.
______________areobservationsthatfallbelow𝑄1 − 1.5×𝐼𝑄𝑅orabove𝑄3 + 1.5×𝐼𝑄𝑅,whereIQRistheinterquartilerange,definedas𝐼𝑄𝑅 = 𝑄3– 𝑄1.
1. Labelthecomponentsoftheboxplotbelow.
0 5 10 15 20 25 30
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2. Considerthefollowingboxplotsandtheirdata’srespectivehistograms.i. Writethreeobservationstocompareandcontrastthefollowingdistributions.
ii. Provideexamplesofreal-lifescenariosthatcouldberepresentedbyeachdistribution.
3. UsetheTI-84calculatoroutputontherighttoconstructaboxplotforthesorteddatasetontheleft.
12 25 35 37
38 39 40 41
41 42 44 45
46 46 51 64
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4. Matcheachhistogram(A,B,C,D,andE)toitscorrespondingboxplot(I,II,III,IV,andV).
765432
8
6
4
2
086420
24
18
12
6
06543210
4
3
2
1
0
6543210
8
6
4
2
06543210
6.0
4.5
3.0
1.5
0.0
AFrequency
B C
D E
VIVIIIIII
9
8
7
6
5
4
3
2
1
0
Frequency
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References
BureauofTransportationStatistics.(n.d.).AirlineOn-TimeStatistics:DetailedStatisticsDepartures.Retrievedfromhttps://www.transtats.bts.gov/ONTIME/Departures.aspx
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