Section 4: Representing Quantitaive Data1. The histograms below summarize the same parking expense data from the previous question. Explain the differences between Histogram A and

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.

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

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.


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.


Total

ii. Howmanynursesspentlessthan$15?


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


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


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


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)


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.


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


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.


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?


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?


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.


[$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%


1. Constructafrequencyhistogramandrelativefrequencyhistogramfromtheparkingexpensedata.

Frequency Histogram of Amounts Spent on Parking

Relative Frequency Histogram of Amounts Spent on Parking


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


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)


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


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


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


References

BureauofTransportationStatistics.(n.d.).AirlineOn-TimeStatistics:DetailedStatisticsDepartures.Retrievedfromhttps://www.transtats.bts.gov/ONTIME/Departures.aspx


Documents

Section 4: Representing Quantitaive Data1. The histograms below summarize the same parking expense data from the previous question. Explain the differences between Histogram A and