Graphical Representation of Statistical Data

GRAPHICAL REPRESENTATION: Tabulation is a good method of condensing and representing data in a readily understandable form, but many people have no taste for figures. They would prefer a way of representation where figures could be avoided. This purpose is achieved by the presentation of statistical data in a visual form. The visual display of statistical data in the form of points, lines, areas and other geometrical forms and symbols, is the most general terms known as Graphical Representation. Statistical data can be studied with this method without going through figures, presented in the form of tables.Such visual representation can be described in the sections that follow. The basic difference between a graph and a diagram is that a graph is a representation of data by a continuous curve, usually shown on a graph paper while a diagram is any other one, two or three-dimensional form of visual representation

1) Simple Bar Diagrams-Simple Bar Chart: Simple bar diagrams are made to represent geographical, historical, numerical and the qualitative data. The vertical or horizontal bars are made to represent the data when the difference between different quantities is not very large. The different quantities may be arranged in ascending or descending order but the time series data (A time series consists of numerical data collected, observed or recorded at more or less regular intervals of time each hour, day, month, quarter or year.) are not arranged.

Suggestions for Constructing Bar Chart:1. The bars should be constructed horizontally when the categorized

observations are the outcomes of a categorical variable. The bars should be constructed vertically when the categorized observations are outcomes of a numerical variable.

2. All bars should have the same width so as not to mislead the reader. Only the length should differ.

3. Spaces between bars should range from one-half the width of a bar to the width of a bar

4. Scales and guidelines are useful aids in reading a chart and should be included. The zero point or origin should be indicated.

5. The axes of the chart should be indicated.6. Any “keys” to interpreting the chart may be included within the

body of the chart or below the body of the chart.7. Footnotes or source notes, when appropriate, are presented after

the title of the chart or at the bottom edge of the chart’s frame.

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a) Categorical Data:-1) Example: - Squash World Open Champions from 1976-2001.

Country No. of timesPakistan 14Australia 6

United Kingdom 2Canada 2

Newzeland 1

Squash World Open Champions

14

6

2

2

1

0 5 10 15

Pakistan

Australia

United Kingdom

canada

Newzeland

Co

un

trie

s

No.of times

No of times

2) Example: - Worlds Largest Deserts by Area in Km2.Source: -

Microsoft Encarta

Deserts Sq KmSahara 9100000Gobi 1300000

Patagonian 670000Rub al Khali 650000Great Sandy 390500

Great Victoria 390500

2

2

Area

9100000

1300000

670000

650000

390500

390500

0 2000000 4000000 6000000 8000000 10000000

Sahara

Gobi

Pantagonian

Rub al Khali

Great Sandy

Great VictoriaD

eser

ts

Arears

Area

b) Numerical Data:-1) Example: - The Following data indicates Consumption of Raw Material by Industry in Pakistan during 1999-2004.

Source:-APTMAperiod Consumption ('000' Kgs)1999-00 1,566,3482000-01 1,673,2802001-02 1,755,6692002-03 1,943,1972003-04 1,938,678

CONSUMPTION OF RAW MATERIAL

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

1999-00

2000-01

2001-02

2002-03

2003-04

Period

Conum

tion(0

00 K

gs

)

Consuption ('000'Kgs)

2) Example: - The following data shows Pakistan’s share of World Trade in Cotton Yarn. Data shown in percent of world trade. Source: - World Textile Demand

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3

Year Share (%)1999 26.12000 27.32001 26.92002 272003 23.8

Share

26.1

27.326.9 27

23.8

22

23

24

25

26

27

28

1999 2000 2001 2002 2003

Share

1. The following table gives the birth rate per thousand of different countries over a certain period of time.

COUNTRY POPULATION RATE IN THOUSANDS

INDIA 33GERMANY 15UK 20CHINA 40DENMARK 30SWEDEN 15

DIAGRAM:

33

1520

40

30

15

0

10

20

30

40

50

INDIA GERMANY UK CHINA DENMARK SWEDEN

2. The following table shows the price list of various brands of cars with 2 Liter engine.

4

4

DIAGRAM:

1554000 17340002500000

5800000

33000002015000

01000000200000030000004000000500000060000007000000

CARS

VA

LU

E I

N R

UP

EE

S

Data Array: The arrangement of raw data by observations in either ascending or descending order.

2) Multiple Bar Diagrams:-A multiple bar chart shows two or more characteristic corresponding to the values

of a common variable in the form of grouped bars, whose lengths are proportional to the values of the characteristics, and each of which is shaded or coloured differently to aid identification. This is a good device for the comparison of two or more kinds of information. For example, imports, exports and productions of a country can be compared from year to year by grouping the three bars together.

1) Example: - The Following data shows production of yarn by Punjab and Sindh from 2000-04. Source: - APTMA

Production (000 Kgs)Period Punjab Sindh2000-01 1,239,358 365,0292001-02 1,283,964 391,492

5

5

CAR VALUE IN RUPEESTOYOTA 1554000NISSAN 1734000HONDA 2500000MERCEDES 5800000BMW 3300000MITSUBISHI 2015000

2002-03 1,353,027 417,5132003-04 1,411,817 428,411

0

500,000

1,000,000

1,500,000

2000-01 2001-02 2002-03 2003-04

Punjab

Sindh

2) Example: - The Following data shows Marks obtained position holders boys and girls in Federal Board HSSC-2 examination 2005. Source: - www.fbise.edu.pk

Discipline Boys GirlsPre-Engineering 978 1003

Pre-Medical 974 979Humanities 896 909Commerce 844 854

Top position in Federal Board

978 974

896

844

1003979

909

854

750

800

850

900

950

1000

1050

Pre-Engeneering

Pre-Medical Humanities Commerce

Disipline

Ma

rks o

bta

ine

d

Boys

Girls

1. The table below gives data relating to the exports and imports of a certain country X (in thousands of dollars) during the four years ending in 1930 - 31.

YEAR EXPORT IMPORT2000-01 315 2492001-02 423 2342002-03 123 3002003-04 278 2502004-05 350 190

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6

DIAGRAM:

315

423

123

278

350

249 234

300250

190

0

100

200

300

400

500

2000-01 2001-02 2002-03 2003-04 2004-05

YEARS

$ I

N T

HO

US

AN

DS

EXPORT

IMPORT

2. The table given below shows the values of shares of P.T.C.L.A, OGDCL, & PPL in KSE on trading days of a week.

SHARES MON TUES WED THURS FRIP.T.C.L.A 62.58 63.15 65.69 66.70 68.85OGDCL 115.45 118.90 119.95 123.30 125.50PPL 208.30 204.15 202.36 205.70 210.95

DIAGRAM:

62.58 63.15 65.69 66.7 68.85

115.45 118.9 119.95 123.3 125.5

208.3 204.15 202.36 205.7 210.95

0

50

100

150

200

250

MON TUES WED THURS FRI

TRADING DAYS

VA

LU

E I

N R

UP

EE

S

P.T.C.L.A

OGDCL

PPL

3) Component Bar Chart:-A component bar chart is an effective technique in which each bar is divided into

two or more sections, proportional in size to the component parts of a total being displayed by each bar. The various component parts shown as sections of the bar are shaded or coloured differently to increase the overall effectiveness of the diagram. The

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component bar charts are used to present the cumulation of the various components of data and the percentages. They are also known as sub-divided bars.

1) Example: - The following data shows rate of Birth, death and migrants per 1000 in different regions of world.

Source:-Microsoft Encarta

RegionRate Per 1000

Births Deaths MigrantsWorld 21.2 8.9 1.5Africa 35.9 14.2 1.7Asia 20.7 7.6 2

Europe 10.2 11.4 1.1

World Population rates

0%

20%

40%

60%

80%

100%

World Africa Asia Europe

Regions

Migrants

deaths

Births

2) Example:-The Following data show World war two casualties of some countries. Source:-Microsoft Encarta

Country

CasualtiesMilitary killed

Military wounded

Prisoners of War

India 24,250 64,250 79,500Newzeland 12,250 19,250 8,500Australia 23,250 39,750 26,250Canada 37,500 53,250 9,750

World War 2 Casualties

0%

20%

40%

60%

80%

100%

Countries

Prisoners of war

Military wounded

Military killed

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1. The following table shows the number of students in various technologies in certain years.

YEAR SPINNING WEAVING PROCESSING GARMENTS2001 40 45 48 552002 30 32 35 382003 35 40 46 502004 50 55 62 65

DIAGRAM:

40 30 35 5032 40

5548

3546

6255

3850

65

45

0

50

100

150

200

250

300

350

2001 2002 2003 2004YEARS

NO

. O

F S

TU

DE

NT

S

GARMENTS

PROCESSING

WEAVING

SPINNING

188

135171

232

2. The following table shows the salary packages of university staff.

SALARY TEACHER

ASSIT. PROFESSOR

PROFESSOR

DEAN

VICE CHANCELLOR

BASIC PAY 25 30 40 50 60MEDICAL 5 5 5 5 5TRANSPORT

4 4 4 4 4

HOUSE RENT

8 12 14 16 20

TOTAL 42 51 63 75 89

DIAGRAM:

9

9

2540

50605

55

5

5

812

14

16

20

44

4

4

4

30

0

20

40

60

80

100

TEACHER ASSISTANTPROFESSOR

PROFESSOR DEAN VICE CHANCELLOR

STAFF

SALA

RY

IN T

HO

SAN

DS

TADA

HOMERENT

MEDICAL

BASICSALARY

42

63

51

75

89

4) Pie Chart:-A pie-diagram, also known as sector diagram, is a graphic device consisting of a circle divided into sectors or pie-shaped pieces whose areas are proportional to the various parts into which the whole quantity is divided. The sectors are shaded or coloured differently to show the relationship of parts to the whole.

Procedure for construction of pie chart: Draw a circle of any convenient radius. As a circle consisting of , the whole quantity to be displayed is equated to 360. the proportion that each component part or category bears to the whole quantity will be the corresponding proportion of . These corresponding proportions, i.e. angles, are calculated by

Then divided the circle into different sectors by constructing angles at the centre by means of a protector and draw the corresponding radii.

1) Example: - The Data Shows Soccer world cup won by different teams from 1930-1998. Source:-www.google.com

CountryNo. of times

won PercentagesOut of 3600

Uruguay 2 13 46.8Italy 3 19 68.4

West Germany 3 19 68.4Brazil 4 24 86.4

England 1 6 21.6Argentina 2 13 46.8

France 1 6 21.6

10

10

No.of times won

13%

19%

19%24%

6%

13%6%

Uruguay

Italy

WestGermanyBrazil

England

Argentina

France

2) Example:-The give data show Exports of USA in Billion dollars from 1975-2000. Source:-Microsoft Encarta

YearExports(Billion

Dollars) Percentage (%)Out of 3600

1975 132.6 4 14.41980 271.8 9 32.41985 288.8 9 32.41990 537.2 17 61.21995 794.2 26 93.62000 1065.7 35 126

Exports(Billion Dollars)19754%

19809%

19859%

199017%

199526%

200035%

1. The following table shows the yearly expenditure of a Mr. Ted, a college undergraduate in various categories.

CATeGORIES EXPENDITURE %age Out Of 360TUITION FEES 6000 20 72BOOKS AND LAB 2000 6.66 24CLOTHS CLEANING

2000 6.66 24

11

11

ROOM AND BIARDING

12000 40 144

TRANSPORTATION 3000 10 36INSURANCE 1000 3.33 12SINDRY EXPENCES 4000 13.33 48TOTAL 30000

DIAGRAM:

6000, 20%

2000, 7%

12000, 40%

3000, 10%

1000, 3%

2000, 7%

4000, 13%

TUITION FEES

BOOKS AND LAB

CLOTHS/CLEANING

ROOM ANDBOARDING

TRANSPORTATION

INSURANCE

SINDRY EXPENCES

2. The pie chart below shows the fractions of dogs in a dog competition in seven different groups of dog breeds. Suppose 1000 dogs entered the competition in all.

GROUPS NO. OF DOGS %age out of 360SPORTING GROUP 240 24 86.4WORKING GROUP 210 21 75.6HOUND GROUP 160 16 57.6TERRIER GROUP 160 16 57.6TOY GROUP 120 12 43.2NON-SPORTING GROUP

50 5 18

HERDING GROUP 60 6 21.6TOTAL 1000

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12

DIAGRAM:

240, 24%

210, 21%

160, 16%

160, 16%

120, 12%

50, 5%

60, 6%

SPORTING GROUP

WORKING GROUP

HOUND GROUP

TERRIER GROUP

TOY GROUP

NON-SPORTING GROUP

HERDING GROUP

5) Frequency Polygons:-

1) Example: -The Following data shows marks obtained by students of a class in a quiz.

Marks No. of Students10 5

11 7

12 11

13 15

14 24

15 16

16 11

17 6

18 3

19 2

20 10

13

13

Result of quiz

57

11

15

24

16

11

63 2

10

0

5

10

15

20

25

30

10 11 12 13 14 15 16 17 18 19 20

Marks

Fre

qu

ency

No.ofStudents

2) Example: -The following data displays monthly wages of person in Dollars.

Wages No. of persons50 1255 360 770 875 19

Weekly w ages

12

3

7 8

19

0

5

10

15

20

50 55 60 70 75

Wage

No

.of

pe

rso

ns

1. The following table shows the Frequency distribution for quantity of Glucose in 100 people.

QUANTITY OF GLUCOSE FREQUENCY64 068 172 076 080 284 888 5

14

14

92 1496 18100 11104 18108 6112 8116 5120 3124 1128 0132 0136 0140 0144 0

DIAGRAM:

0

5

10

15

20

0 20 40 60 80 100 120 140 160

VALUE OF GLUCOSE

FR

EQ

UE

NC

Y

2. The following chart shows the number of lunatics and their frequency.

No. of lunatics Frequency0 025 2100 6200 2300 0400 3500 1550 0

DIAGRAM:

15

15

01234567

0 100 200 300 400 500 600

NO. OF LUNATICS

FR

EQ

UE

NC

Y

6) Cumulative Frequency Polygons:-1) Example: - The data below is a frequency table

Midpoints(x) Frequency(f)Cumulative

Frequency(C.F)25 18 1835 25 4345 44 8755 88 17565 91 26675 97 363

Culmulative Frequecy plygon

18 4387

175266

363

0100200300400

25 35 45 55 65 75

2) Example: -The data shows wickets taken by a cricket player in his debut.Wickets taken(x) No. of times(f)

Cumulative Frequency(c.f)

1 22 222 19 413 20 614 12 735 9 826 3 85

16

16

Culmulative frequency

22

41

6173

82 85

0

20

40

60

80

100

1 2 3 4 5 6

Wickets

Fre

qu

en

cy

1. The following table shows the frequency distribution for quantity of Glucose in 100 people.

QUANTITY OF GLUCOSE

FREQUENCY CUMMULATIVE FREQUENCY

64 0 068 1 172 0 176 0 180 2 384 8 1188 5 1692 14 3096 18 48100 11 59104 18 77108 6 83112 8 91116 5 96120 3 99124 1 100128 0 100132 0 100136 0 100140 0 100144 0 100

DIAGRAM:

17

17

0

50

100

150

0 50 100 150 200QUANTITY OF GLUCOSE

CU

MM

ULA

TIV

E

FRE

QU

EN

CY

2. The following chart shows the number of lunatics and their frequency

No.of lunatics Frequency CUMMULATIVE FREQUNCY

0 0 025 2 2100 6 8200 2 10300 0 10400 3 13500 1 14550 0 14DIAGRAM:

0

5

10

15

0 100 200 300 400 500 600

NUMBER OF LUNATICS

CO

MM

UL

AT

IVE

F

RE

QU

EN

CY

18

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7) Pareto Diagrams:-

Definition: A bar graph used to arrange information in such a way that priorities for process improvement can be established.

A Pareto diagram is used to determine what characteristic is the major contributor in a process. The diagram is constructed by ranking the data in frequency of occurrence and plotting the bars in descending order. 

Purposes:

 To display the relative importance of data.  To direct efforts to the biggest improvement opportunity by

highlighting the vital few in contrast to the useful many.

Pareto diagrams are named after Vilfredo Pareto, an Italian sociologist and economist, who invented this method of information presentation toward the end of the 19th century. The chart is similar to the histogram or bar chart, except that the bars are arranged in decreasing order from left to right along the abscissa. The fundamental idea behind the use of Pareto diagrams for quality improvement is that the first few (as presented on the diagram) contributing causes to a problem usually account for the majority of the result.   Thus, targeting these "major causes" for elimination results in the most cost-effective improvement scheme.

How to Construct:

1. Determine the categories and the units for comparison of the data, such as frequency, cost, or time.

2. Total the raw data in each category, then determine the grand total by adding the totals of each category.

3. Re-order the categories from largest to smallest. 4. Determine the cumulative percent of each category (i.e., the sum of

each category plus all categories that precede it in the rank order, divided by the grand total and multiplied by 100).

5. Draw and label the left-hand vertical axis with the unit of comparison, such as frequency, cost or time.

6. Draw and label the horizontal axis with the categories. List from left to right in rank order.

7. Draw and label the right-hand vertical axis from 0 to 100 percent. The 100 percent should line up with the grand total on the left-hand vertical axis.

8. Beginning with the largest category, draw in bars for each category representing the total for that category.

9. Draw a line graph beginning at the right-hand corner of the first bar to represent the cumulative percent for each category as measured on the right-hand axis.

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10. Analyze the chart.  Usually the top 20% of the categories will comprise roughly 80% of the cumulative total.

Tips:

 Create before and after comparisons of Pareto charts to show impact of improvement efforts.

 Construct Pareto charts using different measurement scales, frequency, cost or time.

 Pareto charts are useful displays of data for presentations. Use objective data to perform Pareto analysis rather than team

members opinions. If there is no clear distinction between the categories -- if all bars

are roughly the same height or half of the categories are required to account for 60 percent of the effect -- consider organizing the data in a different manner and repeating Pareto analysis.

 Pareto analysis is most effective when the problem at hand is defined in terms of shrinking the PV to a customer target. For example, reducing defects or elimination the non-value added time in a process.

1) Example: - The Following data shows Price of Polyester Staple Fiber at Karachi in Rs/Kgs. Source:-APTMA

Months Price Percentage (%)Cumulative Percentage

May 111.72 17.8 17.8April 111.55 17.78 35.58

March 106.95 17.05 52.63January 103.5 16.5 69.13February 103.5 16.5 85.63

June 90.15 14.37 100

Polyester rate

17.8 17.78 17.05 16.5 16.5 14.37

0

20

40

60

80

100

120

May April March January Feburary June

2) Example: - The following data shows areas of continents in million sq. miles.

Continent Area(million Percentage cumulative

20

20

sq miles) (%) PercentageAsia 17.1 35 35

Africa 11.7 24 59North America 9.4 19 78South America 6.9 14 92

Europe 3.9 8 100

Area of continents

0

20

40

60

80

100

120

Asia Africa NorthAmerica

SouthAmerica

Europe

Continents

Are

a (

% )

1. The table given below shows the problems with the computer.

PROBLEMS FREQUENCY FRACTIONSSetup Difficulty 80 0.296296Not Easy to Use 45 0.166667Unspecified 25 0.092593Not Fast Enough 22 0.081481Too Slow 15 0.055556Incompatible 10 0.037037Internet Inoperative 10 0.037037Too Heavy 9 0.033333Too loud 6 0.022222Too Small 6 0.022222Power Inop 5 0.018519Bad Color 4 0.014815Screen Small 3 0.011111Dim Screen 3 0.011111Cord too short 3 0.011111Slow Internet 3 0.011111Too Fast 3 0.011111Floppy Slow 3 0.011111Too Big 2 0.007407Case smells 2 0.007407No Printer 2 0.007407No Books 2 0.007407Pwr Btn Stiff 1 0.003704Won't Start 1 0.003704Won't Print 1 0.003704

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Didn't talk 1 0.003704No Movies 1 0.003704No Help 1 0.003704No Manuals 1 0.003704

DIAGRAM:

2. The following table shows the defect in production in a factory.

TYPE SUBTOTAL % OF TOTALHIGH TURN ON SPEED 18 14.754HIGH RIPPLE CURRENT 38 31.147HIGH LEAKAGE 12 9.836

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Pareto Chart

0

50

100

150

200

250

Group

# O

bser

vatio

ns

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

LOW OUTPUT AT LOW SPEED 15 12.295LOW OUTPUT AT HIGH SPEED 7 5.737DEAD UNIT 4 3.278BAD REGULATOR 22 18.032BAD VOLTAGE SETPOINT 6 4.918

DIAGRAM:

8) Pictographs:-1) Example: - The following data shows Bikes assembled by a company on different week days from Monday to Saturday.Scale: - 1 Bike picture represents 5 bikes

Day ProductionMonday 12Tuesday 23

Wednesday 17Thursday 27

Friday 10

23

23

Saturday 22

Production of bikes

12

23

17

27

10

22

0 5 10 15 20 25 30

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Wee

k d

ays

Proction

2) Example: - The following data shows Gold medals won by different nations in Athens Olympics 2004.

Source: -www.olympic.org

Nation Gold wonUSA 35

China 32Russia 27

Australia 17Japan 16

Gold medals won in Athens Olympics

35

32

27

17

16

0 10 20 30 40

USA

China

Russia

Australia

Japan

Co

un

trie

s

Medals w on

1 gold = 4 units

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9) Fishbone diagram:-Dr. Kaoru Ishikawa, a Japanese quality control statistician, invented the fishbone diagram. Therefore, it may be referred to as the Ishikawa diagram. The fishbone diagram is an analysis tool that provides a systematic way of looking at effects and the causes that create or contribute to those effects. Because of the function of the fishbone diagram, it may be referred to as a cause-and-effect diagram. The design of the diagram looks much like the skeleton of a fish. Therefore, it is often referred to as the fishbone diagram.Whatever name you choose, remember that the value of the fishbone diagram is to assist teams in categorizing the many potential causes of problems or issues in an orderly way and in identifying root causes.When should a fishbone diagram be used?Does the team...

Need to study a problem/issue to determine the root cause? Want to study all the possible reasons why a process is beginning to have

difficulties, problems, or breakdowns? Need to identify areas for data collection? Want to study why a process is not performing properly or producing the desired

results?

1) Example: - This diagram indicates Cause and effect of Heavy School bags. The effect is back pain.

2) Example: - This diagram indicates some possible reasons of high reject rate of machine parts.

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EXAMPLE # 1: Draw a fishbone diagram of doorknob by showing its parts.

GRAPH:

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EXAMPLE # 2: Draw a fishbone diagram of Biological Warfare Disease.

GRAPH:

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1. The CEO of a call center wants to know the cause that why all calls is not answered and also wants to improve the ability to handle calls.

DIAGRAM:

2. The following fish bone diagram illustrates that what were the causes that project deadline was not met

DIAGRAM:

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10) Histogram:-

1) Example: - The following data shows the number of workers in different factories.No of workers Factories h

70-75 15 575-80 9 580-85 25 585-90 18 590-95 27 5

2) Example: - The following data shows marks obtained by 50 students in Physics exam.

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Marks obtained Number of Students h0-10 3 10

10-20 6 1020-30 12 1030-40 18 1040-50 11 10

11) Stem and Leaf Diagram:-A stem-and-leaf display separates data entries into “leading digits” or “stem” and “trailing digits” or “leaves.” For example, since the annual cost (in $000) in the private institution data set all have two-digit integer numbers, the tens and units columns would be the leading digits, and the remaining column (the tenths column) would be the trailing digit. Thus, an entry of 26.4 (corresponding to $26,400) has a stem of 26 and a trailing digit or leaf of 4.The following figure depicts the stem-and-leaf display of the annual cost of attending the 50 sampled private colleges and universities.

Steps to follow in constructing a Stem and Leaf Display

1. Divide each observation in the data set into two parts, the Stem and the Leaf.

2. List the stems in order in a column, starting with the smallest stem and ending with the largest.

3. Proceed through the data set, placing the leaf for each observation in the appropriate stem row.

Depending on the data, a display can use one, two or five lines per stem. Among the different stems, two-line stems are widely used.

Advantages of a stem and leaf display over a frequency distribution (considered in the next section):

1. the original data are preserved.

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2. a stem and leaf display arranges the data in an orderly fashion and makes it easy to determine certain numerical characteristics to be discussed in the following chapter.

the classes and numbers falling in them are quickly determined once we have selected the digits that we want to use for the stems and leaves

1) Example: - The following data shows the sugar levels of 50 patients.

122

132

128

140

110

170

165

112

145

167

176

189

190

156

188

166

165

145

178

133

118

192

185

173

169

154

148

122

113

111

170

122

132

137

109

165

197

129

167

178

109

118

187

176

151

176

145

189

145

123

Blood Sugar LevelFrequency Stem Leaf (=<100)

2 10 996 11 0123886 12 2223894 13 22376 14 0555584 15 14467 16 55567798 17 003666884 18 89593 19 027

2) Example: - The following data shows price of different goods in Pakistani rupees.10 55 63 36 29 10 20 35 18 1014 15 20 22 23 60 58 30 85 9062 75 20 35 40 10 15 17 18 29

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Price of Items in a Utility storeFrequency Stem Leaf (=<10 )

10 1 00004557887 2 00023994 3 05561 4 02 5 583 6 0231 7 51 8 51 9 0

NO#2

A visual display of the five number summary. The box-and-whisker plot is a simplified boxplot taught to beginners . It does not show outliers. The whiskers extending all the way to the minimum and maximum values regardless of how far out they may be.

 

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A box and whisker graph is used to display a set of dataso that you can easily see where most of the numbers are.

For example, suppose you were to catch and measurethe length of 13 fish in a lake:

A box and whisker plot is based on medians. The first step is to rewrite the data in order, from smallest length to largest:

Now find the median of all the numbers. Notice that since there are 13 numbers, the middle one will be the seventh number:

This must be the median (middle number) because there are six numbers on each side.

The next step is to find the lower median. This is the middle of the lower six numbers. The exact centre is half-way between 8 and 9 ... which would be 8.5Now find the upper median. This is the middle of the upper six numbers. The exact centre is half-way between 14 and 14 ... which must be 14

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Now you are ready to construct the actual box & whisker graph. First you will need to draw an ordinary number line that extends far enough in both directions to include all the numbers in your data:

First, locate the main median 12 using a vertical line just above your number line:

Now locate the lower median 8.5 and the upper median 14 with similar vertical lines:

Next, draw a box using the lower and upper median lines as endpoints:

Finally, the whiskers extend out to the data's smallest number 5 and largest number 20:

This is a box & whisker plot!

But what does it mean? What information about the data does this 34

34

graph give you?

Well, it's obvious from the graph that the lengths of the fish were as small as 5 cm, and as long as 20 cm. This gives you the range of the data ... 15.You also know the median, or middle value was 12 cm. Since the medians (three of them) represent the middle points, they split the data into four equal parts. In other words:

one quarter of the data numbers are less than 8.5 one quarter of the data numbers are between 8.5 and

12 one quarter of the data numbers are between 12 and

14

one quarter of the data numbers are greater than 14

The shading below, as an example, shows the quarter of the numbers that are between 12 and 14:

Here is a picture of the quarter of the data that is between 8.5 and 12. Notice that the data is more spread out here:

This picture is showing where half the data numbers are. Half of all the fish caught had a length between 8.5 and 14 centimetres:

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Graphical representation of Textile Related data

1. SIMPLE BAR DIAGRAM

a) Exported man made fiber

Years

Quantity of fiber

Exported1993-94 25,4221994-95 61,4851995-96 28,7141996-97 48,4841997-98 34,0151998-99 34,5151999-00 22,7162000-01 28,5242001-02 45,6652002-03 66,6532003-04 54,878

EXPORTED MAN MADE FIBER

010,00020,00030,00040,00050,00060,00070,000

1993

-94

1994

-95

1995

-96

1996

-97

1997

-98

1998

-99

1999

-00

2000

-01

2001

-02

2002

-03

2003

-04

YEARS

QU

AN

TIT

Y O

F F

IBE

RS

36

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b) Consumption of cotton

Year Cotton1990-91 1,128,9781991-92 1,257,3991992-93 1,318,8921993-94 1,511,6101994-95 1,412,7321995-96 1,509,9551996-97 1,444,3681997-98 1,471,1691998-99 1,441,9231999-00 1,566,3482000-01 1,673,2802001-02 1,755,6692002-03 1,943,1972003-04 1,938,678

consumption of cotton

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

1990

-91

1992

-93

1994

-95

1996

-97

1998

-99

2000

-01

2002

-03

cotton

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2.MULTIPLE BAR DIAGRAM

a) Production of cloth province wise

PERIOD PANJAB SIND N.W.F.P BALUCHISTAN

1993-94 223,789 158,693 6,255 0

1994-95 174,293 143,720 3,828 0

1995-96 189,559 137,422 0 0

1996-97 208,107 125,388 0 0

1997-98 212,813 256,258 8 2

1998-99 230,018 154,519 24 0

1999-00 292,536 144,634 20 0

2000-01 309,634 180,530 0 0

2001-02 355,370 256,952 0 2

2002-03 317,981 264,164 0 0

2003-04 358,962 255,963 0 0

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Cloth Prduction Province Vise

0100,000200,000300,000400,000

Quantity

Yea

rs

Punjab

Sind

N.W.F.P

Baluchistan

b) Textile related exports

Year %age of textile material exportedyarn cloth others

1971-72 55.6 35.5 91979-80 32.87 38.9 28.141989-90 33.2 22.32 44.351996-97 28.09 25.12 46.771997-98 23.7 25.56 50.721998-99 20.72 24.45 551999-00 20.95 21.44 57.592000-01 20.6 19.8 59.592001-02 16 20 642002-03 13 19 682003-04 14 21 65

textile related exports

0

20

40

60

80

1971

-72

1979

-80

1989

-90

1996

-97

1997

-98

1998

-99

1999

-00

2000

-01

2001

-02

2002

-03

2003

-04

yarn

cloth

others

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3. COMPONENT BAR DIAGRAM

a) Production of yarn

Year %age of yarn producedBlended Coarse Medium Fine

1997-98 23 45 27 41998-99 26 47 24 21999-00 25 49 23 22000-01 25 50 22 32001-02 26 49 21 32002-03 25 46 25 42003-04 20 50 25 5

Yarn Production

0%20%40%60%80%

100%

1997

-98

1999

-00

2001

-02

2003

-04

Years

Qu

anti

ty %

Fine

Medium

Coarse

Blended

40

40

B) Production of cloth

Year %age of cloth producedgrey bleached Dyed & printed blended

1971-72 64 17 19 -1981-82 61 10 16 131991-92 52 6 21 211997-98 61 4 19 161998-99 51 7 25 171999-00 60 3 23 142000-01 57 4 25 142001-02 56 3 27 142002-03 51 5 28 162003-04 49 6 30 15

production of cloth

0%20%40%60%80%

100%

Blended

Dyed &printed

Bleached

Grey

4. FREQUENCY POLYGONES

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a) Export of Wool yarn

Years Quantity

1995-96 455,693

1996-97 212,452

1997-98 512,862

1998-99 421,481

1999-00 512,971

2000-01 512,467

2001-02 544,217

2002-03 519,329

2003-04 458,962

Wool Yarn Export

0200,000400,000600,000

Year

Qu

an

tity

B) Export of cloth

Year Quantity of cloth exported1971-72 409808

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1979-80 5457681989-90 10178681996-97 12574301997-98 12712721998-99 13551661999-00 15748762001-01 17358242001-02 19573532002-03 20363212003-04 2378900

export of cloth

0500000

1000000150000020000002500000

5. CUMMULATIVE FREQUENCY POLYGONES

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A) Export of cotton yarn

YearQuantity

Kgs%age of cotton yarn exported

 cumulative %

1991-92 505,863 7.527587673 7.5275876731992-93 555,294 8.263154785 15.790742461993-94 578,648 8.61067829 24.401420751994-95 522,091 7.76907142 32.170492171995-96 535,889 7.974395104 40.144887271996-97 508,188 7.562185264 47.707072541997-98 461,919 6.873670876 54.580743411998-99 421,481 6.271925758 65.580743411999-00 512,971 7.633359578 68.486028752000-01 545,134 8.111967032 76.597995782001-02 544,217 8.098321444 84.696317222002-03 519,329 7.72797097 92.424288192003-04 509,097 7.575711806 100

export of coton yarn

050

100150

quantity

cum

mul

ativ

e %

b) Export of cloth

Year Quantity of cloth exported % Quantity Cumulative %

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44

1996-97 1257430 9.26820107 9.268201071997-98 1271272 9.370226979 18.638428051998-99 1355166 9.988588606 28.627016661999-00 1574876 11.6080159 40.235032562001-01 1735824 12.79432323 53.029355792001-02 1957353 14.42715791 67.45651372002-03 2036321 15.00921123 82.465724932003-04 2378900 17.53427509 100

export of cloth

0

50

100

150

1996-97

1997-98

1998-99

1999-00

2001-01

2001-02

2002-03

2003-04

year

cum

mul

ativ

e %

6. PARETO DIAGRAM

A) Export of yarn

Year Quantity of yarn exported % quantity Cumulative %1996-97 1,411,519 18.89990509 18.89990509

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1997-98 1,153,542 15.44565416 34.345559252003-04 1,141,219 15.28065212 49.626211372000-01 1,076,600 14.41541901 64.041630381999-00 1,071,616 14.34868443 78.390314812002-03 926,358 12.40371421 90.794029021998-99 345,169 4.621731157 95.415760182001-02 342,369 4.58423982 100

export of yarn

020406080

100120

1996-97

1997-98

2003-04

2000-01

1999-00

2002-03

1998-99

2001-02

year

%q

uan

tity

cum

mu

lati

ve

%

% quantity Cumulative %

Cotton prices month wise in year 2004-05

Month Rs per maund % value Cumulative % value

Jul 2615 10.06040088 10.06040088

Aug 2273 8.744662024 18.8050629

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46

Mar 2272 8.740814835 27.54587774

June 2251 8.660023853 36.20590159

Apr 2233 8.590774439 44.79667603

May 2217 8.529219405 53.32589544

Sep 2205 8.48305313 61.80894857

Feb 2164 8.325318355 70.13426692

Jan 2035 7.829030893 77.96329781

Oct 1940 7.463547878 85.42684569

Nov 1923 7.398145655 92.82499135

Dec 1865 7.175008656 100

cotton prices month wise in year 2004-05

0

20

40

60

80

100

120

month

%valu

e

cu

mm

ula

tive %

valu

e

% value Cumulative % value

7.PIE CHART

a) World wide fiber production in year 2004(Quantity measured in 1000 tones)

Type of fiber Quantity of fiber produced % quantity Quantity out of 360Man made fiber 34560 56.86 204.72

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Cotton 24900 40.97 147.50Wool 1215 1.99 7.19Silk 97 0.16 0.57

world wide fiber production in year 2004

56.86

40.97

1.99

0.16

Man made fiber

Cotton

Wool

Silk

B) Production of cloth province wise in year 1998-99 Province Quantity of

cloth produced% quantity Angle of sectors

in degreesPunjab 230,018 59.81 215.32Sindh 154,519 40.18 144.65N.W.F.P 24 0.0062 0.022Baluchistan 0 0 0

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production of cloth province wise in year 1998-99

59.81

40.18

0.0062

0 Punjab

Sindh

N.W.F.P

Baluchistan

Multiple Choice Questions1. Consider the following output from DataDesk when analyzing the pH values of

the 1986 data collected on precipitation events.

2.Which of the

following is NOT CORRECT? a. The 25th percentile is about 5.9. b. Some outliers appear to be present below a pH of 5.4. c. About 95% of the observations have pH values in the approximate range

6±1. d. About 10% of the values are in the range 5.8 to 6.0. e. About 75% of the values are less than 6.4. (d)

3. The following is a histogram showing the actual frequency of the closing prices on the New York exchange of a particular stock.

Based on the above frequency histogram for New York Stock exchange, the class that contains the 80th percentile is:

a. 20-30

Summary stats for 1986NumNumeric = 55NumNonNumeric = 0NumCases = 55Mean = 6.0673Median = 6.1000Std Deviation = 0.47339Range = 2.4000Minimum = 4.6000Maximum = 775-th %ile = 6.4000

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b. 10-20 c. 40-50 d. 50-60 (e)e. 30-40

4. A histogram of the heights of 39 plants is as follows:

The 75th percentile of the height distribution is approximately: a. 9.4 b. 9.7 c. 7.7 d. 7.5 (b)e. 10.0

5. The weights of the male and female students in a class are summarized in the following boxplots:

Which of the following is NOT correct? a. About 50% of the male students have weights between 150 and 185 lbs. b. About 25% of female students have weights more than 130 lbs. c. The median weight of male students is about 162 lbs. d. The mean weight of female students is about 120 because of symmetry. e. The male students have less variability than the female students.

(e)

6. Consider the following box plots of the grades in a course in statistics for each sex drawn according to the convention that the whiskers reach the 10th and 90th percentiles.

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50

Which of the following is correct? a. The mean grade of the female students is about 72. b. The median grade for the male students is about 68. c. About 25% of female students get grades above 72. d. About 10% of male students get grades below 60. (e)e. About 50% of female students get grades between 62 and 82.

7. Consider the following box-plot of the yield of barley drawn using the convention that the wiskers reach the 10 and 90th percentiles.

Which of the following is NOT correct? a. The mean is about 200 g/400 m2. b. the median is about 180 g/400 m2. c. About 50% of the yields are between 160 and 220 g/400 m2. d. About 25% of the yields are below 220 g/400 m2. (d)e. About 10% of the yields are below 130 g/400 m2.

8. Consider the following ogive of the scores of students in an introductory statistics course:

A grade of C or C+ is assigned to a student who scores between 55 and 70. The percentage of students that obtained a grade of C or C+ is:

a. 25% b. 30% c. 20% d. 50% (c)

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e. 15%

9. Which of the following is NOT correct about constructing histograms? a. The approximate number of classes is 1+3.3log(n). b. All class intervals should be of equal width. c. The bars of the histogram are centred over the class mark (midpoint). d. The first and last classes should be open-ended to account for extreme

points. e. There should be no spaces between bars. (d)

10. Forest companies routinely take samples from tracts that have been replanted to monitor the growth of the trees. Suppose that in a recent sample of two tracts, the diameter of the trees was measured with the following results:

Tract A BTrees 75 210Range 232-315 215-250 (mm)

Histograms to compare the two groups are to be constructed. Which of the following is not recommended.

a. The number of classes in Group A and Group B should be around 7 or 8. b. The class width of both groups will be 10 mm. c. The class bounds will be 232-242 mm, 242-252 mm, etc. for Group A and

215- 225 mm, 225-235 mm, etc for Group B. d. The vertical scale for both groups should be relative frequency (%). e. The two histograms will be stacked and alligned so that the vertical axes

are the same and the horizontal axes are identical. (c)

11. Consider the following SAS procedure to construct a histogram.

PROC CHART DATA=BARLEY; VBAR YIELD/ TYPE=PERCENT MIDPOINTS=90 TO 300 BY 30;

Which of the following statements is correct?

a. The first bar will contain only values from 75 to 105. b. All yields below 90 will not be used. c. The vertical axis will be labelled with the actual frequency in each class. d. Several charts will be produced, one for each distinct year. e. The last class will include values above 315. (e)

12. For each student in a class, the sex and weight (in kilograms) are recorded Consider the following SAS program:

DATA STUDENTS; INPUT SEX $ WEIGHT; DATALINES;

F 62

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. . (<-- more data here) .M 78;

PROC SORT DATA=STUDENTS; BY SEX; PROC CHART DATA=STUDENTS; VBAR WEIGHT/TYPE=PERCENT MIDPOINTS=55 TO 95 BY 10; BY SEX;

Which of the following statements is correct?

a. The first class will contain weights from 55 to 65 kg. b. All weights below 55 kilograms will be discarded. c. One vertical bar chart will be produced; it will contain the weights of all of

the students, and the males and females will be indistinguishable. d. The vertical axis will be labelled "frequency". e. Two separate vertical bar charts will be produced -- one for males and one

for females. (e)

13. A single stem-and-leaf plot is a useful tool because: a. it includes the average and the standard deviation. b. it shows the percentage distribution of the data values. c. it enables us to examine the data values for the presence of trends, cycles,

and seasonal variation. d. it enables us to locate the centre of the data, see the overall shape of the

distribution, and look to marked deviations from the overall shape. e. it enables us to compare this dataset against others of a similar kind. (d)

14. Forty students wrote a Statistics examination having a maximum of 50 marks. The mark distribution is given in the following stem-and-leaf plot:

0|28 1|2245 2|01333358889 3|001356679 4|22444466788 5|000

The third quartile of the mark distribution is equal to:

a. 75 b. 44 c. 32 d. 37.5 (b)e. 30

15. Thirty students wrote a statistics examination having a maximum of 50 marks. The mark distribution is given in the following stem-and-leaf plot:

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0|9 1|225 2|013335889 3|00136679 4|02244478 5|0

The median mark is equal to:

a. 30.5 b. 30.0 c. 25.0 d. 28.5 (a)e. 44.0

16. Rainwater was collected in water collectors at thirty different sites near an industrial basin and the amount of acidity (pH level) was measured. The following stem-and- leaf diagram shows the pH values that ranged from 2.6 to 6.3.

Stems Leaves2 6793 2377894 12224468995 05567886 0233

The median acidity is:

a. 4.2 b. 4.4 c. 4.5 d. 4.6 (c)e. Average of 15 and 16.

17. Refer to the previous question. Which of the following box-plots is correct:

(e)

18. Refer to the previous question. The interquartile range is: a. 7.75 b. 23.25

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c. 5.625 d. 3.77 (e)e. 1.855

19. The following is a stem-plot of the birth weights of male babies born to the smoking group. The stems are in units of kg.

Stems Leaves2 3,4,6,7,7,8,8,8,93 2,2,3,4,6,7,8,94 1,2,2,3,45 3,5,5,6

The median birth weight is:

a. 13.5 b. 3.2 c. 3.5 d. 3.7 e. Average of 13 and 14. (c)

20. Refer to the previous question. The first quartile (25th) percentile of the weights is a. 2.3 b. 2.7 c. .25 d. 6.5 (e)e. 2.8

21. Which one of the following statements is FALSE? a. Pie charts are better than bar graphs for comparing relative sizes. b. Data that are nominal scale are presented using frequency tables. c. Means and standard deviation of ordinal data are meaningless. d. The scatter-plot is the basic graphic tool for investigating relationships

between two interval or ratio scaled variables. e. Box-plots are a good choice for comparing the distribution of values

among groups. (a)

22. Which of the following is NOT CORRECT? a. The scatterplot is the basic graphical tool for investigating relationships

between two continuous interval or ratio scaled variables. b. The frequency table is useful for summarizing data from a nominal scaled

variable. c. Means and standard deviations of nominal or ordinal scaled variables are

useful summary measures. d. Pie charts don’t perform well because people have difficulty in accurately

quantifying angles. (c)e. Boxplots perform well for comparing groups because it is relatively

straightforward to see how the mean and median change over the groups.

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23. The following is a display comparing the favorite TV shows selected from a specified set by gender. Each person had to select one preferred show from the three shows given below:

<--------------------- Percent -------------------> Sample10 20 30 40 50 60 70 80 90 100 Size|----|----|----|----|----|----|----|----|----|----|Male: sssssssssssssssssssssfffffffffffffffbbbbbbbbbbbbbbb 100

Female:sssssssssssffffffffffffffffffffffffffffffbbbbbbbbbb 300

Star Trek; ffff= Friends; bbbb = Baywatch;

Which of the following is FALSE:

a. A greater percentage of males have StarTrek as their favorite show than females.

b. More females (in absolute number) selected Baywatch as their favorite show than males.

c. About 1/5 of females surveyed selected StarTrek as their favorite show from the three given.

d. The modal favorite show (from the three specified) for males is StarTrek. e. About twice as many females (in relative numbers) enjoy Star Trek than

males. (e)

24. An experiment was conducted to investigate the effect of a new weed killer to suppress weed germination in onion crops. Two chemicals were used, the standard week killer (C) and the new chemical (W). Both chemicals were tested at high and low concentrations. Measurements are made on each of 50 plots for each treatment combinations of the % weed germination. Here are some box-plots of the results where the whiskers extend to the min and max of the data.

0 10 20 30 40 50 |---------|--------|---------|---------|---------| | | _______________W-low conc. | -----------|________|______|-------- | | ___________C-low conc. | ----|_____|_____|---- | | ______________W-high conc | -----|_______|______|---------- * * * | | _________C-high conc. | --|____|____|---

Which of the following is NOT a feature of this data:

a. At either high or low concentrations, the new chemical (W) gives better control of weed germination than the control (C).

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b. Fewer weeds germinate at higher concentrations of both chemicals. c. The results from the control chemical are less variable than the new

chemical. d. High or low concentrations of either chemical have approximately the

same effects on weed germination. e. Some of the results from the low concentration of weed killer W have

fewer weeds germinating than some of the results from the high concentration of W. (d)

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