Graphical Presentation I

Types of Graphs:

(a)    Histogram

(b)   Frequency Polygon

(c)    Relative Frequency Histogram and Polygon

(d)   Cumulative Frequency Polygon or Ogive

(e)    Frequency Curves and Smoothed Ogive

(a)   Histogram:

1.   A histogram consists of a set of adjacent rectangles having bases along x-axis (marked off by class boundaries) and areas proportional to class frequencies.

2.    To adjust the heights of rectangles in a frequency distribution with unequal class interval sizes, each class frequency is divided by its class interval size.

Class boundaries

Frequency

109.5-119.5

1

119.5-129.5

4

129.5-139.5

17

139.5-149.5

28

149.5-159.5

25

159.5-169.5

18

169.5-179.5

13

179.5-189.5

6

189.5-199.5

5

199.5-209.5

2

209.5-219.5

1

S f

120

Class Interval

f

Class Boundaries

Size

Adjusted Frequency

10-11

4

9.5-11.5

2

4 / 2 = 2

12-14

12

11.5-14.5

3

12 / 3 = 4

15-19

25

14.5-19.5

5

25 / 5 = 5

20-29

60

19.5-29.5

10

6

30-34

25

29.5-34.5

5

5

35-39

15

34.5-39.5

5

3

40-42

6

39.5-42.5

3

2

 

147

 

 

 

(b)   Frequency Polygon:

1.    It is constructed by plotting the class frequencies against their corresponding class marks (mid-points) and then joining the resulting points by means of straight lines. 

2.   The ends of the graph so drawn do not meet the ends of x-axis.  A polygon is a many sided closed figure.  Therefore, extra classes are to be added at both ends of the frequency distribution with zero frequencies.

3.   The frequency polygon can also be obtained by joining the mid-points of the tops of rectangles of histogram.

(c)   Relative Frequency Histogram and Polygon: Same as described above.

(d)    Cumulative Frequency Polygon or Ogive:

1.   The graph showing the cumulative frequencies plotted against the upper class boundaries is called a ‘cumulative frequency polygon’ or ‘ogive’.

2.   The graph corresponding to a less than or a more than cumulative frequency distributions are called ‘less-than’ and ‘more-than ogives’ respectively.

Class Boundaries

Frequency

Less than

Cumulative

Frequency

More than

Cumulative

Frequency

109.5-119.5

1

1

119

119.5-129.5

4

5

115

129.5-139.5

17

22

98

139.5-149.5

28

50

70

149.5-159.5

25

75

45

159.5-169.5

18

93

27

169.5-179.5

13

106

14

179.5-189.5

6

112

8

189.5-199.5

5

117

3

199.5-209.5

2

119

1

209.5-219.5

1

120

0

S f

120

 

 

 

(e)   Frequency Curves and Smoothed Ogives:

Types of Frequency Distribution and Curves:

(a)    Symmetrical Distribution,

(b)   Moderately Skewed or Asymmetrical Distribution,

(c)    Extremely Skewed or J-Shaped Distribution,

(d)   U-Shaped Distribution, and

(e)    Multi-Modal Distribution.

(a)   Symmetrical Distribution: A frequency distribution is said to be symmetrical if the frequencies equidistant from the maximum are equal.

Class interval

0-9

10-19

20-29

30-39

40-49

50-59

60-69

Frequency

2

5

9

12

9

5

2

(b)  Moderately Skewed or Asymmetrical Distribution: A frequency distribution is said to be skewed when it departs from symmetry, i.e., when the frequencies tend to pile up in one end or the other end of a distribution.

Asymmetrical distributions are of two types, i.e.:

(i)                  Positively skewed, and

(ii)                Negatively skewed.

(i)                 Positively Skewed:

Class interval

0-9

10-19

20-29

30-39

40-49

50-59

60-69

Frequency

2

5

12

9

7

4

1

(ii)               Negatively Skewed:

Class interval

0-9

10-19

20-29

30-39

40-49

50-59

60-69

Frequency

1

4

7

9

12

5

2

(c)   Extremely Skewed or J-Shaped Distribution:

Income

0-1999

2000-3999

4000-5999

6000-7999

8000-9999

10000-11999

12000-13999

No. of persons

4000

3000

2500

1500

500

350

150

(d)    U-Shaped Distribution: In such a distribution, the maximum frequencies occur at both ends and a minimum in the centre.

Class interval

1-5

6-10

11-15

16-20

21-25

26-30

Frequency

45

30

18

12

24

40

(e)   Multi-Modal Distribution:

1.    Frequency distributions with more than one maximum are called ‘multi-modal distribution’.

2.    A distribution with two maxima is called a ‘bimodal distribution’.

 

 

 

 

 

 

 

 

Types of Charts:

(a)    Simple Bar Chart,

(b)   Multiple Bar Chart,

(c)    Component Bar Chart,

(d)   Percentage Component Bar Chart, and

(e)    Pie Chart. 

(a)   Simple Bar Chart:

1.    Simple bar chart consists of vertical or horizontal bars of equal width.

2.    The length of the bars is taken proportionately to the magnitude of the values represented.  The width of the bars has no significance.

3.    Vertical bars are used to represent quantitative data or chronological data.  Whereas, the horizontal bars are represented for qualitative data or geographical data.

4.    If the data do not relate to time, then they should be arranged in ascending or descending order of magnitude.

Exports of Pakistan (in US $ million)

Year

Exports

1948

138

1951

406

1961

378

1971

683

1981

2958

1991

6168

2001

9202

2005

14410

(b)   Multiple Bar Chart:

1.    Multiple bar chart is an extension of simple bar chart.

2.    Grouped bars are used to represent related sets of data. For example, imports and exports of a country together are shown in multiple bar chart.

3.    Each bar in a group is shaded or coloured differently for the sake of distinction.

Years

Imports

Exports

Rs. (billion)

Rs. (billion)

1982-83

68.15

34.44

1983-84

76.71

37.33

1984-85

89.78

37.98

1985-86

90.95

49.59

1986-87

92.43

63.35

1987-88

111.38

78.44

(c)    Component Bar Chart:

1.      This chart consists of bars which are sub-divided into two or more parts.

2.      The length of the bars is proportional to the totals.

3.      The component bars are shaded or coloured differently.

Current and Development Expenditure – Pakistan (All figures in Rs. Billion)

Years

Current

Expenditure

Development

Expenditure

Total

Expenditure

1988-89

153

48

201

1989-90

166

56

222

1990-91

196

65

261

1991-92

230

91

321

1992-93

272

76

348

1993-94

294

71

365

1994-95

346

82

428

(d)   Percentage Component Bar Chart:

1.    Component bar charts may also be drawn on percentage basis by expressing the components as percentages of their respective totals.

2.    All the bars are of equal length showing the 100%.  These bars are sub-divided into component bars in proportion to the percentages of their components.

Areas Under Crop Production (1985-90)

(‘000 hectors)

Year

Wheat

Rice

Others

Total

1985-86

7403

1863

1926

11192

1986-87

7706

2066

1906

11678

1987-88

7308

1963

1612

10883

1988-89

7730

2042

1966

11738

1989-90

7759

2107

1970

11836

Percentage Areas Under Production

Year

Wheat

Rice

Others

Total

1985-86

66.2%

16.6%

17.2%

100%

1986-87

66.0

17.7

16.3

100

1987-88

67.2

18.0

14.8

100

1988-89

65.9

17.4

16.7

100

1989-90

65.6

17.8

16.6

100

(e)   Pie Chart:

1.    Pie chart is used to compare the relation between the whole and its components.

2.    The difference between the component bar chart and pie chart is that in case of component bar chart the length of the bars are used while in case of a pie chart the area of the sector of a circle is used.

3.    In pie chart, the circle is drawn with radii proportional to the square root of the quantities to be represented because the area of a circle is given by 2pr2.

4.    The sectors are coloured and shaded differently.

5.    To construct a pie chart, we draw a circle with some suitable radius (square root of the total).  The angles are calculated for each sector as follows:

Angles for each sector =          Component Part           ×          360o

                                                   Total

Development Expenditure (1994-95)

Provinces

Development

Expenditure

(In Rs. Million)

Angles of Sectors

(In Degrees)

Cumulative

Angle

Balochistan

4874

56o

N.W.F.P.

7861

147o

Punjab

12954

297o

Sindh

5500

360o

Total

31189

360o

 

Continued

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