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 xaxis (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 
109.5119.5 
1 
119.5129.5 
4 
129.5139.5 
17 
139.5149.5 
28 
149.5159.5 
25 
159.5169.5 
18 
169.5179.5 
13 
179.5189.5 
6 
189.5199.5 
5 
199.5209.5 
2 
209.5219.5 
1 
S
f 
120 
Class
Interval 
f 
Class
Boundaries 
Size 
Adjusted
Frequency 
1011 
4 
9.511.5 
2 
4 / 2 = 2 
1214 
12 
11.514.5 
3 
12 / 3 = 4 
1519 
25 
14.519.5 
5 
25 / 5 = 5 
2029 
60 
19.529.5 
10 
6 
3034 
25 
29.534.5 
5 
5 
3539 
15 
34.539.5 
5 
3 
4042 
6 
39.542.5 
3 
2 

147 



(b)
Frequency
Polygon:
1.
It is constructed by plotting the class frequencies against their
corresponding class marks (midpoints) 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 xaxis. 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 midpoints 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 ‘lessthan’ and ‘morethan ogives’
respectively.
Class
Boundaries 
Frequency 
Less than Cumulative Frequency 
More
than Cumulative Frequency 
109.5119.5 
1 
1 
119 
119.5129.5 
4 
5 
115 
129.5139.5 
17 
22 
98 
139.5149.5 
28 
50 
70 
149.5159.5 
25 
75 
45 
159.5169.5 
18 
93 
27 
169.5179.5 
13 
106 
14 
179.5189.5 
6 
112 
8 
189.5199.5 
5 
117 
3 
199.5209.5 
2 
119 
1 
209.5219.5 
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 JShaped Distribution,
(d) UShaped Distribution, and
(e)
MultiModal Distribution.
(a)
Symmetrical
Distribution: A frequency distribution
is said to be symmetrical if the frequencies equidistant from the maximum are
equal.
Class
interval 
09 
1019 
2029 
3039 
4049 
5059 
6069 
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 
09 
1019 
2029 
3039 
4049 
5059 
6069 
Frequency 
2 
5 
12 
9 
7 
4 
1 
(ii)
Negatively Skewed:
Class
interval 
09 
1019 
2029 
3039 
4049 
5059 
6069 
Frequency 
1 
4 
7 
9 
12 
5 
2 
(c)
Extremely
Skewed or JShaped Distribution:
Income 
01999 
20003999 
40005999 
60007999 
80009999 
1000011999 
1200013999 
No. of
persons 
4000 
3000 
2500 
1500 
500 
350 
150 
(d)
UShaped
Distribution: In such a distribution,
the maximum frequencies occur at both ends and a minimum in the centre.
Class
interval 
15 
610 
1115 
1620 
2125 
2630 
Frequency 
45 
30 
18 
12 
24 
40 
(e)
MultiModal
Distribution:
1. Frequency distributions with more than one maximum are called ‘multimodal 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) 

198283 
68.15 
34.44 
198384 
76.71 
37.33 
198485 
89.78 
37.98 
198586 
90.95 
49.59 
198687 
92.43 
63.35 
198788 
111.38 
78.44 
(c)
Component
Bar Chart:
1. This chart consists of bars which are subdivided 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 
198889 
153 
48 
201 
198990 
166 
56 
222 
199091 
196 
65 
261 
199192 
230 
91 
321 
199293 
272 
76 
348 
199394 
294 
71 
365 
199495 
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 subdivided into component bars in proportion to the
percentages of their components.
Areas
Under Crop Production (198590)
(‘000
hectors)
Year 
Wheat 
Rice 
Others 
Total 
198586 
7403 
1863 
1926 
11192 
198687 
7706 
2066 
1906 
11678 
198788 
7308 
1963 
1612 
10883 
198889 
7730 
2042 
1966 
11738 
198990 
7759 
2107 
1970 
11836 
Percentage Areas Under Production
Year 
Wheat 
Rice 
Others 
Total 
198586 
66.2% 
16.6% 
17.2% 
100% 
198687 
66.0 
17.7 
16.3 
100 
198788 
67.2 
18.0 
14.8 
100 
198889 
65.9 
17.4 
16.7 
100 
198990 
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 2pr^{2}.
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 × 360^{o}
Total
Development
Expenditure (199495)
Provinces 
Development Expenditure (In
Rs. Million) 
Angles
of Sectors (In
Degrees) 
Cumulative Angle 
Balochistan 
4874 

56^{o} 
N.W.F.P. 
7861 

147^{o} 
Punjab 
12954 

297^{o} 
Sindh 
5500 

360^{o} 
Total 
31189 
360^{o} 
