(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:
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