OGIVES
Cumulative histogram also known as ogives, are graph that can be used to determine how many data
values lie above or below a particular value in a data set .ogive has the shape
of an elongated 'S' and it is sometimes called a double curve with one portion being concave
and the other being convex. while constructing, it is necessary to first
the frequency table. To plot an ogive we need class boundaries and the
cumulative frequencies. For grouped data ,ogive is formed by plotting the cumulative frequency
against tbe upper boundary of the class. For ungrouped data cumulative
frequency is plotted on the y-axis against the data which is on
the x-axis
Note : ogive is used to
study the growth rate of the data as it
shows the accumulation of frequency
There are mainly two types of ogive and they are less than type ogive
and more than type ogive.
Less than type ogive
Following steps are necessary to plot a less than type ogive curve.
1.
Start
from the upper limit of the class intervals and then add class frequency to the
cumulative frequency distribution.
2.
Take
upper limit in the x-axis direction cumulative frequencies along the y- axis
direction.
3.
Plot
the points (x,y),where x is the upper limit
of the class and y is the corresponding cumulative frequency.
4.
Join
the points by a smooth curve.
The less than ogive type look like an
elongated S and starts at the
lowest boundary on the axis and ends at the highest boundary which corresponds
to the total frequency of the distribution.
More than type ogive Following
steps are necessary to plot a more than
type ogive curve:
1.
Starts
from the lower limit of the class intervals and the total frequency is
subtracted from the frequency to get the cumulative frequency distribution.
2.
In
the graph, consider the lower limit x-
axis direction and cumulative frequencies along y- axis direction.
3.
Plot
the points (x,y),where 'x' is the lower limit of the class and 'y' is the corresponding
cumulative frequency.
4.
Join
the points by a smooth curve.
More than
type ogive look like an elongated S turned upside down.
Advantages and
disadvantages
Advantages
|
Disadvantages
|
Ogive can :
« Visually approximate raw data
« Summarize a large data set in visual form
« Delineate each interval in the frequency
distribution.
« Clarify rates of change between classes better than
other graphs.
« Provide visual check of accuracy or reasonableness
of calculations.
« Become more smooth as data points or classes added
« Be easily understood due to widespread use in
business and the media.
« Show the number or proportion of the data points
above / below a particular value.
|
Ogive can:
« Be somewhat complicated to prepare.
« Fail to reflect all data points in a data set.
« Reveal little about central tendency , dispersion ,
skew or kurtosis
« Often require additional written or verbal
explanation.
« Be inadequate to describe the attribute
,behavior,or condition of interest.
« Fail to reveal key assumptions , norms ,causes or
effects.
« Be somewhat easily manipulated to yeild false
impressions.
|
SOLVED EXAMPLE
Question: For the data given below , construct a less than
cumulative frequency table and plot its ogive.
Marks
|
0-10
|
10-20
|
20-30
|
30-40
|
40-50
|
50-60
|
60-70
|
70-80
|
80-90
|
90-100
|
Frequency
|
3
|
5
|
6
|
7
|
8
|
9
|
10
|
12
|
6
|
4
|
Marks.
|
Frequency
|
Less than cumilative
frequency
|
0-10
|
3
|
3
|
10-20
|
5
|
8
|
20-30
|
6
|
14
|
30-40
|
7
|
21
|
40-50
|
8
|
29
|
50-60
|
9
|
38
|
60-70
|
10
|
48
|
70-80
|
12
|
60
|
80-90
|
6
|
66
|
90-100
|
4
|
70
|
Plot the points having abscissa as upper limis and ordinates as the
cumulative
frequencies:(10,3),(20,8),(30,14),(40,21),(50,29),(60,38),(70,48),(80,60),(90,66),(100,70)
and join the points by a smooth curve
SOLVED EXAMPLE
Question: For the data given below ,construct more than cumulative frequency and plot its
ogive.
Marks
|
0-5
|
5-10
|
10-15
|
15-20
|
20-25
|
25-30
|
30-35
|
35-40
|
40-45
|
45-50
|
Frequency
|
3
|
5
|
7
|
8
|
10
|
11
|
14
|
19
|
15
|
13
|
SOLUTION :
Marks
|
Frequency
|
cumulative frequency
|
0-5
|
3
|
95
|
5-10
|
5
|
95 - 3 = 92
|
10-15
|
7
|
92 -5 = 87
|
15-20
|
8
|
87 -7 = 80
|
20-25
|
10
|
80 - 8 = 72
|
25-30
|
11
|
72 - 10 = 62
|
30-35
|
14
|
62 - 11 = 51
|
35-40
|
19
|
51 - 14 = 37
|
40-45
|
15
|
37 - 19 =18
|
45-50
|
13
|
18 - 15 = 3
|
On the graph ,plot the points (0,95),
(5,92),(10,87), (15,80), (20,72), (25,62),(30,51),(35,37),(40,18),(45,3),and
join the point by smooth curve.
References
Ø https://www.statisticshowto.datasciencecentral.com/ogive-graph
Ø /https://www.statisticshowto.datasciencecentral.com/ogive-graph/
Ø http://www.mathcaptain.com/statistics/ogive.html
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